Topospaces - User contributions [en]

MediaWiki:Sitenotice

2024-10-07T17:48:22Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-10-07T17:48:07Z

Vipul:

MediaWiki:Sitenotice

2024-10-07T01:13:45Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.''' 
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

MediaWiki:Sitenotice

2024-09-06T01:58:28Z

Vipul:

Topospaces:Enabling site search autocompletion

2024-09-06T01:57:55Z

Vipul:

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Topospaces).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

Topospaces:Enabling site search autocompletion

2024-09-06T01:57:32Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Companal). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Companal).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

File:Site search autocompletion working.png

2024-09-06T01:57:21Z

Vipul:

File:Site search autocompletion broken.png

2024-09-06T01:57:02Z

Vipul:

Topospaces:429 Too Many Requests error

2024-09-06T01:54:36Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

MediaWiki:Sitenotice

2024-09-06T01:53:13Z

Vipul: Blanked the page

User:Vipul/Sandbox

2024-09-06T01:51:54Z

Vipul:

<math>\sqrt{\frac{e^{\pi^32}}{\sigma^3 + e^\sigma + 1 + \sqrt{243}}}</math>

<math>2^7 - 1 =127</math>

<math>2^{8 - 1} = 128</math>

<math>\sqrt{7 + 2}!! + 0 = 720</math>

<math>6^{2 + 1} = 216</math>

<math>5^{1 + 2} = 125</math>

<math>(3 + 4)^3 = 343</math>

User:Vipul/Sandbox

2024-09-06T01:46:19Z

Vipul:

User:Vipul/Sandbox

2024-09-06T01:42:53Z

Vipul:

MediaWiki:Sitenotice

2024-09-06T01:36:22Z

Vipul:

'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''

User:Vipul/Sandbox

2024-07-05T23:35:33Z

Vipul:

<math>\sqrt{\frac{e^{\pi^32}}{\sigma^3 + e^\sigma + 1 + \sqrt{243}}}</math>

<math>2^7 - 1 =127</math>

<math>2^{8 - 1} = 128</math>

<math>\sqrt{7 + 2}!! + 0 = 720</math>

User:Vipul/Sandbox

2024-07-05T23:28:27Z

Vipul:

<math>\sqrt{\frac{e^{\pi^32}}{\sigma^3 + e^\sigma + 1 + \sqrt{23}}}</math>

<math>2^7 - 1 =127</math>

<math>2^{8 - 1} = 128</math>

<math>\sqrt{7 + 2}!! + 0 = 720</math>

User:Vipul/Sandbox

2024-05-05T03:01:06Z

Vipul:

<math>\sqrt{\frac{e^{\pi^32}}{\sigma^3 + e^\sigma + 1 + \sqrt{2}}}</math>

<math>2^7 - 1 =127</math>

<math>2^{8 - 1} = 128</math>

<math>\sqrt{7 + 2}!! + 0 = 720</math>

User:Vipul/Sandbox

2024-05-05T02:33:56Z

Vipul:

<math>\sqrt{\frac{e^{\pi^32}}{\sigma^3 + e^\sigma + 1 + \sqrt{2}}}</math>

<math>2^7 - 1 =127</math>

<math>2^{8 - 1} = 128</math>

Main Page

2024-05-04T23:44:44Z

Vipul:

__NOTOC__
{{quotation|Welcome to '''Topospaces (The Topology Wiki)'''. This is a pre-alpha stage topology wiki primarily managed by [[User:Vipul|Vipul Naik]], a Ph.D. in Mathematics at the University of Chicago. We have over 400 articles including some material in basic point-set topology. It is part of a broader subject wikis initiative -- see the [[Ref:Main Page|subject wikis reference guide]] for more details}}

User:Vipul/Sandbox

2024-05-04T23:33:30Z

Vipul:

<math>\sqrt{\frac{e^{\pi^32}}{\sigma^3 + e^\sigma + 1 + \sqrt{2}}}</math>

<math>2^7 - 1 =127</math>

User:Vipul/Sandbox

2024-05-04T23:23:51Z

Vipul:

<math>\sqrt{\frac{e^{\pi^32}}{\sigma^3 + e^\sigma + 1 + \sqrt{2}}}</math>

Contractible space

2023-10-26T22:57:28Z

Vipul: /* Stronger properties */

{{homotopy-invariant topospace property}}

==Definition==

=== Equivalent definitions in tabular format ===

A nonempty [[topological space]] is said to be '''contractible''' if it satisfies the following equivalent conditions. The [[empty space]] is generally excluded from consideration when considering the question of contractibility.

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed contractible if ... !! A topological space <math>X</math> is termed contractible if ...
|-
| 1 || homotopy-equivalent to a point || there is a [[homotopy equivalence of topological spaces]] between the topological space and a [[one-point space]]. || There exist [[continuous map]]s <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math> such that <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 2 || homotopy-equivalent to a point (arbitrary map) || ''any'' pair of maps between the space and a one-point space define a homotopy equivalence of topological spaces. || For ''any'' continuous map <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math>, <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 3 || admits a contracting homotopy || there is a point in the space to which there is a [[contracting homotopy]]. || there exists a point <math>x_0 \in X</math> and a contracting homotopy that contracts <math>X</math> to <math>x_0</math>. Explicitly, there exists a continuous map <math>F: X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>. Note that we do ''not'' assume or require that <math>F(x_0,a) = x_0</math> for all <math>a \in [0,1]</math>.
|-
| 4 || admits a contracting homotopy (arbitrary point) || the space admits a [[contracting homotopy]] to ''any'' point in it. || for ''any'' point <math>x_1 \in X</math>, there is a contracting homotopy that contracts <math>X</math> to <math>x_1</math>.
|-
| 5 || unique homotopy class of maps to it || for any other topological space, there is a unique homotopy class of maps from the other space to it. || for any topological space <math>Y</math>, and any two continuous maps <math>h_1, h_2: Y \to X</math>, <math>h_1</math> and <math>h_2</math> are homotopic. In particular, any map from a topological space to it is nullhomotopic.
|-
| 6 || retract of cone space || it is a [[retract]] of its [[cone space]]. || the inclusion <math>\iota: X \to CX</math> in the cone space <math>CX</math> (as the base) has a one-sided inverse, <math>j: CX \to X</math>, i.e., <math>j</math> is a continuous map such that <math>j \circ \iota</math> is the identity map on <math>X</math>.
|}

==Metaproperties==

{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[contractibility is product-closed]] || If <math>X_i, i \in I</math> form a (finite or infinite) collection of contractible spaces, then the product of the <math>X_i</math>s, equipped with the [[product topology]], is also contractible. In particular, if <math>X_1, X_2</math> are contractible, then <math>X_1 \times X_2</math> is also contractible.
|-
| [[satisfies metaproperty::retract-hereditary property of topological spaces]] || Yes || [[contractibility is retract-hereditary]] || If <math>A \subseteq X</math> and <math>f:X \to A</math> is a continuous [[retraction]], and <math>X</math> is contractible, then <math>A</math> is contractible.
|-
| [[satisfies metaproperty::suspension-closed property of topological spaces]] || Yes || [[contractibility is suspension-closed]] || The [[suspension]] of a contractible space is contractible.
|-
| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[contractibility is not closure-preserved]] || It is possible to have a topological space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is contractible in the [[subspace topology]], but the [[closure]] <math>\overline{A}</math> is not.
|-
| [[dissatisfies metaproperty::connected union-closed property of topological spaces]] || No || [[contractibility is not connected union-closed]] || It is possible to have a topological space <math>X</math> expressible as a union of subsets <math>A,B</math>, both contractible in their subspace topology, with <math>A \cap B</math> nonempty, but <math>X</math> itself not contractible.
|}

== Examples ==

=== Extreme and basic examples ===

* The [[one-point space]] is contractible.
* Any [[Euclidean space]] is contractible.
* The closed unit disk in any dimension is contractible.
* [[Compact manifold]]s in dimension one or more, such as the [[circle]], are not contractible.

=== Intuition behind examples ===

Contractibility is, fundamentally, a ''global'' property of topological spaces. It is not possible to examine a small part of the space and conclude that it is contractible, nor does examining a small part of the space allow us to rule ''out'' the possibiilty that it is contractible. For the intuition behind the former, note that we can attach non-contractible pieces (like [[circle]]s) far off from the part of the space we are looking at. For the intuition behind the latter claim, note that we can embed ''any'' topological space as a closed subspace of a [[contractible space]], namely, its [[cone space]].

For this reason, when looking for examples or counterexamples, we need to focus on the global structure.

=== Examples from topological construction ===

One thing to keep in mind is that since the definition of contractibility invokes the closed unit interval, it is likely that any effort to construct contractible spaces will invariably involve dealing with the real numbers. The most topologically general way of constructing a contractible space is as the [[cone space]] of an arbitrary topological space. One way of thinking of this cone space is as a literal cone that fills in between the space and a point. Up until the very tip of the cone, the cross-sections look homeomorphic to the topological space.

=== Examples from geometry ===

A [[topologically star-like space]] is a classic example of a contractible space. A topological space is termed topologically star-like if it is homeomorphic to a star-like subset of Euclidean space. A star-like subset of Euclidean space is a subset for which there exists a point in it such that for every other point in it, the line segment joining the points is completely inside the space.

A topologically star-like space is contractible, and can in fact be contracted to any point relative to which it is a star through a straight-line homotopy, i.e., moving each point toward the center in a straight line. The contracting homotopy fixes the center, and therefore, the space is in fact a [[SDR-contractible space]].

Note that, if also compact, a topologically star-like space is homeomorphic to the cone space of its boundary. Otherwise, the space is still ''almost'' a cone space: it is a subspace of the cone space that contains the full complement of the base and an arbitrary subset of the base. Nonetheless, it is important to note that the condition of being star-like also carries various geometric implications (in particular, from being a [[sub-Euclidean space]]) that are not satisfied for arbitrary cone spaces.

A [[nonempty topologically convex space]] is a nonempty space that is homeomorphic to a [[convex subset of Euclidean space]]. Any nonempty topologically convex space is topologically star-like, and ''any'' point can be taken as the center. An example of a topologically star-like space that is not a nonempty topologically convex space is a [[pair of intersecting lines]].

It is possible to construct spaces that are not topologically star-like, but still contractible. For instance, any geometric realization of a tree is contractible, but if the tree has more than one point with degree greater than two, it is not topologically star-like. As a related example, a set of parallel lines combined with one line that intersects all of them form a contractible space that is not topologically star-like.

==Relation with other properties==

=== Incomparable properties ===

Contractibility is incomparable with most of the interesting separation and compactness properties.

'''Broad argument for why contractibility cannot imply any meaningful separation or compactness property''': The [[cone space]] over ''any'' topological space is contractible. In particular, since any topological space arises as a closed subspace of its cone space (namely, the "base" of the space), every topological space arises as a closed subspace of a contractible space. Therefore, contractible cannot imply any nontrivial property that is [[subspace-hereditary property of topological spaces|subspace-hereditary]] or even [[weakly hereditary property of topological spaces|weakly hereditary]] (inherited by closed subsets).

'''Broad argument for why meaningful separation or compactness properties cannot imply contractibility''': Most meaningful separation and compactness properties are satisfied by all [[compact manifold]]s. However, compact manifolds of dimension greater than one are not contractible. The simplest counterexample is generally the [[circle]].

Some incomparable properties:

* [[T0 space]]
* [[T1 space]]
* [[Hausdorff space]]
* [[regular space]]
* [[normal space]]
* [[metrizable space]]
* [[paracompact space]]
* [[compact space]]

The property of being a contractible space is also incomparable with the property of being a [[locally contractible space]]. A contractible space need not be locally contractible. In fact, it need not even be locally connected! An example of a contractible space that is not locally contractible is the [[comb space]]. An example of a space that is locally contractible but not contractible is the [[circle]] (or, more generally, any compact manifold).

===Stronger properties===

Note that we may need to assume nonemptiness on top of the provided definitions in some case.

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[cone space]] over some topological space || || [[cone space implies contractible]] || ||
|-
| [[Weaker than::topologically star-like space]] || || || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Weaker than::nonempty topologically convex space]] || homeomorphic to a [[convex subset of Euclidean space]] || [[convex implies star-like|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Weaker than::suddenly contractible space]] || has a [[contracting homotopy]] that is also a [[sudden homotopy]] || || ||{{intermediate notions short|contractible space|suddenly contractible space}}
|-
| [[Weaker than::SDR-contractible space]] || has a [[contracting homotopy]] that is also a [[deformation retraction]] || || || {{intermediate notions short|contractible space|SDR-contractible space}}
|-
| [[Weaker than::equiconnected space]] || (complicated, described in some context as "contractible mod diagonal") || [[equiconnected implies contractible]] || [[contractible not implies equiconnected]] || {{intermediate notions short|contractible space|equiconnected space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || [[contractible implies weakly contractible]] || [[weakly contractible not implies contractible]] || {{intermediate notions short|weakly contractible space|contractible space}}
|-
| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || the <math>n</math>-sphere <math>S^n</math> is <math>(n-1)</math>-connected but not <math>n</math>-connected. || {{intermediate notions short|multiply connected space|contractible space}}
|-
| [[Stronger than::simply connected space]] || [[path-connected space]], [[fundamental group]] is [[trivial group|trivial]] || || || {{intermediate notions short|simply connected space|contractible space}}
|-
| [[Stronger than::path-connected space]] || there is a [[path]] between any two points || || || {{intermediate notions short|path-connected space|contractible space}}
|-
| [[Stronger than::connected space]] || cannot be partitioned into disjoint nonempty subsets || || || {{intermediate notions short|connected space|contractible space}}
|-
| [[Stronger than::acyclic space]] || homology groups over <math>\mathbb{Z}</math> all zero except zeroth homology group which is <math>\mathbb{Z}</math> || || || {{intermediate notions short|acyclic space|contractible space}}
|-
| [[Stronger than::rationally acyclic space]] || homology groups over <matH>\mathbb{Q}</math> all zero except zeroth homology group which is <math>\mathbb{Q}</math> || || || {{intermediate notions short|rationally acyclic space|contractible space}}
|-
| [[Stronger than::space with Euler characteristic one]] || [[Euler characteristic]] of the space is one. || ([[acyclic implies Euler charcateristic one|via acyclic]])|| [[Euler characteristic one not implies acyclic]] || {{intermediate notions short|space with Euler characteristic one|contractible space}}
|}

=== Conjunction with other properties ===

* [[Contractible manifold]]: Contractible as well as a [[manifold]]
* [[Contractible polyhedron]]: Contractible as well as a [[polyhedron]], i.e., the geometric realization of a [[simplicial complex]]

==References==

===Textbook references===

* {{booklink|Munkres}}, Page 330, Exercise 3 (definition introduced in exercise)
* {{booklink|SingerThorpe}}, Page 51 (formal definition)
* {{booklink|Rotman}}, Page 18 (formal definition)
* {{booklink|Hatcher}}, Page 4 (formal definition)
* {{booklink|Spanier}}, Page 25 (definition in paragraph)

Nonempty topologically convex space

2023-10-26T22:57:06Z

Vipul: /* Weaker properties */

{{topospace property}}

==Definition==

A '''nonempty topologically convex space''' is a nonempty [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[convex subset of Euclidean space]].

==Relation with other properties==

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::topologically star-like space]] || || follows from [[convex implies star-like]] || the [[pair of intersecting lines]] is topologically star-like but not topologically convex || {{intermediate notions short|topologically star-like space|topologically convex space}}
|-
| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Stronger than::semi-suddenly contractible space]] || has a [[semi-sudden homotopy|semi-sudden]] [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|semi-suddenly contractible space|topologically convex space}}
|-
| [[Stronger than::SDR-contractible space]] || has a [[contracting homotopy]] that is a [[deformation retraction]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|SDR-contractible space|topologically convex space}}
|-
| [[Stronger than::equiconnected space]] || (complicated, described in one context as "contractible mod diagonal") || [[nonempty topologically convex implies equiconnected]] || || {{intermediate notions short|equiconnected space|nonempty topologically convex space}}
|}

Contractible space

2023-10-26T22:56:34Z

Vipul: /* Stronger properties */

{{homotopy-invariant topospace property}}

==Definition==

=== Equivalent definitions in tabular format ===

A nonempty [[topological space]] is said to be '''contractible''' if it satisfies the following equivalent conditions. The [[empty space]] is generally excluded from consideration when considering the question of contractibility.

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed contractible if ... !! A topological space <math>X</math> is termed contractible if ...
|-
| 1 || homotopy-equivalent to a point || there is a [[homotopy equivalence of topological spaces]] between the topological space and a [[one-point space]]. || There exist [[continuous map]]s <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math> such that <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 2 || homotopy-equivalent to a point (arbitrary map) || ''any'' pair of maps between the space and a one-point space define a homotopy equivalence of topological spaces. || For ''any'' continuous map <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math>, <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 3 || admits a contracting homotopy || there is a point in the space to which there is a [[contracting homotopy]]. || there exists a point <math>x_0 \in X</math> and a contracting homotopy that contracts <math>X</math> to <math>x_0</math>. Explicitly, there exists a continuous map <math>F: X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>. Note that we do ''not'' assume or require that <math>F(x_0,a) = x_0</math> for all <math>a \in [0,1]</math>.
|-
| 4 || admits a contracting homotopy (arbitrary point) || the space admits a [[contracting homotopy]] to ''any'' point in it. || for ''any'' point <math>x_1 \in X</math>, there is a contracting homotopy that contracts <math>X</math> to <math>x_1</math>.
|-
| 5 || unique homotopy class of maps to it || for any other topological space, there is a unique homotopy class of maps from the other space to it. || for any topological space <math>Y</math>, and any two continuous maps <math>h_1, h_2: Y \to X</math>, <math>h_1</math> and <math>h_2</math> are homotopic. In particular, any map from a topological space to it is nullhomotopic.
|-
| 6 || retract of cone space || it is a [[retract]] of its [[cone space]]. || the inclusion <math>\iota: X \to CX</math> in the cone space <math>CX</math> (as the base) has a one-sided inverse, <math>j: CX \to X</math>, i.e., <math>j</math> is a continuous map such that <math>j \circ \iota</math> is the identity map on <math>X</math>.
|}

==Metaproperties==

{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[contractibility is product-closed]] || If <math>X_i, i \in I</math> form a (finite or infinite) collection of contractible spaces, then the product of the <math>X_i</math>s, equipped with the [[product topology]], is also contractible. In particular, if <math>X_1, X_2</math> are contractible, then <math>X_1 \times X_2</math> is also contractible.
|-
| [[satisfies metaproperty::retract-hereditary property of topological spaces]] || Yes || [[contractibility is retract-hereditary]] || If <math>A \subseteq X</math> and <math>f:X \to A</math> is a continuous [[retraction]], and <math>X</math> is contractible, then <math>A</math> is contractible.
|-
| [[satisfies metaproperty::suspension-closed property of topological spaces]] || Yes || [[contractibility is suspension-closed]] || The [[suspension]] of a contractible space is contractible.
|-
| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[contractibility is not closure-preserved]] || It is possible to have a topological space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is contractible in the [[subspace topology]], but the [[closure]] <math>\overline{A}</math> is not.
|-
| [[dissatisfies metaproperty::connected union-closed property of topological spaces]] || No || [[contractibility is not connected union-closed]] || It is possible to have a topological space <math>X</math> expressible as a union of subsets <math>A,B</math>, both contractible in their subspace topology, with <math>A \cap B</math> nonempty, but <math>X</math> itself not contractible.
|}

== Examples ==

=== Extreme and basic examples ===

* The [[one-point space]] is contractible.
* Any [[Euclidean space]] is contractible.
* The closed unit disk in any dimension is contractible.
* [[Compact manifold]]s in dimension one or more, such as the [[circle]], are not contractible.

=== Intuition behind examples ===

Contractibility is, fundamentally, a ''global'' property of topological spaces. It is not possible to examine a small part of the space and conclude that it is contractible, nor does examining a small part of the space allow us to rule ''out'' the possibiilty that it is contractible. For the intuition behind the former, note that we can attach non-contractible pieces (like [[circle]]s) far off from the part of the space we are looking at. For the intuition behind the latter claim, note that we can embed ''any'' topological space as a closed subspace of a [[contractible space]], namely, its [[cone space]].

For this reason, when looking for examples or counterexamples, we need to focus on the global structure.

=== Examples from topological construction ===

One thing to keep in mind is that since the definition of contractibility invokes the closed unit interval, it is likely that any effort to construct contractible spaces will invariably involve dealing with the real numbers. The most topologically general way of constructing a contractible space is as the [[cone space]] of an arbitrary topological space. One way of thinking of this cone space is as a literal cone that fills in between the space and a point. Up until the very tip of the cone, the cross-sections look homeomorphic to the topological space.

=== Examples from geometry ===

A [[topologically star-like space]] is a classic example of a contractible space. A topological space is termed topologically star-like if it is homeomorphic to a star-like subset of Euclidean space. A star-like subset of Euclidean space is a subset for which there exists a point in it such that for every other point in it, the line segment joining the points is completely inside the space.

A topologically star-like space is contractible, and can in fact be contracted to any point relative to which it is a star through a straight-line homotopy, i.e., moving each point toward the center in a straight line. The contracting homotopy fixes the center, and therefore, the space is in fact a [[SDR-contractible space]].

Note that, if also compact, a topologically star-like space is homeomorphic to the cone space of its boundary. Otherwise, the space is still ''almost'' a cone space: it is a subspace of the cone space that contains the full complement of the base and an arbitrary subset of the base. Nonetheless, it is important to note that the condition of being star-like also carries various geometric implications (in particular, from being a [[sub-Euclidean space]]) that are not satisfied for arbitrary cone spaces.

A [[nonempty topologically convex space]] is a nonempty space that is homeomorphic to a [[convex subset of Euclidean space]]. Any nonempty topologically convex space is topologically star-like, and ''any'' point can be taken as the center. An example of a topologically star-like space that is not a nonempty topologically convex space is a [[pair of intersecting lines]].

It is possible to construct spaces that are not topologically star-like, but still contractible. For instance, any geometric realization of a tree is contractible, but if the tree has more than one point with degree greater than two, it is not topologically star-like. As a related example, a set of parallel lines combined with one line that intersects all of them form a contractible space that is not topologically star-like.

==Relation with other properties==

=== Incomparable properties ===

Contractibility is incomparable with most of the interesting separation and compactness properties.

'''Broad argument for why contractibility cannot imply any meaningful separation or compactness property''': The [[cone space]] over ''any'' topological space is contractible. In particular, since any topological space arises as a closed subspace of its cone space (namely, the "base" of the space), every topological space arises as a closed subspace of a contractible space. Therefore, contractible cannot imply any nontrivial property that is [[subspace-hereditary property of topological spaces|subspace-hereditary]] or even [[weakly hereditary property of topological spaces|weakly hereditary]] (inherited by closed subsets).

'''Broad argument for why meaningful separation or compactness properties cannot imply contractibility''': Most meaningful separation and compactness properties are satisfied by all [[compact manifold]]s. However, compact manifolds of dimension greater than one are not contractible. The simplest counterexample is generally the [[circle]].

Some incomparable properties:

* [[T0 space]]
* [[T1 space]]
* [[Hausdorff space]]
* [[regular space]]
* [[normal space]]
* [[metrizable space]]
* [[paracompact space]]
* [[compact space]]

The property of being a contractible space is also incomparable with the property of being a [[locally contractible space]]. A contractible space need not be locally contractible. In fact, it need not even be locally connected! An example of a contractible space that is not locally contractible is the [[comb space]]. An example of a space that is locally contractible but not contractible is the [[circle]] (or, more generally, any compact manifold).

===Stronger properties===

Note that we may need to assume nonemptiness on top of the provided definitions in some case.

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[cone space]] over some topological space || || [[cone space implies contractible]] || ||
|-
| [[Weaker than::topologically star-like space]] || || || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Weaker than::nonempty topologically convex space]] || homeomorphic to a [[convex subset of Euclidean space]] || [[convex implies star-like|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Weaker than::suddenly contractible space]] || has a [[contracting homotopy]] that is also a [[sudden homotopy]] || || ||{{intermediate notions short|contractible space|suddenly contractible space}}
|-
| [[Weaker than::SDR-contractible space]] || has a [[contracting homotopy]] that is also a [[deformation retraction]] || || || {{intermediate notions short|contractible space|SDR-contractible space}}
|-
| [[Weaker than::equiconnected space]] || (complicated, described in some context as a "homotopy mod diagonal") || [[equiconnected implies contractible]] || [[contractible not implies equiconnected]] || {{intermediate notions short|contractible space|equiconnected space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || [[contractible implies weakly contractible]] || [[weakly contractible not implies contractible]] || {{intermediate notions short|weakly contractible space|contractible space}}
|-
| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || the <math>n</math>-sphere <math>S^n</math> is <math>(n-1)</math>-connected but not <math>n</math>-connected. || {{intermediate notions short|multiply connected space|contractible space}}
|-
| [[Stronger than::simply connected space]] || [[path-connected space]], [[fundamental group]] is [[trivial group|trivial]] || || || {{intermediate notions short|simply connected space|contractible space}}
|-
| [[Stronger than::path-connected space]] || there is a [[path]] between any two points || || || {{intermediate notions short|path-connected space|contractible space}}
|-
| [[Stronger than::connected space]] || cannot be partitioned into disjoint nonempty subsets || || || {{intermediate notions short|connected space|contractible space}}
|-
| [[Stronger than::acyclic space]] || homology groups over <math>\mathbb{Z}</math> all zero except zeroth homology group which is <math>\mathbb{Z}</math> || || || {{intermediate notions short|acyclic space|contractible space}}
|-
| [[Stronger than::rationally acyclic space]] || homology groups over <matH>\mathbb{Q}</math> all zero except zeroth homology group which is <math>\mathbb{Q}</math> || || || {{intermediate notions short|rationally acyclic space|contractible space}}
|-
| [[Stronger than::space with Euler characteristic one]] || [[Euler characteristic]] of the space is one. || ([[acyclic implies Euler charcateristic one|via acyclic]])|| [[Euler characteristic one not implies acyclic]] || {{intermediate notions short|space with Euler characteristic one|contractible space}}
|}

=== Conjunction with other properties ===

* [[Contractible manifold]]: Contractible as well as a [[manifold]]
* [[Contractible polyhedron]]: Contractible as well as a [[polyhedron]], i.e., the geometric realization of a [[simplicial complex]]

==References==

===Textbook references===

* {{booklink|Munkres}}, Page 330, Exercise 3 (definition introduced in exercise)
* {{booklink|SingerThorpe}}, Page 51 (formal definition)
* {{booklink|Rotman}}, Page 18 (formal definition)
* {{booklink|Hatcher}}, Page 4 (formal definition)
* {{booklink|Spanier}}, Page 25 (definition in paragraph)

Nonempty topologically convex space

2023-10-26T22:50:05Z

Vipul: /* Weaker properties */

{{topospace property}}

==Definition==

A '''nonempty topologically convex space''' is a nonempty [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[convex subset of Euclidean space]].

==Relation with other properties==

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::topologically star-like space]] || || follows from [[convex implies star-like]] || the [[pair of intersecting lines]] is topologically star-like but not topologically convex || {{intermediate notions short|topologically star-like space|topologically convex space}}
|-
| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Stronger than::semi-suddenly contractible space]] || has a [[semi-sudden homotopy|semi-sudden]] [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|semi-suddenly contractible space|topologically convex space}}
|-
| [[Stronger than::SDR-contractible space]] || has a [[contracting homotopy]] that is a [[deformation retraction]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|SDR-contractible space|topologically convex space}}
|-
| [[Stronger than::equiconnected space]] || (complicated) || [[nonempty topologically convex implies equiconnected]] || || {{intermediate notions short|equiconnected space|nonempty topologically convex space}}
|}

Nonempty topologically convex implies equiconnected

2023-10-26T22:46:05Z

Vipul:

{{topospace property implication|stronger = nonempty topologically convex space|weaker = equiconnected space}}

==Statement==

===Topological version===

Any [[nonempty topologically convex space]] is an [[equiconnected space]].

===Realized version===

Any nonempty [[convex subset of Euclidean space]] is (topologically) an [[equiconnected space]].

==Definitions used==

===Nonempty topologically convex space===

{{further|[[nonempty topologically convex space]]}}

A topological space is called a nonempty topologically convex space if it is nonempty and homeomorphic to a [[convex subset of Euclidean space]].

===Convex subset of Euclidean space===

{{further|[[convex subset of Euclidean space]]}}

A convex subset of Euclidean space is a subset in <math>\R^n</math> for some <math>n</math>, with the property that given any two points in the subset, the line segment joining those two points also lies completely within the subset.

The definition may also apply to infinite-dimensional Euclidean spaces, and the proof would work even in that case.

===Equiconnected space===

{{further|[[equiconnected space]]}}

A nonempty topological space <math>X</math> is said to be '''equiconnected''' if there is a continuous map <math>k:X \times [0,1] \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x,y \in X</math>.

==Proof==

'''Given''': A convex subset <math>X</math> of <math>\R^n</math>

'''To prove''': There is a continuous map <math>k:X \times [0,1] \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x,y \in X</math>

'''Proof''':

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation !! Geometric interpretation
|-
| 1 || Define <math>k: X \times [0,1] \times X \to X</math> as follows: <math>k(x,t,y) := (1 - t)x + ty</math>. Here, the scalar multiplication addition is as vectors in <math>\R^n</math>. || || <math>X</math> is a convex subset of <math>\R^n</math> || || <math>k(x,t,y)</math> is a convex combination of <math>x</math> and <math>y</math>, so by convexity of <math>X</math>, it is inside <math>X</math>. || <math>k</math> defines the point on the line segment joining the points <math>x</math> and <math>y</math> that divides the line segment in the ratio <math>t:(1-t)</math>. By convexity of <math>X</math>, this point is inside <math>X</math>. As <math>t</math> increases from 0 to 1, <math>k(x,t,y)</math> moves at constant speed along the line segment from <math>x</math> to <math>y</math>.
|-
| 2 || <math>k</math> is continuous. || || || Step (1) || This follows from addition and scalar multiplication being continuous functions, and the fact that the composite of continuous functions is continuous. || This is tricky to visualize (even for something simple such as a unit interval), so we shall skip the description.
|-
| 3 || <math>k(x,t,x) = x</math> for all <math>x \in X, t \in [0,1]</math>. || || || Step (1) || This follows by plugging in <math>y = x</math> in the definition in Step (1). || When <math>x = y</math>, the "line segment" joining <math>x</math> and <math>y</math> becomes a point, so for all <math>t</math> we stay at that point.
|-
| 4 || <math>k(x,0,y) = x</math> for all <math>x,y \in X</math>. || || || Step (1) || This follows pretty directly by plugging <math>t = 0</math> in the expression for <math>k</math> in Step (1). || This is the "start" (at <math>t = 0</math>) of the traversal along the line segment from <math>x</math> to <math>y</math>, so it is naturally at <math>x</math>.
|-
| 5 || <math>k(x,1,y) = y</math> for all <math>x,y \in X</math>. || | ||| Step (1) || This follows pretty directly by plugging <math>t = 1</math> in the expression for <math>k</math> in Step (1). || This is the "end" (at <math>t = 1</math>) of the traversal along the line segment from <math>x</math> to <math>y</math>, so it is naturally at <math>y</math>.
|-
| 6 || <math>k</math> is the desired function. || || || Steps (1), (2), (3), (4), (5) || Step (1) defines <math>k</math> with the correct domain and co-domain. Steps (2), (3), (4), and (5) establish the criteria that <math>k</math> needs to satisfy. ||
|}

{{tabular proof format}}

Equiconnected implies contractible

2023-10-26T22:26:17Z

Vipul: /* Proof */

{{topospace property implication|stronger = equiconnected space|weaker = contractible space}}

==Statement==

Any [[equiconnected space]] is [[contractible space|contractible]].

==Definitions used==

===Equiconnected space===

{{further|[[equiconnected space]]}}

A nonempty topological space <math>X</math> is said to be '''equiconnected''' if there is a continuous map <math>k:X \times [0,1] \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x</math> and <math>y</math>.

===Contractible space===

{{further|[[contractible space]]}}

A nonempty topological space <math>X</math> is said to be contractible if there exists a point <math>x_0 \in X</math> and a continuous map <math>F: X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>. (Another equivalent definition states that this is true for ''every'' <math>x_0 \in X</math>).

==Facts used==

# Slice inclusion is continuous: For a product of topological space <math>A \times B</math> and a point <math>b \in B</math>, the map <math>A \to A \times B</math> given by <math>a \mapsto (a,b)</math> is continuous.
# The composition of continuous maps is continuous
==Proof==

'''Given''': A nonempty topological space <math>X</math>, a continuous map <math>k:X \times [0,1] \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x</math> and <math>y</math>

'''To prove''': There exists a point <math>x_0 \in X</math> and a continuous map <math>F:X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>.

'''Proof''':

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || Choose any point <math>x_0 \in X</math> || || <math>X</math> is nonempty || ||
|-
| 2 || Define <math>F: X \times [0,1] \to X</math> as <math>F(x,t) := k(x,t,x_0)</math>. || || <math>k:X \times [0,1] \times X \to X</math> exists || Step (1) ||
|-
| 3 || <math>F</math> is continuous. || Facts (1), (2) || <math>k</math> is continuous || Step (2) || <math>F</math> is the composite of <math>k</math> applied on the slice inclusion map <math>X \times [0,1] \to X \times [0,1] \times X</math> defined by (x,t) \mapsto (x,t,x_0)</math>. By Fact (1), the slice inclusion is continuous, so by Fact (2), the composite is continuous.
|-
| 4 || <math>F(x,0) = x</math> for all <math>x \in X</math>. || || <math>k(x,0,y) = x</math> for all <math>x, y \in X</math> || Step (2) || <math>F(x,0) = k(x,0,x_0)</math> by Step (2). Combining this with the given data, we get that <math>F(x,0) = x</math>.
|-
| 5 || <math>F(x,1) = x_0</math> for all <math>x \in X</math>. || || <math>k(x,1,y) = y</math> for all <math>x, y \in X</math> || Step (2) || <math>F(x,1) = k(x,1,x_0)</math> by Step (2). Combining this with the given data, we get that <math>F(x,1) = x_0</math>.
|-
| 6 || <math>x_0</math> from Step (1) and <math>F</math> from Step (2) are as desired. || || || Steps (1), (2), (3), (4), (5) || <math>x_0 \in X</math> by Step (1) and <math>F</math> has the correct domain and co-domain by Step(2). Steps (3), (4), (5) establish the three additional things that need to be established: continuity, value at 0, and value at 1. This completes the proof.
|}

{{tabular proof format}}

Nonempty topologically convex implies equiconnected

2023-10-26T22:23:18Z

Vipul: /* Equiconnected space */

Pair of intersecting lines

2023-10-26T22:21:59Z

Vipul: /* Properties */

{{particular topospace}}

{{standard counterexample}}

== Definition ==

A '''pair of intersecting lines''' is the underlying [[topological space]] of any subset of [[Euclidean space]] (of dimension two or higher) that comprises two distinct and intersecting lines. An example is the set:

<math>\{ (x,y) \in \R^2 \mid xy = 0 \}</math>

This is the pair of the <math>x</math>-axis and the <math>y</math>-axis in <math>\R^2</math>.

== Properties ==

{| class="sortable" border="1"
! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied
|-
! colspan="4" style="text-align:center; background: white"| Similarity to Euclidean space
|-
| [[satisfies property::closed sub-Euclidean space]] || Yes || || satisfies: [[satisfies property::sub-Euclidean space]], [[satisfies property::completely metrizable space]]
|-
| [[dissatisfies property::locally Euclidean space]] || No || Not Euclidean around point of intersection || dissatisfies: [[dissatisfies property::manifold]]
|-
! colspan="4" style="text-align:center; background: white"| Separation and metrizability
|-
| [[satisfies property::metrizable space]] || Yes || || satisfies: [[satisfies property::binormal space]], [[satisfies property::normal space]], [[satisfies property::perfectly normal space]], [[satisfies property::completely normal space]], [[satisfies property::regular space]], [[satisfies property::Hausdorff space]], [[satisfies property::paracompact Hausdorff space]], [[satisfies property::paracompact space]]
|-
| [[satisfies property::CW-space]] || Yes || ||
|-
| [[satisfies property::polyhedron]] || Yes || ||
|-
! colspan="4" style="text-align:center; background: white"| Connectedness
|-
| [[satisfies property::SDR-contractible space]] || Yes || || satisfies: [[satisfies property::contractible space]], [[satisfies property::simply connected space]], [[satisfies property::path-connected space]]
|-
| [[satisfies property::locally contractible space]] || Yes || ||satisfies: [[satisfies property::locally path-connected space]], [[satisfies property::locally simply connected space]], [[satisfies property::semilocally simply connected space]], [[satisfies property::locally connected space]]
|-
! colspan="4" style="text-align:center; background: white"| Compactness
|-
| [[satisfies property::locally compact space]] || Yes || ||
|-
| [[dissatisfies property::compact space]] || No || ||
|-
| [[satisfies property::paracompact space]] || Yes || via metrizability ||
|-
! colspan="4" style="text-align:center; background: white"| Countability
|-
| [[satisfies property::second-countable space]] || Yes || || satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]
|-
! colspan="4" style="text-align:center; background: white"| Uniformness and self-similarity
|-
| [[dissatisfies property::homogeneous space]] || No || point of intersection is not similar to other points ||
|-
| [[dissatisfies property::uniformly based space]] || No || basis sets containing the point of intersection look different from basis sets far away || dissatisfies: [[dissatisfies property::self-based space]]
|-
! colspan="4" style="text-align:center; background: white"| Topological equivalence to convex spaces
|-
| [[dissatisfies property::nonempty topologically convex space]] || No || not convex around point of intersection: the rays emerging from it must be put in pairs, and convexity is violated for points on two unpaired rays ||
|-
| [[satisfies property::topologically star-like space]] || Yes || star-like relative to the point of intersection || satisfies: [[satisfies property::SDR-contractible space]], [[satisfies property::contractible space]]
|}

Space in which every retraction is a deformation retraction

2023-10-26T22:21:07Z

Vipul: /* Stronger properties */

==Definition==

A '''space in which every retraction is a deformation retraction''' is a [[topological space]] <math>X</math> with the property that any [[retraction]] <math>r:X \to Y</math> for a subspace <math>Y</math> of <math>X</math> (with the [[subspace topology]]) arises as a [[deformation retraction]], i.e., there is a homotopy from the identity map to that retraction that restricts to <math>r</math> on the subspace <math>Y</math> at all time.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::nonempty topologically convex space]] || || || || {{intermediate notions short|space in which every retraction is a deformation retraction|topologically convex space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::contractible space]] || || || || {{intermediate notions short|contractible space|space in which every retraction is a deformation retraction}}
|-
| [[Stronger than::SDR-contractible space]] || || || || {{intermediate notions short|SDR-contractible space|space in which every retraction is a deformation retraction}}
|}

Space in which every retraction is a deformation retraction

2023-10-26T22:20:59Z

Vipul: /* Stronger properties */

==Definition==

A '''space in which every retraction is a deformation retraction''' is a [[topological space]] <math>X</math> with the property that any [[retraction]] <math>r:X \to Y</math> for a subspace <math>Y</math> of <math>X</math> (with the [[subspace topology]]) arises as a [[deformation retraction]], i.e., there is a homotopy from the identity map to that retraction that restricts to <math>r</math> on the subspace <math>Y</math> at all time.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::nonempty topologically convex space]] || || || || {{intermediate notions|space in which every retraction is a deformation retraction|topologically convex space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::contractible space]] || || || || {{intermediate notions short|contractible space|space in which every retraction is a deformation retraction}}
|-
| [[Stronger than::SDR-contractible space]] || || || || {{intermediate notions short|SDR-contractible space|space in which every retraction is a deformation retraction}}
|}

Space in which every retraction is a deformation retraction

2023-10-26T22:20:53Z

Vipul: /* Definition */

==Definition==

A '''space in which every retraction is a deformation retraction''' is a [[topological space]] <math>X</math> with the property that any [[retraction]] <math>r:X \to Y</math> for a subspace <math>Y</math> of <math>X</math> (with the [[subspace topology]]) arises as a [[deformation retraction]], i.e., there is a homotopy from the identity map to that retraction that restricts to <math>r</math> on the subspace <math>Y</math> at all time.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::topologically convex space]] || || || || {{intermediate notions|space in which every retraction is a deformation retraction|topologically convex space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::contractible space]] || || || || {{intermediate notions short|contractible space|space in which every retraction is a deformation retraction}}
|-
| [[Stronger than::SDR-contractible space]] || || || || {{intermediate notions short|SDR-contractible space|space in which every retraction is a deformation retraction}}
|}

Weakly contractible space

2023-10-26T22:20:37Z

Vipul: /* Stronger properties */

{{topospace property}}

{{variation of|contractible space}}

==Definition==

=== Equivalent definitions in tabular format ===

A nonempty [[topological space]] is said to be '''weakly contractible''' if it satisfies the following equivalent conditions. The [[empty space]] is generally excluded from consideration when considering the question of weak contractibility.

As we see below, each of the definitions (implicitly or explicitly) implies that the space is a [[path-connected space]].

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed weakly contractible if ... !! A topological space <math>X</math> is termed weakly contractible if ...
|-
| 1 || weakly homotopy-equivalent to a point || it is weakly homotopy-equivalent to a one-point space. Note that the definition in particular implies that the space is a [[path-connected space]]. || <math>X</math> is path-connected, and it is in the same equivalence class as the [[one-point space]] under weak homotopy equivalence.
|-
| 2 || weakly homotopy-equivalent to a point (unique map to the point) || it is path-connected and the unique map from it to the [[one-point space]] is a [[weak homotopy equivalence of topological spaces]] (regardless of choice of basepoint). || (fix any point <math>x_0 \in X</math>; the truth of this fact does not depend on the choice of <math>x_0</math>) <math>X</math> is path-connected, and the unique map <math>f: X \to \{ * \}</math> to the one-point space (which is continuous by definition) induces isomorphisms <math>\pi_f: \pi_n(X,x_0) \to \pi_n(\{ * \}, *)</math> on all the [[homotopy groups]].
|-
| 3 || weakly homotopy-equivalent to a point (arbitrary choice of basepoint) || it is path-connected and any map from a one-point space to it induces a weak homotopy equivalence of topological spaces || (fix any point <math>x_0 \in X</math>; the truth of this fact does not depend on the choice of <math>x_0</math>) Suppose <math>g</math> is the (continuous by definition) map from the one-point space <math>\{ * \}</math> to <math>X</math> that sends the point <math>*</math> to <math>x_0</math>. Then, <math>g</math> induces isomorphisms <math>\pi_g: \pi_n(\{ * \}, *) \to \pi_n(X, x_0</math> on all the homotopy groups.
|-
| 4 || all homotopy groups are trivial || it is path-connected (i.e., its [[set of path components]] has size one) and all its [[homotopy group]]s are trivial. || (fix any point <math>x_0 \in X</math>; the truth of this fact does not depend on the choice of <math>x_0</math>) <math>X</math> is path-connected and <math>\pi_n(X, x_0)</math> is the trivial group for all positive integers <math>n</math>.
|-
| 5 || any map from a sphere is nullhomotopic || it is path-connected and any continuous map from a [[sphere]] (of any finite dimension) to it is nullhomotopic. || <math>X</math> is path-connected, and any continuous map <math>f: S^n \to X</math> for any <math>n \ge 1</math> is nullhomotopic, i.e., we can construct a homotopy from that map to a map that sends <math>S^n</math> to a single point. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
|-
| 6 || any map from a connected CW-space is nullhomotopic || it is path-connected and any continuous map from a connected [[CW-space]] (the underlying space of a [[CW-complex]]) to it is nullhomotopic. || <math>X</math> is path-connected and for any connected CW-space <math>Y</math> and any continuous map <math>f: Y \to X</math>, <math>f</math> is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
|-
| 7 || any map from a connected polyhedron is nullhomotopic || it is path-connected and any continuous map from a connected [[polyhedron]] (the geometric realization of a simplicial complex) to it is nullhomotopic. || <math>X</math> is path-connected and for any connected polyhedron <math>Y</math> and any continuous map <math>f: Y \to X</math>, <math>f</math> is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
|-
| 8 || any map from a connected manifold is nullhomotopic || it is path-connected and any continuous map from a connected manifold to it is nullhomotopic. || <math>X</math> is path-connected and for any connected manifold <math>Y</math> and any continuous map <math>f: Y \to X</math>, <math>f</math> is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
|-
| 9 || any map from a manifold is nullhomotopic || it is path-connected and any continuous map from a manifold to it is nullhomotopic. || <math>X</math> is path-connected and for any manifold <math>Y</math> and any continuous map <math>f: Y \to X</math>, <math>f</math> is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
|}

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::contractible space]] || admits a [[contracting homotopy]] || (direct from definition) || [[weakly contractible not implies contractible]] (however, [[Whitehead's theorem]] says that weakly contractible equals contractible when we restrict to [[CW-space]]s) || {{intermediate notions short|weakly contractible space|contractible space}}
|-
| [[Weaker than::nonempty topologically convex space]] || nonempty and homeomorphic to a [[convex subset of Euclidean space]] || (via contractible) || (via contractible) || {{intermediate notions short|weakly contractible space|topologically convex space}}
|}

===Weaker properties ===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::acyclic space]] || all the homology groups are the same as those of a point || [[weakly contractible implies acyclic]] || || {{intermediate notions short|acyclic space|weakly contractible space}}
|-
| [[Stronger than::rationally acyclic space]] || all the homology groups over the rationals are the same as those of a point || || || {{intermediate notions short|rationally acyclic space|weakly contractible space}}
|-
| [[Stronger than::simply connected space]] || the [[fundamental group]] is trivial || || || {{intermediate notions short|simply connected space|weakly contractible space}}
|-
| [[Stronger than::simple space]] || the [[fundamental group]] is abelian and it acts trivially on all the higher homotopy groups || || || {{intermediate notions short|simple space|weakly contractible space}}
|-
| [[Stronger than::path-connected space]] || any two points can be connected by a path || || || {{intermediate notions short|path-connected space|weakly contractible space}}
|}

==Metaproperties==

{{DP-closed}}

Since the homotopy group of the product of two spaces is the product of their homotopy groups, the product of two weakly contractible spaces is again weakly contractible.

{{retract-hereditary}}

Any retract, and more generally, any [[homotopically injective subspace]] of a weakly contractible space is again weakly contractible.

Nonempty topologically convex implies equiconnected

2023-10-26T22:20:18Z

Vipul: /* Topologically convex space */

Nonempty topologically convex implies equiconnected

2023-10-26T22:20:04Z

Vipul:

Nonempty topologically convex implies equiconnected

2023-10-26T22:19:44Z

Vipul:

{{topospace property implication|stronger = nonempty topologically convex space|weaker = equiconnected space}}

==Statement==

===Topological version===

Any nonempty [[topologically convex space]] is an [[equiconnected space]].

===Realized version===

Any nonempty [[convex subset of Euclidean space]] is (topologically) an [[equiconnected space]].

==Definitions used==

===Topologically convex space===

{{further|[[topologically convex space]]}}

A topological space is called a topologically convex space if it is homeomorphic to a [[convex subset of Euclidean space]].

===Convex subset of Euclidean space===

{{further|[[convex subset of Euclidean space]]}}

A convex subset of Euclidean space is a subset in <math>\R^n</math> for some <math>n</math>, with the property that given any two points in the subset, the line segment joining those two points also lies completely within the subset.

The definition may also apply to infinite-dimensional Euclidean spaces, and the proof would work even in that case.

===Equiconnected space===

Topologically convex implies equiconnected

2023-10-26T22:19:35Z

Vipul: Vipul moved page Topologically convex implies equiconnected to Nonempty topologically convex implies equiconnected

#REDIRECT [[Nonempty topologically convex implies equiconnected]]

Nonempty topologically convex implies equiconnected

2023-10-26T22:19:34Z

Vipul: Vipul moved page Topologically convex implies equiconnected to Nonempty topologically convex implies equiconnected

{{topospace property implication|stronger = topologically convex space|weaker = equiconnected space}}

==Statement==

===Topological version===

Any nonempty [[topologically convex space]] is an [[equiconnected space]].

===Realized version===

Any nonempty [[convex subset of Euclidean space]] is (topologically) an [[equiconnected space]].

==Definitions used==

===Topologically convex space===

{{further|[[topologically convex space]]}}

A topological space is called a topologically convex space if it is homeomorphic to a [[convex subset of Euclidean space]].

===Convex subset of Euclidean space===

{{further|[[convex subset of Euclidean space]]}}

A convex subset of Euclidean space is a subset in <math>\R^n</math> for some <math>n</math>, with the property that given any two points in the subset, the line segment joining those two points also lies completely within the subset.

The definition may also apply to infinite-dimensional Euclidean spaces, and the proof would work even in that case.

===Equiconnected space===

Equiconnected space

2023-10-26T22:19:25Z

Vipul: /* Stronger properties */

{{topospace property}}

==Definition==

A nonempty [[topological space]] <math>X</math> is said to be '''equiconnected''' if there is a continuous map <math>k:X \times [0,1] \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x,y \in X</math>.

Roughly, speaking, at any given time <math>t</math>, we get a map <math>X \times X</math> to <math>X</math>. At time <math>0</math>, it is projection on the first coordinate, and at time 1, it is projection on the second coordinate. For elements on the diagonal, it always remains the value at the diagonal.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::nonempty topologically convex space]] || nonempty space that is homeomorphic to a [[convex subset of Euclidean space]] || [[topologically convex implies equiconnected]] || || {{intermediate notions short|equiconnected space|topologically convex space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[equiconnected implies contractible]] || [[contractible not implies equiconnected]] || {{intermediate notions short|contractible space|equiconnected space}}
|}

==References==

* {{mathoverflow|number = 457103|title = Spaces that are contractible mod diagonal}}: This describes the property and asks for its name

Contractible space

2023-10-26T22:19:09Z

Vipul: /* Stronger properties */

{{homotopy-invariant topospace property}}

==Definition==

=== Equivalent definitions in tabular format ===

A nonempty [[topological space]] is said to be '''contractible''' if it satisfies the following equivalent conditions. The [[empty space]] is generally excluded from consideration when considering the question of contractibility.

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed contractible if ... !! A topological space <math>X</math> is termed contractible if ...
|-
| 1 || homotopy-equivalent to a point || there is a [[homotopy equivalence of topological spaces]] between the topological space and a [[one-point space]]. || There exist [[continuous map]]s <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math> such that <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 2 || homotopy-equivalent to a point (arbitrary map) || ''any'' pair of maps between the space and a one-point space define a homotopy equivalence of topological spaces. || For ''any'' continuous map <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math>, <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 3 || admits a contracting homotopy || there is a point in the space to which there is a [[contracting homotopy]]. || there exists a point <math>x_0 \in X</math> and a contracting homotopy that contracts <math>X</math> to <math>x_0</math>. Explicitly, there exists a continuous map <math>F: X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>. Note that we do ''not'' assume or require that <math>F(x_0,a) = x_0</math> for all <math>a \in [0,1]</math>.
|-
| 4 || admits a contracting homotopy (arbitrary point) || the space admits a [[contracting homotopy]] to ''any'' point in it. || for ''any'' point <math>x_1 \in X</math>, there is a contracting homotopy that contracts <math>X</math> to <math>x_1</math>.
|-
| 5 || unique homotopy class of maps to it || for any other topological space, there is a unique homotopy class of maps from the other space to it. || for any topological space <math>Y</math>, and any two continuous maps <math>h_1, h_2: Y \to X</math>, <math>h_1</math> and <math>h_2</math> are homotopic. In particular, any map from a topological space to it is nullhomotopic.
|-
| 6 || retract of cone space || it is a [[retract]] of its [[cone space]]. || the inclusion <math>\iota: X \to CX</math> in the cone space <math>CX</math> (as the base) has a one-sided inverse, <math>j: CX \to X</math>, i.e., <math>j</math> is a continuous map such that <math>j \circ \iota</math> is the identity map on <math>X</math>.
|}

==Metaproperties==

{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[contractibility is product-closed]] || If <math>X_i, i \in I</math> form a (finite or infinite) collection of contractible spaces, then the product of the <math>X_i</math>s, equipped with the [[product topology]], is also contractible. In particular, if <math>X_1, X_2</math> are contractible, then <math>X_1 \times X_2</math> is also contractible.
|-
| [[satisfies metaproperty::retract-hereditary property of topological spaces]] || Yes || [[contractibility is retract-hereditary]] || If <math>A \subseteq X</math> and <math>f:X \to A</math> is a continuous [[retraction]], and <math>X</math> is contractible, then <math>A</math> is contractible.
|-
| [[satisfies metaproperty::suspension-closed property of topological spaces]] || Yes || [[contractibility is suspension-closed]] || The [[suspension]] of a contractible space is contractible.
|-
| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[contractibility is not closure-preserved]] || It is possible to have a topological space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is contractible in the [[subspace topology]], but the [[closure]] <math>\overline{A}</math> is not.
|-
| [[dissatisfies metaproperty::connected union-closed property of topological spaces]] || No || [[contractibility is not connected union-closed]] || It is possible to have a topological space <math>X</math> expressible as a union of subsets <math>A,B</math>, both contractible in their subspace topology, with <math>A \cap B</math> nonempty, but <math>X</math> itself not contractible.
|}

== Examples ==

=== Extreme and basic examples ===

* The [[one-point space]] is contractible.
* Any [[Euclidean space]] is contractible.
* The closed unit disk in any dimension is contractible.
* [[Compact manifold]]s in dimension one or more, such as the [[circle]], are not contractible.

=== Intuition behind examples ===

Contractibility is, fundamentally, a ''global'' property of topological spaces. It is not possible to examine a small part of the space and conclude that it is contractible, nor does examining a small part of the space allow us to rule ''out'' the possibiilty that it is contractible. For the intuition behind the former, note that we can attach non-contractible pieces (like [[circle]]s) far off from the part of the space we are looking at. For the intuition behind the latter claim, note that we can embed ''any'' topological space as a closed subspace of a [[contractible space]], namely, its [[cone space]].

For this reason, when looking for examples or counterexamples, we need to focus on the global structure.

=== Examples from topological construction ===

One thing to keep in mind is that since the definition of contractibility invokes the closed unit interval, it is likely that any effort to construct contractible spaces will invariably involve dealing with the real numbers. The most topologically general way of constructing a contractible space is as the [[cone space]] of an arbitrary topological space. One way of thinking of this cone space is as a literal cone that fills in between the space and a point. Up until the very tip of the cone, the cross-sections look homeomorphic to the topological space.

=== Examples from geometry ===

A [[topologically star-like space]] is a classic example of a contractible space. A topological space is termed topologically star-like if it is homeomorphic to a star-like subset of Euclidean space. A star-like subset of Euclidean space is a subset for which there exists a point in it such that for every other point in it, the line segment joining the points is completely inside the space.

A topologically star-like space is contractible, and can in fact be contracted to any point relative to which it is a star through a straight-line homotopy, i.e., moving each point toward the center in a straight line. The contracting homotopy fixes the center, and therefore, the space is in fact a [[SDR-contractible space]].

Note that, if also compact, a topologically star-like space is homeomorphic to the cone space of its boundary. Otherwise, the space is still ''almost'' a cone space: it is a subspace of the cone space that contains the full complement of the base and an arbitrary subset of the base. Nonetheless, it is important to note that the condition of being star-like also carries various geometric implications (in particular, from being a [[sub-Euclidean space]]) that are not satisfied for arbitrary cone spaces.

A [[nonempty topologically convex space]] is a nonempty space that is homeomorphic to a [[convex subset of Euclidean space]]. Any nonempty topologically convex space is topologically star-like, and ''any'' point can be taken as the center. An example of a topologically star-like space that is not a nonempty topologically convex space is a [[pair of intersecting lines]].

It is possible to construct spaces that are not topologically star-like, but still contractible. For instance, any geometric realization of a tree is contractible, but if the tree has more than one point with degree greater than two, it is not topologically star-like. As a related example, a set of parallel lines combined with one line that intersects all of them form a contractible space that is not topologically star-like.

==Relation with other properties==

=== Incomparable properties ===

Contractibility is incomparable with most of the interesting separation and compactness properties.

'''Broad argument for why contractibility cannot imply any meaningful separation or compactness property''': The [[cone space]] over ''any'' topological space is contractible. In particular, since any topological space arises as a closed subspace of its cone space (namely, the "base" of the space), every topological space arises as a closed subspace of a contractible space. Therefore, contractible cannot imply any nontrivial property that is [[subspace-hereditary property of topological spaces|subspace-hereditary]] or even [[weakly hereditary property of topological spaces|weakly hereditary]] (inherited by closed subsets).

'''Broad argument for why meaningful separation or compactness properties cannot imply contractibility''': Most meaningful separation and compactness properties are satisfied by all [[compact manifold]]s. However, compact manifolds of dimension greater than one are not contractible. The simplest counterexample is generally the [[circle]].

Some incomparable properties:

* [[T0 space]]
* [[T1 space]]
* [[Hausdorff space]]
* [[regular space]]
* [[normal space]]
* [[metrizable space]]
* [[paracompact space]]
* [[compact space]]

The property of being a contractible space is also incomparable with the property of being a [[locally contractible space]]. A contractible space need not be locally contractible. In fact, it need not even be locally connected! An example of a contractible space that is not locally contractible is the [[comb space]]. An example of a space that is locally contractible but not contractible is the [[circle]] (or, more generally, any compact manifold).

===Stronger properties===

Note that we may need to assume nonemptiness on top of the provided definitions in some case.

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[cone space]] over some topological space || || [[cone space implies contractible]] || ||
|-
| [[Weaker than::topologically star-like space]] || || || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Weaker than::nonempty topologically convex space]] || homeomorphic to a [[convex subset of Euclidean space]] || [[convex implies star-like|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Weaker than::suddenly contractible space]] || has a [[contracting homotopy]] that is also a [[sudden homotopy]] || || ||{{intermediate notions short|contractible space|suddenly contractible space}}
|-
| [[Weaker than::SDR-contractible space]] || has a [[contracting homotopy]] that is also a [[deformation retraction]] || || || {{intermediate notions short|contractible space|SDR-contractible space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || [[contractible implies weakly contractible]] || [[weakly contractible not implies contractible]] || {{intermediate notions short|weakly contractible space|contractible space}}
|-
| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || the <math>n</math>-sphere <math>S^n</math> is <math>(n-1)</math>-connected but not <math>n</math>-connected. || {{intermediate notions short|multiply connected space|contractible space}}
|-
| [[Stronger than::simply connected space]] || [[path-connected space]], [[fundamental group]] is [[trivial group|trivial]] || || || {{intermediate notions short|simply connected space|contractible space}}
|-
| [[Stronger than::path-connected space]] || there is a [[path]] between any two points || || || {{intermediate notions short|path-connected space|contractible space}}
|-
| [[Stronger than::connected space]] || cannot be partitioned into disjoint nonempty subsets || || || {{intermediate notions short|connected space|contractible space}}
|-
| [[Stronger than::acyclic space]] || homology groups over <math>\mathbb{Z}</math> all zero except zeroth homology group which is <math>\mathbb{Z}</math> || || || {{intermediate notions short|acyclic space|contractible space}}
|-
| [[Stronger than::rationally acyclic space]] || homology groups over <matH>\mathbb{Q}</math> all zero except zeroth homology group which is <math>\mathbb{Q}</math> || || || {{intermediate notions short|rationally acyclic space|contractible space}}
|-
| [[Stronger than::space with Euler characteristic one]] || [[Euler characteristic]] of the space is one. || ([[acyclic implies Euler charcateristic one|via acyclic]])|| [[Euler characteristic one not implies acyclic]] || {{intermediate notions short|space with Euler characteristic one|contractible space}}
|}

=== Conjunction with other properties ===

* [[Contractible manifold]]: Contractible as well as a [[manifold]]
* [[Contractible polyhedron]]: Contractible as well as a [[polyhedron]], i.e., the geometric realization of a [[simplicial complex]]

==References==

===Textbook references===

* {{booklink|Munkres}}, Page 330, Exercise 3 (definition introduced in exercise)
* {{booklink|SingerThorpe}}, Page 51 (formal definition)
* {{booklink|Rotman}}, Page 18 (formal definition)
* {{booklink|Hatcher}}, Page 4 (formal definition)
* {{booklink|Spanier}}, Page 25 (definition in paragraph)

Contractible space

2023-10-26T22:18:42Z

Vipul: /* Examples from geometry */

{{homotopy-invariant topospace property}}

==Definition==

=== Equivalent definitions in tabular format ===

A nonempty [[topological space]] is said to be '''contractible''' if it satisfies the following equivalent conditions. The [[empty space]] is generally excluded from consideration when considering the question of contractibility.

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed contractible if ... !! A topological space <math>X</math> is termed contractible if ...
|-
| 1 || homotopy-equivalent to a point || there is a [[homotopy equivalence of topological spaces]] between the topological space and a [[one-point space]]. || There exist [[continuous map]]s <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math> such that <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 2 || homotopy-equivalent to a point (arbitrary map) || ''any'' pair of maps between the space and a one-point space define a homotopy equivalence of topological spaces. || For ''any'' continuous map <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math>, <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 3 || admits a contracting homotopy || there is a point in the space to which there is a [[contracting homotopy]]. || there exists a point <math>x_0 \in X</math> and a contracting homotopy that contracts <math>X</math> to <math>x_0</math>. Explicitly, there exists a continuous map <math>F: X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>. Note that we do ''not'' assume or require that <math>F(x_0,a) = x_0</math> for all <math>a \in [0,1]</math>.
|-
| 4 || admits a contracting homotopy (arbitrary point) || the space admits a [[contracting homotopy]] to ''any'' point in it. || for ''any'' point <math>x_1 \in X</math>, there is a contracting homotopy that contracts <math>X</math> to <math>x_1</math>.
|-
| 5 || unique homotopy class of maps to it || for any other topological space, there is a unique homotopy class of maps from the other space to it. || for any topological space <math>Y</math>, and any two continuous maps <math>h_1, h_2: Y \to X</math>, <math>h_1</math> and <math>h_2</math> are homotopic. In particular, any map from a topological space to it is nullhomotopic.
|-
| 6 || retract of cone space || it is a [[retract]] of its [[cone space]]. || the inclusion <math>\iota: X \to CX</math> in the cone space <math>CX</math> (as the base) has a one-sided inverse, <math>j: CX \to X</math>, i.e., <math>j</math> is a continuous map such that <math>j \circ \iota</math> is the identity map on <math>X</math>.
|}

==Metaproperties==

{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[contractibility is product-closed]] || If <math>X_i, i \in I</math> form a (finite or infinite) collection of contractible spaces, then the product of the <math>X_i</math>s, equipped with the [[product topology]], is also contractible. In particular, if <math>X_1, X_2</math> are contractible, then <math>X_1 \times X_2</math> is also contractible.
|-
| [[satisfies metaproperty::retract-hereditary property of topological spaces]] || Yes || [[contractibility is retract-hereditary]] || If <math>A \subseteq X</math> and <math>f:X \to A</math> is a continuous [[retraction]], and <math>X</math> is contractible, then <math>A</math> is contractible.
|-
| [[satisfies metaproperty::suspension-closed property of topological spaces]] || Yes || [[contractibility is suspension-closed]] || The [[suspension]] of a contractible space is contractible.
|-
| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[contractibility is not closure-preserved]] || It is possible to have a topological space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is contractible in the [[subspace topology]], but the [[closure]] <math>\overline{A}</math> is not.
|-
| [[dissatisfies metaproperty::connected union-closed property of topological spaces]] || No || [[contractibility is not connected union-closed]] || It is possible to have a topological space <math>X</math> expressible as a union of subsets <math>A,B</math>, both contractible in their subspace topology, with <math>A \cap B</math> nonempty, but <math>X</math> itself not contractible.
|}

== Examples ==

=== Extreme and basic examples ===

* The [[one-point space]] is contractible.
* Any [[Euclidean space]] is contractible.
* The closed unit disk in any dimension is contractible.
* [[Compact manifold]]s in dimension one or more, such as the [[circle]], are not contractible.

=== Intuition behind examples ===

Contractibility is, fundamentally, a ''global'' property of topological spaces. It is not possible to examine a small part of the space and conclude that it is contractible, nor does examining a small part of the space allow us to rule ''out'' the possibiilty that it is contractible. For the intuition behind the former, note that we can attach non-contractible pieces (like [[circle]]s) far off from the part of the space we are looking at. For the intuition behind the latter claim, note that we can embed ''any'' topological space as a closed subspace of a [[contractible space]], namely, its [[cone space]].

For this reason, when looking for examples or counterexamples, we need to focus on the global structure.

=== Examples from topological construction ===

One thing to keep in mind is that since the definition of contractibility invokes the closed unit interval, it is likely that any effort to construct contractible spaces will invariably involve dealing with the real numbers. The most topologically general way of constructing a contractible space is as the [[cone space]] of an arbitrary topological space. One way of thinking of this cone space is as a literal cone that fills in between the space and a point. Up until the very tip of the cone, the cross-sections look homeomorphic to the topological space.

=== Examples from geometry ===

A [[topologically star-like space]] is a classic example of a contractible space. A topological space is termed topologically star-like if it is homeomorphic to a star-like subset of Euclidean space. A star-like subset of Euclidean space is a subset for which there exists a point in it such that for every other point in it, the line segment joining the points is completely inside the space.

A topologically star-like space is contractible, and can in fact be contracted to any point relative to which it is a star through a straight-line homotopy, i.e., moving each point toward the center in a straight line. The contracting homotopy fixes the center, and therefore, the space is in fact a [[SDR-contractible space]].

Note that, if also compact, a topologically star-like space is homeomorphic to the cone space of its boundary. Otherwise, the space is still ''almost'' a cone space: it is a subspace of the cone space that contains the full complement of the base and an arbitrary subset of the base. Nonetheless, it is important to note that the condition of being star-like also carries various geometric implications (in particular, from being a [[sub-Euclidean space]]) that are not satisfied for arbitrary cone spaces.

A [[nonempty topologically convex space]] is a nonempty space that is homeomorphic to a [[convex subset of Euclidean space]]. Any nonempty topologically convex space is topologically star-like, and ''any'' point can be taken as the center. An example of a topologically star-like space that is not a nonempty topologically convex space is a [[pair of intersecting lines]].

It is possible to construct spaces that are not topologically star-like, but still contractible. For instance, any geometric realization of a tree is contractible, but if the tree has more than one point with degree greater than two, it is not topologically star-like. As a related example, a set of parallel lines combined with one line that intersects all of them form a contractible space that is not topologically star-like.

==Relation with other properties==

=== Incomparable properties ===

Contractibility is incomparable with most of the interesting separation and compactness properties.

'''Broad argument for why contractibility cannot imply any meaningful separation or compactness property''': The [[cone space]] over ''any'' topological space is contractible. In particular, since any topological space arises as a closed subspace of its cone space (namely, the "base" of the space), every topological space arises as a closed subspace of a contractible space. Therefore, contractible cannot imply any nontrivial property that is [[subspace-hereditary property of topological spaces|subspace-hereditary]] or even [[weakly hereditary property of topological spaces|weakly hereditary]] (inherited by closed subsets).

'''Broad argument for why meaningful separation or compactness properties cannot imply contractibility''': Most meaningful separation and compactness properties are satisfied by all [[compact manifold]]s. However, compact manifolds of dimension greater than one are not contractible. The simplest counterexample is generally the [[circle]].

Some incomparable properties:

* [[T0 space]]
* [[T1 space]]
* [[Hausdorff space]]
* [[regular space]]
* [[normal space]]
* [[metrizable space]]
* [[paracompact space]]
* [[compact space]]

The property of being a contractible space is also incomparable with the property of being a [[locally contractible space]]. A contractible space need not be locally contractible. In fact, it need not even be locally connected! An example of a contractible space that is not locally contractible is the [[comb space]]. An example of a space that is locally contractible but not contractible is the [[circle]] (or, more generally, any compact manifold).

===Stronger properties===

Note that we may need to assume nonemptiness on top of the provided definitions in some case.

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[cone space]] over some topological space || || [[cone space implies contractible]] || ||
|-
| [[Weaker than::topologically star-like space]] || || || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Weaker than::topologically convex space]] || homeomorphic to a [[convex subset of Euclidean space]] || [[convex implies star-like|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Weaker than::suddenly contractible space]] || has a [[contracting homotopy]] that is also a [[sudden homotopy]] || || ||{{intermediate notions short|contractible space|suddenly contractible space}}
|-
| [[Weaker than::SDR-contractible space]] || has a [[contracting homotopy]] that is also a [[deformation retraction]] || || || {{intermediate notions short|contractible space|SDR-contractible space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || [[contractible implies weakly contractible]] || [[weakly contractible not implies contractible]] || {{intermediate notions short|weakly contractible space|contractible space}}
|-
| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || the <math>n</math>-sphere <math>S^n</math> is <math>(n-1)</math>-connected but not <math>n</math>-connected. || {{intermediate notions short|multiply connected space|contractible space}}
|-
| [[Stronger than::simply connected space]] || [[path-connected space]], [[fundamental group]] is [[trivial group|trivial]] || || || {{intermediate notions short|simply connected space|contractible space}}
|-
| [[Stronger than::path-connected space]] || there is a [[path]] between any two points || || || {{intermediate notions short|path-connected space|contractible space}}
|-
| [[Stronger than::connected space]] || cannot be partitioned into disjoint nonempty subsets || || || {{intermediate notions short|connected space|contractible space}}
|-
| [[Stronger than::acyclic space]] || homology groups over <math>\mathbb{Z}</math> all zero except zeroth homology group which is <math>\mathbb{Z}</math> || || || {{intermediate notions short|acyclic space|contractible space}}
|-
| [[Stronger than::rationally acyclic space]] || homology groups over <matH>\mathbb{Q}</math> all zero except zeroth homology group which is <math>\mathbb{Q}</math> || || || {{intermediate notions short|rationally acyclic space|contractible space}}
|-
| [[Stronger than::space with Euler characteristic one]] || [[Euler characteristic]] of the space is one. || ([[acyclic implies Euler charcateristic one|via acyclic]])|| [[Euler characteristic one not implies acyclic]] || {{intermediate notions short|space with Euler characteristic one|contractible space}}
|}

=== Conjunction with other properties ===

* [[Contractible manifold]]: Contractible as well as a [[manifold]]
* [[Contractible polyhedron]]: Contractible as well as a [[polyhedron]], i.e., the geometric realization of a [[simplicial complex]]

==References==

===Textbook references===

* {{booklink|Munkres}}, Page 330, Exercise 3 (definition introduced in exercise)
* {{booklink|SingerThorpe}}, Page 51 (formal definition)
* {{booklink|Rotman}}, Page 18 (formal definition)
* {{booklink|Hatcher}}, Page 4 (formal definition)
* {{booklink|Spanier}}, Page 25 (definition in paragraph)

Topologically star-like space

2023-10-26T22:18:06Z

Vipul: /* Stronger properties */

{{topospace property}}

==Definition==

A non-empty [[topological space]] is termed '''topologically star-like''' if it is [[defining ingredient::homeomorphism|homeomorphic]] to a [[defining ingredient::star-like subset of Euclidean space]] (Where the Euclidean space is possibly infinite-dimensional), where the latter is endowed with the [[subspace topology]] from Euclidean space.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::nonempty topologically convex space]] || homemorphic to a [[convex subset of Euclidean space]] || follows from [[nonempty and convex implies star-like]] || ? || {{intermediate notions short|topologically star-like space|topologically convex space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[star-like implies contractible]] || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Stronger than::semi-suddenly contractible space]] || has a [[semi-sudden homotopy|semi-sudden]] [[contracting homotopy]] || || [[star-like implies contractible]] || || {{intermediate notions short|semi-suddenly contractible space|topologically star-like space}}
|-
| [[Stronger than::SDR-contractible space]] || has a [[contracting homotopy]] that is a [[deformation retraction]] || [[star-like implies contractible]] || || {{intermediate notions short|SDR-contractible space|topologically star-like space}}
|}

Topologically star-like space

2023-10-26T22:17:56Z

Vipul: /* Stronger properties */

{{topospace property}}

==Definition==

A non-empty [[topological space]] is termed '''topologically star-like''' if it is [[defining ingredient::homeomorphism|homeomorphic]] to a [[defining ingredient::star-like subset of Euclidean space]] (Where the Euclidean space is possibly infinite-dimensional), where the latter is endowed with the [[subspace topology]] from Euclidean space.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::nonempty topologically convex space]] || homemorphic to a [[convex subset of Euclidean space]] || follows from [[convex implies star-like]] || ? || {{intermediate notions short|topologically star-like space|topologically convex space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[star-like implies contractible]] || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Stronger than::semi-suddenly contractible space]] || has a [[semi-sudden homotopy|semi-sudden]] [[contracting homotopy]] || || [[star-like implies contractible]] || || {{intermediate notions short|semi-suddenly contractible space|topologically star-like space}}
|-
| [[Stronger than::SDR-contractible space]] || has a [[contracting homotopy]] that is a [[deformation retraction]] || [[star-like implies contractible]] || || {{intermediate notions short|SDR-contractible space|topologically star-like space}}
|}

Nonempty topologically convex space

2023-10-26T22:17:46Z

Vipul:

Topologically convex space

2023-10-26T22:17:36Z

Vipul: Vipul moved page Topologically convex space to Nonempty topologically convex space

#REDIRECT [[Nonempty topologically convex space]]

Nonempty topologically convex space

2023-10-26T22:17:36Z

Vipul: Vipul moved page Topologically convex space to Nonempty topologically convex space

{{topospace property}}

==Definition==

A '''topologically convex space''' is a [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[convex subset of Euclidean space]].

==Relation with other properties==

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::topologically star-like space]] || || follows from [[convex implies star-like]] || the [[pair of intersecting lines]] is topologically star-like but not topologically convex || {{intermediate notions short|topologically star-like space|topologically convex space}}
|-
| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Stronger than::semi-suddenly contractible space]] || has a [[semi-sudden homotopy|semi-sudden]] [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|semi-suddenly contractible space|topologically convex space}}
|-
| [[Stronger than::SDR-contractible space]] || has a [[contracting homotopy]] that is a [[deformation retraction]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|SDR-contractible space|topologically convex space}}
|}

Contractible space

2023-10-26T22:16:56Z

Vipul: /* Stronger properties */

{{homotopy-invariant topospace property}}

==Definition==

=== Equivalent definitions in tabular format ===

A nonempty [[topological space]] is said to be '''contractible''' if it satisfies the following equivalent conditions. The [[empty space]] is generally excluded from consideration when considering the question of contractibility.

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed contractible if ... !! A topological space <math>X</math> is termed contractible if ...
|-
| 1 || homotopy-equivalent to a point || there is a [[homotopy equivalence of topological spaces]] between the topological space and a [[one-point space]]. || There exist [[continuous map]]s <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math> such that <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 2 || homotopy-equivalent to a point (arbitrary map) || ''any'' pair of maps between the space and a one-point space define a homotopy equivalence of topological spaces. || For ''any'' continuous map <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math>, <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 3 || admits a contracting homotopy || there is a point in the space to which there is a [[contracting homotopy]]. || there exists a point <math>x_0 \in X</math> and a contracting homotopy that contracts <math>X</math> to <math>x_0</math>. Explicitly, there exists a continuous map <math>F: X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>. Note that we do ''not'' assume or require that <math>F(x_0,a) = x_0</math> for all <math>a \in [0,1]</math>.
|-
| 4 || admits a contracting homotopy (arbitrary point) || the space admits a [[contracting homotopy]] to ''any'' point in it. || for ''any'' point <math>x_1 \in X</math>, there is a contracting homotopy that contracts <math>X</math> to <math>x_1</math>.
|-
| 5 || unique homotopy class of maps to it || for any other topological space, there is a unique homotopy class of maps from the other space to it. || for any topological space <math>Y</math>, and any two continuous maps <math>h_1, h_2: Y \to X</math>, <math>h_1</math> and <math>h_2</math> are homotopic. In particular, any map from a topological space to it is nullhomotopic.
|-
| 6 || retract of cone space || it is a [[retract]] of its [[cone space]]. || the inclusion <math>\iota: X \to CX</math> in the cone space <math>CX</math> (as the base) has a one-sided inverse, <math>j: CX \to X</math>, i.e., <math>j</math> is a continuous map such that <math>j \circ \iota</math> is the identity map on <math>X</math>.
|}

==Metaproperties==

{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[contractibility is product-closed]] || If <math>X_i, i \in I</math> form a (finite or infinite) collection of contractible spaces, then the product of the <math>X_i</math>s, equipped with the [[product topology]], is also contractible. In particular, if <math>X_1, X_2</math> are contractible, then <math>X_1 \times X_2</math> is also contractible.
|-
| [[satisfies metaproperty::retract-hereditary property of topological spaces]] || Yes || [[contractibility is retract-hereditary]] || If <math>A \subseteq X</math> and <math>f:X \to A</math> is a continuous [[retraction]], and <math>X</math> is contractible, then <math>A</math> is contractible.
|-
| [[satisfies metaproperty::suspension-closed property of topological spaces]] || Yes || [[contractibility is suspension-closed]] || The [[suspension]] of a contractible space is contractible.
|-
| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[contractibility is not closure-preserved]] || It is possible to have a topological space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is contractible in the [[subspace topology]], but the [[closure]] <math>\overline{A}</math> is not.
|-
| [[dissatisfies metaproperty::connected union-closed property of topological spaces]] || No || [[contractibility is not connected union-closed]] || It is possible to have a topological space <math>X</math> expressible as a union of subsets <math>A,B</math>, both contractible in their subspace topology, with <math>A \cap B</math> nonempty, but <math>X</math> itself not contractible.
|}

== Examples ==

=== Extreme and basic examples ===

* The [[one-point space]] is contractible.
* Any [[Euclidean space]] is contractible.
* The closed unit disk in any dimension is contractible.
* [[Compact manifold]]s in dimension one or more, such as the [[circle]], are not contractible.

=== Intuition behind examples ===

Contractibility is, fundamentally, a ''global'' property of topological spaces. It is not possible to examine a small part of the space and conclude that it is contractible, nor does examining a small part of the space allow us to rule ''out'' the possibiilty that it is contractible. For the intuition behind the former, note that we can attach non-contractible pieces (like [[circle]]s) far off from the part of the space we are looking at. For the intuition behind the latter claim, note that we can embed ''any'' topological space as a closed subspace of a [[contractible space]], namely, its [[cone space]].

For this reason, when looking for examples or counterexamples, we need to focus on the global structure.

=== Examples from topological construction ===

One thing to keep in mind is that since the definition of contractibility invokes the closed unit interval, it is likely that any effort to construct contractible spaces will invariably involve dealing with the real numbers. The most topologically general way of constructing a contractible space is as the [[cone space]] of an arbitrary topological space. One way of thinking of this cone space is as a literal cone that fills in between the space and a point. Up until the very tip of the cone, the cross-sections look homeomorphic to the topological space.

=== Examples from geometry ===

A [[topologically star-like space]] is a classic example of a contractible space. A topological space is termed topologically star-like if it is homeomorphic to a star-like subset of Euclidean space. A star-like subset of Euclidean space is a subset for which there exists a point in it such that for every other point in it, the line segment joining the points is completely inside the space.

A topologically star-like space is contractible, and can in fact be contracted to any point relative to which it is a star through a straight-line homotopy, i.e., moving each point toward the center in a straight line. The contracting homotopy fixes the center, and therefore, the space is in fact a [[SDR-contractible space]].

Note that, if also compact, a topologically star-like space is homeomorphic to the cone space of its boundary. Otherwise, the space is still ''almost'' a cone space: it is a subspace of the cone space that contains the full complement of the base and an arbitrary subset of the base. Nonetheless, it is important to note that the condition of being star-like also carries various geometric implications (in particular, from being a [[sub-Euclidean space]]) that are not satisfied for arbitrary cone spaces.

A [[topologically convex space]] is a (nonempty) space that is homeomorphic to a [[convex subset of Euclidean space]]. Any topologically convex space is topologically star-like, and ''any'' point can be taken as the center. An example of a topologically star-like space that is not topologically convex is a [[pair of intersecting lines]].

It is possible to construct spaces that are not topologically star-like, but still contractible. For instance, any geometric realization of a tree is contractible, but if the tree has more than one point with degree greater than two, it is not topologically star-like. As a related example, a set of parallel lines combined with one line that intersects all of them form a contractible space that is not topologically star-like.

==Relation with other properties==

=== Incomparable properties ===

Contractibility is incomparable with most of the interesting separation and compactness properties.

'''Broad argument for why contractibility cannot imply any meaningful separation or compactness property''': The [[cone space]] over ''any'' topological space is contractible. In particular, since any topological space arises as a closed subspace of its cone space (namely, the "base" of the space), every topological space arises as a closed subspace of a contractible space. Therefore, contractible cannot imply any nontrivial property that is [[subspace-hereditary property of topological spaces|subspace-hereditary]] or even [[weakly hereditary property of topological spaces|weakly hereditary]] (inherited by closed subsets).

'''Broad argument for why meaningful separation or compactness properties cannot imply contractibility''': Most meaningful separation and compactness properties are satisfied by all [[compact manifold]]s. However, compact manifolds of dimension greater than one are not contractible. The simplest counterexample is generally the [[circle]].

Some incomparable properties:

* [[T0 space]]
* [[T1 space]]
* [[Hausdorff space]]
* [[regular space]]
* [[normal space]]
* [[metrizable space]]
* [[paracompact space]]
* [[compact space]]

The property of being a contractible space is also incomparable with the property of being a [[locally contractible space]]. A contractible space need not be locally contractible. In fact, it need not even be locally connected! An example of a contractible space that is not locally contractible is the [[comb space]]. An example of a space that is locally contractible but not contractible is the [[circle]] (or, more generally, any compact manifold).

===Stronger properties===

Note that we may need to assume nonemptiness on top of the provided definitions in some case.

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[cone space]] over some topological space || || [[cone space implies contractible]] || ||
|-
| [[Weaker than::topologically star-like space]] || || || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Weaker than::topologically convex space]] || homeomorphic to a [[convex subset of Euclidean space]] || [[convex implies star-like|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Weaker than::suddenly contractible space]] || has a [[contracting homotopy]] that is also a [[sudden homotopy]] || || ||{{intermediate notions short|contractible space|suddenly contractible space}}
|-
| [[Weaker than::SDR-contractible space]] || has a [[contracting homotopy]] that is also a [[deformation retraction]] || || || {{intermediate notions short|contractible space|SDR-contractible space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || [[contractible implies weakly contractible]] || [[weakly contractible not implies contractible]] || {{intermediate notions short|weakly contractible space|contractible space}}
|-
| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || the <math>n</math>-sphere <math>S^n</math> is <math>(n-1)</math>-connected but not <math>n</math>-connected. || {{intermediate notions short|multiply connected space|contractible space}}
|-
| [[Stronger than::simply connected space]] || [[path-connected space]], [[fundamental group]] is [[trivial group|trivial]] || || || {{intermediate notions short|simply connected space|contractible space}}
|-
| [[Stronger than::path-connected space]] || there is a [[path]] between any two points || || || {{intermediate notions short|path-connected space|contractible space}}
|-
| [[Stronger than::connected space]] || cannot be partitioned into disjoint nonempty subsets || || || {{intermediate notions short|connected space|contractible space}}
|-
| [[Stronger than::acyclic space]] || homology groups over <math>\mathbb{Z}</math> all zero except zeroth homology group which is <math>\mathbb{Z}</math> || || || {{intermediate notions short|acyclic space|contractible space}}
|-
| [[Stronger than::rationally acyclic space]] || homology groups over <matH>\mathbb{Q}</math> all zero except zeroth homology group which is <math>\mathbb{Q}</math> || || || {{intermediate notions short|rationally acyclic space|contractible space}}
|-
| [[Stronger than::space with Euler characteristic one]] || [[Euler characteristic]] of the space is one. || ([[acyclic implies Euler charcateristic one|via acyclic]])|| [[Euler characteristic one not implies acyclic]] || {{intermediate notions short|space with Euler characteristic one|contractible space}}
|}

=== Conjunction with other properties ===

* [[Contractible manifold]]: Contractible as well as a [[manifold]]
* [[Contractible polyhedron]]: Contractible as well as a [[polyhedron]], i.e., the geometric realization of a [[simplicial complex]]

==References==

===Textbook references===

* {{booklink|Munkres}}, Page 330, Exercise 3 (definition introduced in exercise)
* {{booklink|SingerThorpe}}, Page 51 (formal definition)
* {{booklink|Rotman}}, Page 18 (formal definition)
* {{booklink|Hatcher}}, Page 4 (formal definition)
* {{booklink|Spanier}}, Page 25 (definition in paragraph)

Nonempty topologically convex implies equiconnected

2023-10-26T22:15:51Z

Vipul: Created page with "{{topospace property implication|stronger = topologically convex space|weaker = equiconnected space}} ==Statement== ===Topological version=== Any nonempty topologically c..."

Equiconnected space

2023-10-26T22:07:44Z

Vipul: /* Relation with other properties */

{{topospace property}}

==Definition==

A nonempty [[topological space]] <math>X</math> is said to be '''equiconnected''' if there is a continuous map <math>k:X \times [0,1] \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x,y \in X</math>.

Roughly, speaking, at any given time <math>t</math>, we get a map <math>X \times X</math> to <math>X</math>. At time <math>0</math>, it is projection on the first coordinate, and at time 1, it is projection on the second coordinate. For elements on the diagonal, it always remains the value at the diagonal.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::topologically convex space]] || nonempty space that is homeomorphic to a [[convex subset of Euclidean space]] || [[topologically convex implies equiconnected]] || || {{intermediate notions short|equiconnected space|topologically convex space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[equiconnected implies contractible]] || [[contractible not implies equiconnected]] || {{intermediate notions short|contractible space|equiconnected space}}
|}

==References==

* {{mathoverflow|number = 457103|title = Spaces that are contractible mod diagonal}}: This describes the property and asks for its name

Contractible space

2023-10-26T22:06:21Z

Vipul: /* Examples from geometry */

{{homotopy-invariant topospace property}}

==Definition==

=== Equivalent definitions in tabular format ===

A nonempty [[topological space]] is said to be '''contractible''' if it satisfies the following equivalent conditions. The [[empty space]] is generally excluded from consideration when considering the question of contractibility.

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed contractible if ... !! A topological space <math>X</math> is termed contractible if ...
|-
| 1 || homotopy-equivalent to a point || there is a [[homotopy equivalence of topological spaces]] between the topological space and a [[one-point space]]. || There exist [[continuous map]]s <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math> such that <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 2 || homotopy-equivalent to a point (arbitrary map) || ''any'' pair of maps between the space and a one-point space define a homotopy equivalence of topological spaces. || For ''any'' continuous map <math>f: X \to \{ * \}</math> and <math>g: \{ * \} \to X</math>, <math>f \circ g</math> is homotopic to the identity map on <math>\{ * \}</math> and <math>g \circ f</math> is homotopic to the identity map on <math>X</math>. Here, <math>\{ * \}</math> is a one-point space. Also, note that the condition on <math>f \circ g</math> is tautological, and the map <math>f</math> is already uniquely determined, so all the action occurs in the definition of <math>g</math> and the homotopy between <math>g \circ f</math> and the identity map on <math>X</math>.
|-
| 3 || admits a contracting homotopy || there is a point in the space to which there is a [[contracting homotopy]]. || there exists a point <math>x_0 \in X</math> and a contracting homotopy that contracts <math>X</math> to <math>x_0</math>. Explicitly, there exists a continuous map <math>F: X \times [0,1] \to X</math> such that <math>F(x,0) = x</math> for all <math>x</math>, and <math>F(x,1) = x_0</math> for all <math>x</math>. Note that we do ''not'' assume or require that <math>F(x_0,a) = x_0</math> for all <math>a \in [0,1]</math>.
|-
| 4 || admits a contracting homotopy (arbitrary point) || the space admits a [[contracting homotopy]] to ''any'' point in it. || for ''any'' point <math>x_1 \in X</math>, there is a contracting homotopy that contracts <math>X</math> to <math>x_1</math>.
|-
| 5 || unique homotopy class of maps to it || for any other topological space, there is a unique homotopy class of maps from the other space to it. || for any topological space <math>Y</math>, and any two continuous maps <math>h_1, h_2: Y \to X</math>, <math>h_1</math> and <math>h_2</math> are homotopic. In particular, any map from a topological space to it is nullhomotopic.
|-
| 6 || retract of cone space || it is a [[retract]] of its [[cone space]]. || the inclusion <math>\iota: X \to CX</math> in the cone space <math>CX</math> (as the base) has a one-sided inverse, <math>j: CX \to X</math>, i.e., <math>j</math> is a continuous map such that <math>j \circ \iota</math> is the identity map on <math>X</math>.
|}

==Metaproperties==

{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[contractibility is product-closed]] || If <math>X_i, i \in I</math> form a (finite or infinite) collection of contractible spaces, then the product of the <math>X_i</math>s, equipped with the [[product topology]], is also contractible. In particular, if <math>X_1, X_2</math> are contractible, then <math>X_1 \times X_2</math> is also contractible.
|-
| [[satisfies metaproperty::retract-hereditary property of topological spaces]] || Yes || [[contractibility is retract-hereditary]] || If <math>A \subseteq X</math> and <math>f:X \to A</math> is a continuous [[retraction]], and <math>X</math> is contractible, then <math>A</math> is contractible.
|-
| [[satisfies metaproperty::suspension-closed property of topological spaces]] || Yes || [[contractibility is suspension-closed]] || The [[suspension]] of a contractible space is contractible.
|-
| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[contractibility is not closure-preserved]] || It is possible to have a topological space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is contractible in the [[subspace topology]], but the [[closure]] <math>\overline{A}</math> is not.
|-
| [[dissatisfies metaproperty::connected union-closed property of topological spaces]] || No || [[contractibility is not connected union-closed]] || It is possible to have a topological space <math>X</math> expressible as a union of subsets <math>A,B</math>, both contractible in their subspace topology, with <math>A \cap B</math> nonempty, but <math>X</math> itself not contractible.
|}

== Examples ==

=== Extreme and basic examples ===

* The [[one-point space]] is contractible.
* Any [[Euclidean space]] is contractible.
* The closed unit disk in any dimension is contractible.
* [[Compact manifold]]s in dimension one or more, such as the [[circle]], are not contractible.

=== Intuition behind examples ===

Contractibility is, fundamentally, a ''global'' property of topological spaces. It is not possible to examine a small part of the space and conclude that it is contractible, nor does examining a small part of the space allow us to rule ''out'' the possibiilty that it is contractible. For the intuition behind the former, note that we can attach non-contractible pieces (like [[circle]]s) far off from the part of the space we are looking at. For the intuition behind the latter claim, note that we can embed ''any'' topological space as a closed subspace of a [[contractible space]], namely, its [[cone space]].

For this reason, when looking for examples or counterexamples, we need to focus on the global structure.

=== Examples from topological construction ===

One thing to keep in mind is that since the definition of contractibility invokes the closed unit interval, it is likely that any effort to construct contractible spaces will invariably involve dealing with the real numbers. The most topologically general way of constructing a contractible space is as the [[cone space]] of an arbitrary topological space. One way of thinking of this cone space is as a literal cone that fills in between the space and a point. Up until the very tip of the cone, the cross-sections look homeomorphic to the topological space.

=== Examples from geometry ===

A [[topologically star-like space]] is a classic example of a contractible space. A topological space is termed topologically star-like if it is homeomorphic to a star-like subset of Euclidean space. A star-like subset of Euclidean space is a subset for which there exists a point in it such that for every other point in it, the line segment joining the points is completely inside the space.

A topologically star-like space is contractible, and can in fact be contracted to any point relative to which it is a star through a straight-line homotopy, i.e., moving each point toward the center in a straight line. The contracting homotopy fixes the center, and therefore, the space is in fact a [[SDR-contractible space]].

Note that, if also compact, a topologically star-like space is homeomorphic to the cone space of its boundary. Otherwise, the space is still ''almost'' a cone space: it is a subspace of the cone space that contains the full complement of the base and an arbitrary subset of the base. Nonetheless, it is important to note that the condition of being star-like also carries various geometric implications (in particular, from being a [[sub-Euclidean space]]) that are not satisfied for arbitrary cone spaces.

A [[topologically convex space]] is a (nonempty) space that is homeomorphic to a [[convex subset of Euclidean space]]. Any topologically convex space is topologically star-like, and ''any'' point can be taken as the center. An example of a topologically star-like space that is not topologically convex is a [[pair of intersecting lines]].

It is possible to construct spaces that are not topologically star-like, but still contractible. For instance, any geometric realization of a tree is contractible, but if the tree has more than one point with degree greater than two, it is not topologically star-like. As a related example, a set of parallel lines combined with one line that intersects all of them form a contractible space that is not topologically star-like.

==Relation with other properties==

=== Incomparable properties ===

Contractibility is incomparable with most of the interesting separation and compactness properties.

'''Broad argument for why contractibility cannot imply any meaningful separation or compactness property''': The [[cone space]] over ''any'' topological space is contractible. In particular, since any topological space arises as a closed subspace of its cone space (namely, the "base" of the space), every topological space arises as a closed subspace of a contractible space. Therefore, contractible cannot imply any nontrivial property that is [[subspace-hereditary property of topological spaces|subspace-hereditary]] or even [[weakly hereditary property of topological spaces|weakly hereditary]] (inherited by closed subsets).

'''Broad argument for why meaningful separation or compactness properties cannot imply contractibility''': Most meaningful separation and compactness properties are satisfied by all [[compact manifold]]s. However, compact manifolds of dimension greater than one are not contractible. The simplest counterexample is generally the [[circle]].

Some incomparable properties:

* [[T0 space]]
* [[T1 space]]
* [[Hausdorff space]]
* [[regular space]]
* [[normal space]]
* [[metrizable space]]
* [[paracompact space]]
* [[compact space]]

The property of being a contractible space is also incomparable with the property of being a [[locally contractible space]]. A contractible space need not be locally contractible. In fact, it need not even be locally connected! An example of a contractible space that is not locally contractible is the [[comb space]]. An example of a space that is locally contractible but not contractible is the [[circle]] (or, more generally, any compact manifold).

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[cone space]] over some topological space || || [[cone space implies contractible]] || ||
|-
| [[Weaker than::topologically star-like space]] || || || || {{intermediate notions short|contractible space|topologically star-like space}}
|-
| [[Weaker than::topologically convex space]] || homeomorphic to a [[convex subset of Euclidean space]] || [[convex implies star-like|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}}
|-
| [[Weaker than::suddenly contractible space]] || has a [[contracting homotopy]] that is also a [[sudden homotopy]] || || ||{{intermediate notions short|contractible space|suddenly contractible space}}
|-
| [[Weaker than::SDR-contractible space]] || has a [[contracting homotopy]] that is also a [[deformation retraction]] || || || {{intermediate notions short|contractible space|SDR-contractible space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || [[contractible implies weakly contractible]] || [[weakly contractible not implies contractible]] || {{intermediate notions short|weakly contractible space|contractible space}}
|-
| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || the <math>n</math>-sphere <math>S^n</math> is <math>(n-1)</math>-connected but not <math>n</math>-connected. || {{intermediate notions short|multiply connected space|contractible space}}
|-
| [[Stronger than::simply connected space]] || [[path-connected space]], [[fundamental group]] is [[trivial group|trivial]] || || || {{intermediate notions short|simply connected space|contractible space}}
|-
| [[Stronger than::path-connected space]] || there is a [[path]] between any two points || || || {{intermediate notions short|path-connected space|contractible space}}
|-
| [[Stronger than::connected space]] || cannot be partitioned into disjoint nonempty subsets || || || {{intermediate notions short|connected space|contractible space}}
|-
| [[Stronger than::acyclic space]] || homology groups over <math>\mathbb{Z}</math> all zero except zeroth homology group which is <math>\mathbb{Z}</math> || || || {{intermediate notions short|acyclic space|contractible space}}
|-
| [[Stronger than::rationally acyclic space]] || homology groups over <matH>\mathbb{Q}</math> all zero except zeroth homology group which is <math>\mathbb{Q}</math> || || || {{intermediate notions short|rationally acyclic space|contractible space}}
|-
| [[Stronger than::space with Euler characteristic one]] || [[Euler characteristic]] of the space is one. || ([[acyclic implies Euler charcateristic one|via acyclic]])|| [[Euler characteristic one not implies acyclic]] || {{intermediate notions short|space with Euler characteristic one|contractible space}}
|}

=== Conjunction with other properties ===

* [[Contractible manifold]]: Contractible as well as a [[manifold]]
* [[Contractible polyhedron]]: Contractible as well as a [[polyhedron]], i.e., the geometric realization of a [[simplicial complex]]

==References==

===Textbook references===

* {{booklink|Munkres}}, Page 330, Exercise 3 (definition introduced in exercise)
* {{booklink|SingerThorpe}}, Page 51 (formal definition)
* {{booklink|Rotman}}, Page 18 (formal definition)
* {{booklink|Hatcher}}, Page 4 (formal definition)
* {{booklink|Spanier}}, Page 25 (definition in paragraph)