Contractible space: Difference between revisions

Revision as of 18:56, 7 July 2011

This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces

View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces

Definition

Symbol-free definition

A topological space is said to be contractible if it satisfies the following equivalent conditions:

It is in the same homotopy class as a point
The identity map from the space to itself, is homotopic to a constant map, from the space to a particular point in that space
There is a single point which is a homotopy retract
It has a contracting homotopy

Definition with symbols

A topological space $X$ is said to be contractible if it has a contracting homotopy, viz a continuous map $f:X\times [0,1]\to X$ and a point $x_{0}\in X$ such that $f(x,0)=x$ and $f(x,1)=x_{0}$ for all $x$ .

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
product-closed property of topological spaces	Yes	contractibility is product-closed	If $X_{i},i\in I$ form a (finite or infinite) collection of contractible spaces, then the product of the $X_{i}$ s, equipped with the product topology, is also contractible. In particular, for contractible spaces $X_{1},X_{2}$ , $X_{1}\times X_{2}$ is contractible.
retract-hereditary property of topological spaces	Yes	contractibility is retract-hereditary	If $A\subseteq X$ and $f:X\to A$ is a continuous retraction, and $X$ is contractible, then $A$ is contractible.
closure-preserved property of topological spaces	No	contractibility is not closure-preserved	It is possible to have a topological space $X$ and a subset $A$ of $X$ such that $A$ is contractible in the subspace topology, but the closure ${\overline {A}}$ is not.
interior-preserved property of topological spaces	No	contractibility is not interior-preserved	It is possible to have a topological space $X$ and a subset $A$ of $X$ such that $A$ is contractible in the subspace topology, but the interior of $A$ is not.
intersection-closed property of topological spaces	No	contractibility is not intersection-closed	It is possible to have a topological space $X$ and subsets $A,B$ of $X$ such that $A,B$ are both contractible in their respective subspace topology but $A\cap B$ is not.
connected union-closed property of topological spaces	No	contractibility is not connected union-closed	It is possible to have a topological space $X$ expressible as a union of subsets $A,B$ , both contractible in their subspace topology, with $A\cap B</mathnonempty,but<math>X$ itself not contractible.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Intermediate notions
cone space over some topological space		cone space implies contractible
topologically star-like space			\|FULL LIST, MORE INFO
topologically convex space	homeomorphic to a convex subset of Euclidean space	via star-like	Equiconnected space, Space in which every retraction is a deformation retraction, Topologically star-like space\|FULL LIST, MORE INFO
suddenly contractible space	has a contracting homotopy that is also a sudden homotopy		\|FULL LIST, MORE INFO
SDR-contractible space	has a contracting homotopy that is also a deformation retraction		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
weakly contractible space	path-connected space, all homotopy groups vanish	contractible implies weakly contractible	weakly contractible not implies contractible	\|FULL LIST, MORE INFO
multiply connected space	path-connected space, first $k$ homotopy groups vanish for $k\geq 2$		the $n$ -sphere $S^{n}$ is $(n-1)$ -connected but not $n$ -connected.	\|FULL LIST, MORE INFO
simply connected space	path-connected space, fundamental group is trivial			Weakly contractible space\|FULL LIST, MORE INFO
path-connected space	there is a path between any two points			Weakly contractible space\|FULL LIST, MORE INFO
connected space	cannot be partitioned into disjoint nonempty subsets			\|FULL LIST, MORE INFO
acyclic space	homology groups over $\mathbb {Z}$ all zero except zeroth homology group which is $\mathbb {Z}$			Weakly contractible space\|FULL LIST, MORE INFO
rationally acyclic space	homology groups over $\mathbb {Q}$ all zero except zeroth homology group which is $\mathbb {Q}$			Acyclic space, Weakly contractible space\|FULL LIST, MORE INFO
space with Euler characteristic one	Euler characteristic of the space is one.	(via acyclic)	Euler characteristic one not implies acyclic	Acyclic space\|FULL LIST, MORE INFO

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 330, Exercise 3 (definition introduced in exercise)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 51 (formal definition)
An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman^{More info}, Page 18 (formal definition)
Algebraic Topology by Allen Hatcher^{Full text PDF}^{More info}, Page 4 (formal definition)
Algebraic Topology by Edwin H. Spanier^{More info}, Page 25 (definition in paragraph)

@@ Line 15: / Line 15: @@
 A [[topological space]] <math>X</math> is said to be '''contractible''' if it has a [[contracting homotopy]], viz a continuous map <math>f: X \times [0,1] \to X</math> and a point <math>x_0 \in X</math> such that <math>f(x,0) = x</math> and <math>f(x,1) = x_0</math> for all <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.<br>In particular, for contractible spaces <math>X_1, X_2</math>, <math>X_1 \times X_2</math> is 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.
+|-
+| [[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::interior-preserved property of topological spaces]] || No || [[contractibility is not interior-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 [[interior]] of <math>A</math> is not.
+|-
+| [[dissatisfies metaproperty::intersection-closed property of topological spaces]] || No || [[contractibility is not intersection-closed]] || It is possible to have a topological space <math>X</math> and subsets <math>A,B</math> of <math>X</math> such that <math>A,B</math> are both contractible in their respective [[subspace topology]] but <math>A \cap B</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.
+|}
 ==Relation with other properties==
@@ Line 48: / Line 66: @@
 |-
 | [[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}}
 |}
-==Metaproperties==
-{{DP-closed}}
-A direct product of contractible spaces is contractible. {{proofat|[[Contractibility is product-closed]]}}
-{{retract-hereditary}}
-Any [[retract]] of a contractible space is contractible. {{proofat|[[Contractibility is retract-hereditary]]}}
-==Subset-related properties==
-===Closure===
-The closure of a contractible subset need not be contractible; in fact it need not even be path-connected. An example is the [[topologist's sine curve]], which is contractible as it is the image of a path; however, it's closure is not even path-connected.
-===Interior===
-The interior of a contractible subset need not be contractible; in fact, it need not even be connected. An example is a wedge of two discs, whose interior is the disjoint union of their interiors.
-===Intersection===
-An intersection of contractible subsets need not be contractible. There is, however, a relation between the homology of the intersection and the homology of the union, when the contractible subsets are open (or more generally, when they are strong deformation retracts of neighbourhoods).
-An example is the intersection of two semicircular paths on the circle, which is a pair of points.
-===Connected union===
-A connected union of contractible subsets need not be contractible.
 ==References==