Locally contractible space: Difference between revisions

Latest revision as of 05:15, 31 May 2016

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of contractible space. View other variations of contractible space

Definition

Symbol-free definition

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

It has a basis of open subsets each of which is a contractible space under the subspace topology.
For every $x\in X$ and every open subset $V\ni x$ of $X$ , there exists an open subset $U\ni x$ such that $U\subseteq V$ and $U$ is a contractible space in the subspace topology from $X$ .

Formalisms

In terms of the locally operator

This property is obtained by applying the locally operator to the property: contractible space

Note that the locally operator here means the existence of a basis of contractible spaces. It is a stronger condition than merely saying that every point is contained in a contractible open subset; rather, we are claiming that there are arbitrarily small contractible open subsets. The mere condition that every point is contained in a contractible open subset is much weaker.

Relation with other properties

Incomparable properties

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 connected is the comb space. Conversely, a locally contractible space need not be contractible. For instance, any manifold is locally contractible, but manifolds such as the circle are not contractible.

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
locally Euclidean space	has a basis comprising subsets homeomorphic to Euclidean space	follows from Euclidean spaces being contractible	a pair of intersecting lines is locally contractible but not locally Euclidean	\|FULL LIST, MORE INFO
manifold	locally Euclidean of fixed dimension as well as Hausdorff and second-countable	(via locally Euclidean)	(via locally Euclidean)	Locally Euclidean space\|FULL LIST, MORE INFO
CW-space	underlying topological space (up to homeomorphism) of a CW-complex	CW implies locally contractible		\|FULL LIST, MORE INFO
polyhedron	underlying topological space (up to homeomorphism) of the geometric realization of a simplicial complex	(via CW-space)		CW-space\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Intermediate notions
locally simply connected space	has a basis comprising subsets that are simply connected	follows from contractible implies simply connected	\|FULL LIST, MORE INFO
semilocally weakly contractible space			\|FULL LIST, MORE INFO
semilocally simply connected space			\|FULL LIST, MORE INFO
locally path-connected space	has a basis comprising path-connected subsets		\|FULL LIST, MORE INFO
locally connected space	has a basis comprising connected subsets		\|FULL LIST, MORE INFO

@@ Line 10: / Line 10: @@
 # It has a [[basis]] of [[open subset]]s each of which is a [[contractible space]] under the [[subspace topology]].
-# For every <math>x \in X</math> and every [[open subset]] <math>V \ni x</math> of <math>X</math>, there exists an open subset <math>U \ni x </math> such that <math>U \subseteq V</math> and <math>U</math> is a [[contractible space]] in the [[subspace topology]] from <math>V</math>.
+# For every <math>x \in X</math> and every [[open subset]] <math>V \ni x</math> of <math>X</math>, there exists an open subset <math>U \ni x </math> such that <math>U \subseteq V</math> and <math>U</math> is a [[contractible space]] in the [[subspace topology]] from <math>X</math>.
+== Formalisms ==
+{{obtained by applying the|locally operator|contractible space}}
+Note that the locally operator here means the existence of a basis of contractible spaces. It is a stronger condition than merely saying that every point is contained in a contractible open subset; rather, we are claiming that there are arbitrarily small contractible open subsets. The mere condition that every point is contained in a contractible open subset is much weaker.
 ==Relation with other properties==
+=== Incomparable properties ===
+* [[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 connected is the [[comb space]]. Conversely, a locally contractible space need not be contractible. For instance, any manifold is locally contractible, but manifolds such as the [[circle]] are not contractible.
 ===Stronger properties===
-* [[Manifold]]
+{| class="sortable" border="1"
-* [[Locally Euclidean space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[CW-space]]
+|-
+| [[Weaker than::locally Euclidean space]] || has a basis comprising subsets homeomorphic to Euclidean space || follows from Euclidean spaces being contractible || a pair of intersecting lines is locally contractible but not locally Euclidean || {{intermediate notions short|locally contractible space|locally Euclidean space}}
+|-
+| [[Weaker than::manifold]] || locally Euclidean of fixed dimension as well as [[Hausdorff space|Hausdorff]] and [[second-countable space|second-countable]] || (via locally Euclidean) || (via locally Euclidean) || {{intermediate notions short|locally contractible space|manifold}}
+|-
+| [[Weaker than::CW-space]] || underlying topological space (up to homeomorphism) of a [[CW-complex]] || [[CW implies locally contractible]] || || {{intermediate notions short|locally contractible space|CW-space}}
+|-
+| [[Weaker than::polyhedron]] || underlying topological space (up to homeomorphism) of the geometric realization of a [[simplicial complex]] || (via CW-space) || || {{intermediate notions short|locally contractible space|polyhedron}}
+|}
 ===Weaker properties===
-* [[Locally simply connected space]]
+{| class="sortable" border="1"
-* [[Semilocally weakly contractible space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Semilocally simply connected space]]
+|-
-* [[Locally path-connected space]]
+| [[Stronger than::locally simply connected space]] || has a basis comprising subsets that are [[simply connected space|simply connected]] || follows from [[contractible implies simply connected]] || || {{intermediate notions short|locally simply connected space|locally contractible space}}
-* [[Locally connected space]]
+|-
+| [[Stronger than::semilocally weakly contractible space]] || || || || {{intermediate notions short|semilocally weakly contractible space|locally contractible space}}
+|-
+| [[Stronger than::semilocally simply connected space]] || || || || {{intermediate notions short|semilocally simply connected space|locally contractible space}}
+|-
+| [[Stronger than::locally path-connected space]] || has a basis comprising path-connected subsets || || || {{intermediate notions short|locally path-connected space|locally contractible space}}
+|-
+| [[Stronger than::locally connected space]] || has a basis comprising connected subsets || || || {{intermediate notions short|locally connected space|locally contractible space}}
+|}