Locally contractible space: Difference between revisions

Revision as of 01:03, 28 January 2012

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 contractibility. View other variations of contractibility

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 $V$ .

Relation with other properties

Stronger properties

Weaker properties

@@ Line 7: / Line 7: @@
 ===Symbol-free definition===
-A [[topological space]] is said to be ''locally contractible'' if given any point and any open neighbourhood of it, there exists a smaller open neighbourhood containing the point, which is contractible.
+A [[topological space]] <math>X</math> is said to be ''locally contractible'' if it satisfies the following equivalent conditions:
-===Definition with symbols===
+# 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>.
-A [[topological space]] <math>X</math> is said to be ''locally contractible'' if given any <math>x \in U \subset X</math> with <math>U</math> open, there exists a <math>V \ni x</math>, contained in <math>U</math>, such that <math>V</math> is contractible (note <math>V</math> is given the [[subspace topology]]).
 ==Relation with other properties==