Locally simply connected space: Difference between revisions

Latest revision as of 00:53, 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

Definition

A topological space $X$ is termed locally simply connected if it satisfies the following equivalent conditions:

For every point $x\in X$ , and every open subset $V$ of $X$ containing $x$ , there is an open subset $U$ of $X$ contained in $V$ , and which is simply connected in the subspace topology from $X$ .
$X$ has a basis of open subsets each of which is a simply connected space with the subspace topology.

Relation with other properties

Stronger properties

Locally contractible space

Weaker properties

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+A [[topological space]] <math>X</math> is termed '''locally simply connected''' if it satisfies the following equivalent conditions:
-A [[topological space]] is said to be '''locally simply connected''' if given any point and any open subset containing that point, there exists a smaller open set containing the point, which is [[simply connected space|simply connected]] in the [[subspace topology]].
+# For every point <math>x \in X</math>, and every open subset <math>V</math> of <math>X</math> containing <math>x</math>, there is an open subset <math>U</math> of <math>X</math> contained in <math>V</math>, and which is [[simply connected space|simply connected]] in the [[subspace topology]] from <math>X</math>.
+# <math>X</math> has a [[basis]] of [[open subset]]s each of which is a [[simply connected space]] with the [[subspace topology]].
 ==Relation with other properties==
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 ===Weaker properties===
+* [[Semilocally simply connected space]]
 * [[Locally path-connected space]]
 * [[Connected space]]