Semilocally simply connected space: Difference between revisions

Revision as of 21:35, 30 September 2007

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

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Definition

Symbol-free definition

A topological space is said to be semilocally simply connected if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.

Definition with symbols

A topological space $X$ is said to be semilocally simply connected if for any $x\in X$ there exists a neighbourhood $U$ of $x$ such that the homomorphism of fundamental groups induced by the inclusion of $U$ in $X$ , is trivial. In other words, every loop about $x$ contained in $U$ , is nullhomotopic in $X$ .

Relation with other properties

Stronger properties

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 A [[topological space]] is said to be '''semilocally simply connected''' if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.
+===Definition with symbols===
+A [[topological space]] <math>X</math> is said to be '''semilocally simply connected''' if for any <math>x \in X</math> there exists a neighbourhood <math>U</math> of <math>x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math>, is trivial. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>.
 ==Relation with other properties==