Locally connected space: Difference between revisions

Revision as of 17:48, 27 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 a locally connected space if, for every point $x\in X$ , and every open subset $U$ of $X$ containing $x$ , there exists an open subset $V$ of $X$ such that $x\in V$ , $V\subseteq U$ , and $V$ is a connected space with the subspace topology.

Relation with other properties

Related properties

Connected space: Being connected does not imply being locally connected, and being locally connected does not imply being connected. Further information: connected not implies locally connected, locally connected not implies connected

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 ==Definition==
-A [[topological space]] <math>X</math> is termed a '''locally connected space''' if, for every point <math>x \in X</math>, and every open subset <math>U</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>V</math> of <math>X</math> such that <math>x \in V</math>, <math>\overline{V} \subseteq U</math>, and <math>V</math> is a [[connected space]] with the subspace topology.
+A [[topological space]] <math>X</math> is termed a '''locally connected space''' if, for every point <math>x \in X</math>, and every open subset <math>U</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>V</math> of <math>X</math> such that <math>x \in V</math>, <math>V \subseteq U</math>, and <math>V</math> is a [[connected space]] with the subspace topology.
 ==Relation with other properties==