Locally connected space: Difference between revisions

Revision as of 00:44, 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

Equivalent definitions in tabular format

No.	Shorthand	A topological space $X$ is termed locally connected if ...
1	locally connected at every point	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.
2	weakly locally connected at every point	for every point $x \in X$ , and every open subset $U$ of $X$ containing $x$ , there exists a subset $A$ of $X$ such that $x$ is in the interon of $A$ , $A \subseteq U$ , and $A$ is a connected space with the subspace topology.
3	basis of open connected subsets	$X$ has a basis (of open subsets) such that all members of the basis are connected in 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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 | 2 || [[weakly locally connected space at a point|weakly locally connected at]] every point || 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 a subset <math>A</math> of <math>X</math> such that <math>x</math> is in the interon of <math>A</math>, <math>A \subseteq U</math>, and <math>A</math> is a [[connected space]] with the subspace topology.
+|-
+| 3 || basis of open connected subsets || <math>X</math> has a [[basis]] (of open subsets) such that all members of the basis are [[connected space|connected]] in the [[subspace topology]].
 |}