Locally connected space: Difference between revisions

Revision as of 00:42, 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.

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

@@ Line 3: / Line 3: @@
 ==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>V \subseteq U</math>, and <math>V</math> is a [[connected space]] with the subspace topology.
+===Equivalent definitions in tabular format===
+{| class="sortable" border="1"
+! No. !! Shorthand !! A topological space <math>X</math> is termed locally connected if ...
+|-
+| 1 || [[locally connected space at a point|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 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.
+|-
+| 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.
+|}
 ==Relation with other properties==