Locally connected space: Difference between revisions

Latest revision as of 00:51, 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 $V$ of $X$ containing $x$ , there exists an open subset $U$ of $X$ such that $x\in U$ , $U\subseteq V$ , and $U$ 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 $V$ of $X$ containing $x$ , there exists a subset $A$ of $X$ such that $x$ is in the interior of $A$ , $A\subseteq V$ , 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

Incomparable 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

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
locally path-connected space
locally simply connected space
locally contractible space
locally Euclidean space
manifold

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
space in which all connected components are open
space in which the connected components coincide with the quasicomponents

@@ 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>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U</math>, <math>U \subseteq V</math>, and <math>U</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>V</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 interior of <math>A</math>, <math>A \subseteq V</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]].
+|}
 ==Relation with other properties==
-===Related properties===
+===Incomparable properties===
 * [[Connected space]]: Being connected does not imply being locally connected, and being locally connected does not imply being connected. {{further|[[connected not implies locally connected]], [[locally connected not implies connected]]}}
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !!Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::locally path-connected space]] || || || ||
+|-
+| [[Weaker than::locally simply connected space]] || || || ||
+|-
+| [[Weaker than::locally contractible space]] || || || ||
+|-
+| [[Weaker than::locally Euclidean space]] || || || ||
+|-
+| [[Weaker than::manifold]] || || || ||
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !!Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::space in which all connected components are open]] || || || ||
+|-
+| [[Stronger than::space in which the connected components coincide with the quasicomponents]] || || || ||
+|}