Locally connected space - Revision history

Vipul: /* Definition */

2012-01-28T00:51:57Z

Definition

← Older revision		Revision as of 00:51, 28 January 2012
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	! No. !! Shorthand !! A topological space <math>X</math> is termed locally connected if ...		! No. !! Shorthand !! A topological space <math>X</math> is termed locally connected if ...
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	\| 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>V</math> is a [[connected space]] with the subspace topology.		\| 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.		\| 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.

Vipul: /* Definition */

2012-01-28T00:51:45Z

Definition

← Older revision		Revision as of 00:51, 28 January 2012
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	! No. !! Shorthand !! A topological space <math>X</math> is termed locally connected if ...		! 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.		\| 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>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.		\| 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]].		\| 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]].

Vipul: /* Relation with other properties */

2012-01-28T00:47:07Z

Relation with other properties

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 ==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]]}}

Vipul: /* Definition */

2012-01-28T00:44:58Z

Definition

← Older revision		Revision as of 00:44, 28 January 2012
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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.		\| 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]].
	\|}		\|}

Vipul at 00:42, 28 January 2012

2012-01-28T00:42:53Z

← Older revision		Revision as of 00:42, 28 January 2012
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	==Definition==		==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==		==Relation with other properties==

Vipul at 17:48, 27 January 2012

2012-01-27T17:48:54Z

← Older revision		Revision as of 17:48, 27 January 2012
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	==Definition==		==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==		==Relation with other properties==

Vipul: Created page with '{{topospace property}} ==Definition== A topological space $X$ is termed a '''locally connected space''' if, for every point $x \in X$ , and every open …'

2009-12-25T07:11:59Z

Created page with '{{topospace property}} ==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 …'

New page

{{topospace property}}

==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.

==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|[[connected not implies locally connected]], [[locally connected not implies connected]]}}

Locally connected space - Revision history

Vipul: /* Definition */

Vipul: /* Definition */

Vipul: /* Relation with other properties */

Vipul: /* Definition */

Vipul at 00:42, 28 January 2012

Vipul at 17:48, 27 January 2012

Vipul: Created page with '{{topospace property}} ==Definition== A topological space X is termed a '''locally connected space''' if, for every point x \in X, and every open …'

Vipul: Created page with '{{topospace property}} ==Definition== A topological space $X$ is termed a '''locally connected space''' if, for every point $x \in X$ , and every open …'