Normal space - Revision history

DavidB4-bot: /* top */clean up & uniformity

2016-07-13T16:46:35Z

top: clean up & uniformity

2009-06-28T23:36:31Z

2009-04-23T21:45:31Z

2007-04-07T02:17:35Z

2007-04-07T02:17:22Z

2007-04-06T18:31:27Z

2007-04-06T18:29:47Z

2007-04-01T16:54:45Z

2007-04-01T16:54:38Z

2007-03-22T22:08:18Z

← Older revision		Revision as of 16:46, July 13, 2016
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	By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be separated by a [[continuous function]]. The converse also hold.		By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be separated by a [[continuous function]]. The converse also hold.

−	[[~~category~~: Topology]]	+	[[Category:Topology]]

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 A '''Normal space''' is a [[Hausdorff space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the [[Sorgenfrey plane]] is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.  Every subspace of a normal space is a [[completely regular space]].  A normal space which is also T<sub>1</sub> is called T<sub>4</sub>.
-By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]].  The converse also hold.
+By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be separated by a [[continuous function]].  The converse also hold.
 [[category: Topology]]

@@ Line 1: / Line 1: @@
-'''Normal space''' (or '''T<sub>4</sub> space''') is a [[Hausdorff space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the [[Sorgenfrey plane]] is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.  Every subspace of a normal space is a [[completely regular space]].
+A '''Normal space''' is a [[Hausdorff space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the [[Sorgenfrey plane]] is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.  Every subspace of a normal space is a [[completely regular space]].  A normal space which is also T<sub>1</sub> is called T<sub>4</sub>.
 By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]].  The converse also hold.
 [[category: Topology]]

@@ Line 1: / Line 1: @@
-'''Normal space''' (or '''T<sub>4</sub> space''') is a Hausdorff [[topological space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the [[Sorgenfrey plane]] is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.  Every subspace of a normal space is a [[completely regular space]].
+'''Normal space''' (or '''T<sub>4</sub> space''') is a [[Hausdorff space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the [[Sorgenfrey plane]] is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.  Every subspace of a normal space is a [[completely regular space]].
 By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]].  The converse also hold.
 [[category: Topology]]

@@ Line 1: / Line 1: @@
-Normal space is a Hausdorff [[topological space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the [[Sorgenfrey plane]] is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.  Every subspace of a normal space is a [[completely regular space]].
+'''Normal space''' (or '''T<sub>4</sub> space''') is a Hausdorff [[topological space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the [[Sorgenfrey plane]] is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.  Every subspace of a normal space is a [[completely regular space]].
 By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]].  The converse also hold.
 [[category: Topology]]

← Older revision		Revision as of 16:54, April 1, 2007
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	By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]]. The converse also hold.		By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]]. The converse also hold.

−	[[~~Category~~:~~Mathematics~~]]	+	[[category: Topology]]

@@ Line 1: / Line 1: @@
 Normal space is a Hausdorff [[topological space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.  A product of normal spaces is not necessarily normal, the Sorgenfrey plane is an example of a product of normal spaces that is not normal.  On the other hand, every regular space with a countable basis is normal.
-By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be seperated by a [[continous function]].  The converse also hold.
+By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]].  The converse also hold.
 [[Category:Mathematics]]