Hereditarily normal space - Revision history

Vipul at 21:54, 27 January 2012

2012-01-27T21:54:00Z

← Older revision		Revision as of 21:54, 27 January 2012
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	~~{{topospace property}}~~

	~~{{variationof\|normality}}~~

	~~{{T family\|T5}}~~

	==Definition==		==Definition==

	===Symbol-free definition===		===Symbol-free definition===

	A [[topological space]] is said to be '''hereditarily normal''' or '''completely normal''' (sometimes also '''totally normal''') if it ~~is [[T1 space\|T1]] and~~ satisfies the following equivalent conditions:		A [[topological space]] is said to be '''hereditarily normal''' or '''completely normal''' (sometimes also '''totally normal''') if it satisfies the following equivalent conditions:

	* Every subspace of it is [[normal space\|normal]] under the [[subspace topology]]		* Every subspace of it is [[normal space\|normal]] under the [[subspace topology]]
	* Given two [[separated subsets]] of the topological space (viz two subsets such that neither intersects the closure of the other), there exist disjoint open sets containing them		* Given two [[separated subsets]] of the topological space (viz two subsets such that neither intersects the closure of the other), there exist disjoint open sets containing them

			{{topospace property}}

			{{T family\|T5}}

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Relation with other properties */

2009-01-17T01:25:31Z

Relation with other properties

← Older revision		Revision as of 01:25, 17 January 2009
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	===Stronger properties===		===Stronger properties===

	* [[Perfectly normal space]]		* [[Weaker than::Perfectly normal space]]
	* [[Monotonically normal space]]		* [[Weaker than::Monotonically normal space]]
	* [[Hereditarily collectionwise normal space]]		* [[Weaker than::Hereditarily collectionwise normal space]]
	* [[Elastic space]]		* [[Weaker than::Elastic space]]
	* [[Metrizable space]]		* [[Weaker than::Metrizable space]]
	* [[Linearly orderable space]]		* [[Weaker than::Linearly orderable space]]
	* [[CW-space]]		* [[Weaker than::CW-space]]

	===Weaker properties===		===Weaker properties===

	* [[Normal space]]		* [[Stronger than::Normal space]]

	==Metaproperties==		==Metaproperties==

Vipul: 5 revisions

2008-05-11T19:45:46Z

5 revisions

← Older revision	Revision as of 19:45, 11 May 2008
(No difference)

Vipul at 21:15, 21 April 2008

2008-04-21T21:15:18Z

← Older revision		Revision as of 21:15, 21 April 2008
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	By the first definition, it is clear that any subspace of a hereditarily normal space is hereditarily normal.		By the first definition, it is clear that any subspace of a hereditarily normal space is hereditarily normal.

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 205, Exercise 6 (definition introduced in exercise): Introduced using term '''completely normal space'''

Vipul: /* Stronger properties */

2007-12-18T20:25:06Z

Stronger properties

← Older revision		Revision as of 20:25, 18 December 2007
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	* [[Perfectly normal space]]		* [[Perfectly normal space]]
			* [[Monotonically normal space]]
			* [[Hereditarily collectionwise normal space]]
			* [[Elastic space]]
	* [[Metrizable space]]		* [[Metrizable space]]
			* [[Linearly orderable space]]
			* [[CW-space]]

	===Weaker properties===		===Weaker properties===

Vipul: /* Weaker properties= */

2007-12-17T20:49:38Z

Weaker properties=

← Older revision		Revision as of 20:49, 17 December 2007
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	* [[Metrizable space]]		* [[Metrizable space]]

	==Weaker properties===		===Weaker properties===

	* [[Normal space]]		* [[Normal space]]

Vipul: /* Symbol-free definition */

2007-12-17T20:49:26Z

Symbol-free definition

← Older revision		Revision as of 20:49, 17 December 2007
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	===Symbol-free definition===		===Symbol-free definition===

	A [[topological space]] is said to be '''hereditarily normal''' if it is [[T1 space\|T1]] and satisfies the following equivalent conditions:		A [[topological space]] is said to be '''hereditarily normal''' or '''completely normal''' (sometimes also '''totally normal''') if it is [[T1 space\|T1]] and satisfies the following equivalent conditions:

	* Every subspace of it is [[normal space\|normal]] under the [[subspace topology]]		* Every subspace of it is [[normal space\|normal]] under the [[subspace topology]]

Vipul at 09:11, 20 August 2007

2007-08-20T09:11:35Z

New page

{{topospace property}}

{{variationof|normality}}

{{T family|T5}}

==Definition==

===Symbol-free definition===

A [[topological space]] is said to be '''hereditarily normal''' if it is [[T1 space|T1]] and satisfies the following equivalent conditions:

* Every subspace of it is [[normal space|normal]] under the [[subspace topology]]
* Given two [[separated subsets]] of the topological space (viz two subsets such that neither intersects the closure of the other), there exist disjoint open sets containing them

==Relation with other properties==

===Stronger properties===

* [[Perfectly normal space]]
* [[Metrizable space]]

==Weaker properties===

* [[Normal space]]

==Metaproperties==

{{subspace-closed}}

By the first definition, it is clear that any subspace of a hereditarily normal space is hereditarily normal.