Normality is weakly hereditary - Revision history

Vipul at 22:37, 24 January 2012

2012-01-24T22:37:29Z

← Older revision		Revision as of 22:37, 24 January 2012
Line 1:		Line 1:
	{{topospace metaproperty satisfaction}}		{{topospace metaproperty satisfaction\|
			property = normal space\|
	~~{{basic fact~~}}		metaproperty = weakly hereditary property of topological spaces}}

	==Statement==		==Statement==

Vipul: /* References */

2008-07-20T17:52:55Z

References

← Older revision		Revision as of 17:52, 20 July 2008
Line 35:		Line 35:

	===Textbook references===		===Textbook references===
	* {{booklink\|Munkres}}, Page 205 (Exercise 1)		* {{booklink-stated\|Munkres}}, Page 205, Exercise 1, Chapter 4, Section 32

Vipul: 2 revisions

2008-05-11T19:54:35Z

2 revisions

← Older revision	Revision as of 19:54, 11 May 2008
(No difference)

Vipul at 20:25, 21 April 2008

2008-04-21T20:25:06Z

← Older revision		Revision as of 20:25, 21 April 2008
Line 31:		Line 31:
	* Separate them by disjoint open sets in the whole space		* Separate them by disjoint open sets in the whole space
	* Intersect these with the subspace, and use the definition of [[subspace topology]] to conclude that we have a separation by disjoint open sets in the subspace		* Intersect these with the subspace, and use the definition of [[subspace topology]] to conclude that we have a separation by disjoint open sets in the subspace

			==References==

			===Textbook references===
			* {{booklink\|Munkres}}, Page 205 (Exercise 1)

Vipul at 23:44, 26 December 2007

2007-12-26T23:44:57Z

New page

{{topospace metaproperty satisfaction}}

{{basic fact}}

==Statement==

===Verbal statement===

Any [[closed subset]] of a [[normal space]] is also normal, in the [[subspace topology]].

==Definitions used==

===Normal space===

{{further|[[normal space]]}}

===Subspace topology===

{{further|[[subspace topology]]}}

==Proof==

===Proof outline===

Note that the property of being a [[T1 space]] is certainly hereditary to all subspaces, so we only need to check the separation of closed subsets.

We proceed as follows:

* Pick two [[closed subset]]s inside the subspace
* Observe, using the fact that [[closedness is transitive|a closed subspace of a closed subspace is closed]], that both of them are closed in the whole space
* Separate them by disjoint open sets in the whole space
* Intersect these with the subspace, and use the definition of [[subspace topology]] to conclude that we have a separation by disjoint open sets in the subspace