Elastic space - Revision history

Vipul: /* Relation with other properties */

2012-01-24T23:12:32Z

Relation with other properties

← Older revision		Revision as of 23:12, 24 January 2012
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	===Stronger properties===		===Stronger properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::Metrizable space]] \|\| underlying topology of a [[metric space]] \|\| [[metrizable implies elastic]] \|\| [[elastic not implies metrizable]] \|\| {{intermediate notions short\|elastic space\|metrizable space}}		\| [[Weaker than::Metrizable space]] \|\| underlying topology of a [[metric space]] \|\| [[metrizable implies elastic]] \|\| [[elastic not implies metrizable]] \|\| {{intermediate notions short\|elastic space\|metrizable space}}
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	===Weaker properties===		===Weaker properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::~~Paracompact~~ Hausdorff space]] \|\| [[paracompact space\|paracompact]] and [[Hausdorff space\|Hausdorff]] \|\| [[elastic implies paracompact Hausdorff]] \|\| [[paracompact Hausdorff not implies elastic]] \|\| {{intermediate notions short\|paracompact Hausdorff space\|elastic space}}		\| [[Stronger than::paracompact Hausdorff space]] \|\| [[paracompact space\|paracompact]] and [[Hausdorff space\|Hausdorff]] \|\| [[elastic implies paracompact Hausdorff]] \|\| [[paracompact Hausdorff not implies elastic]] \|\| {{intermediate notions short\|paracompact Hausdorff space\|elastic space}}
	\|-		\|-
	\| [[Stronger than::~~Monotonically~~ normal space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|monotonically normal space\|elastic space}}		\| [[Stronger than::monotonically normal space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|monotonically normal space\|elastic space}}
	\|-		\|-
	\| [[Stronger than::~~Hereditarily~~ collectionwise normal space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|hereditarily collectionwise normal space\|elastic space}}		\| [[Stronger than::hereditarily collectionwise normal space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|hereditarily collectionwise normal space\|elastic space}}
	\|-		\|-
	\| [[Stronger than::~~Hereditarily~~ normal space]] \|\| every subspace is [[normal space\|normal]] \|\| \|\| \|\| {{intermediate notions short\|hereditarily normal space\|elastic space}}		\| [[Stronger than::hereditarily normal space]] \|\| every subspace is [[normal space\|normal]] \|\| \|\| \|\| {{intermediate notions short\|hereditarily normal space\|elastic space}}
	\|-		\|-
	\| [[Stronger than::~~Collectionwise~~ normal space]] \|\| <math>T_1</math>, and any discrete collection of closed subsets can be separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|collectionwise normal space\|elastic space}}		\| [[Stronger than::collectionwise normal space]] \|\| <math>T_1</math>, and any discrete collection of closed subsets can be separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|collectionwise normal space\|elastic space}}
	\|-		\|-
	\| [[Stronger than::~~Normal~~ space]] \|\| <math>T_1</math>, and any two disjoint closed subsets are separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|normal space\|elastic space}}		\| [[Stronger than::normal space]] \|\| <math>T_1</math>, and any two disjoint closed subsets are separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|normal space\|elastic space}}
	\|}		\|}

Vipul: /* Relation with other properties */

2009-10-24T22:38:01Z

Relation with other properties

← Older revision		Revision as of 22:38, 24 October 2009
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	===Stronger properties===		===Stronger properties===

	* [[Metrizable space]]		{\| class="wikitable" border="1"
			! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
			\|-
			\| [[Weaker than::Metrizable space]] \|\| underlying topology of a [[metric space]] \|\| [[metrizable implies elastic]] \|\| [[elastic not implies metrizable]] \|\| {{intermediate notions short\|elastic space\|metrizable space}}
			\|-
			\| [[Weaker than::Manifold]] \|\| \|\| (via metrizable) \|\| \|\| {{intermediate notions short\|elastic space\|manifold}}
			\|-
			\| [[Weaker than::Sub-Euclidean space]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::Closed sub-Euclidean space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elastic space\|closed sub-Euclidean space}}
			\|}

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

	* [[Paracompact Hausdorff space]]		{\| class="wikitable" border="1"
	* [[Monotonically normal space]]		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
	* [[Hereditarily collectionwise normal space]]		\|-
	* [[Hereditarily normal space]]		\| [[Stronger than::Paracompact Hausdorff space]] \|\| [[paracompact space\|paracompact]] and [[Hausdorff space\|Hausdorff]] \|\| [[elastic implies paracompact Hausdorff]] \|\| [[paracompact Hausdorff not implies elastic]] \|\| {{intermediate notions short\|paracompact Hausdorff space\|elastic space}}
	* [[Normal space]]		\|-
			\| [[Stronger than::Monotonically normal space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|monotonically normal space\|elastic space}}
			\|-
			\| [[Stronger than::Hereditarily collectionwise normal space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|hereditarily collectionwise normal space\|elastic space}}
			\|-
			\| [[Stronger than::Hereditarily normal space]] \|\| every subspace is [[normal space\|normal]] \|\| \|\| \|\| {{intermediate notions short\|hereditarily normal space\|elastic space}}
			\|-
			\| [[Stronger than::Collectionwise normal space]] \|\| <math>T_1</math>, and any discrete collection of closed subsets can be separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|collectionwise normal space\|elastic space}}
			\|-
			\| [[Stronger than::Normal space]] \|\| <math>T_1</math>, and any two disjoint closed subsets are separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|normal space\|elastic space}}
			\|}

	==References==		==References==

	* ''Paracompactness and elastic spaces'' by Hisahiro Tamano and J. E. Vaughan, ''Proc. Am. Math. Soc., Vol. 28. No. 1 (Apr 1971) pp. 299-303''		* ''Paracompactness and elastic spaces'' by Hisahiro Tamano and J. E. Vaughan, ''Proc. Am. Math. Soc., Vol. 28. No. 1 (Apr 1971) pp. 299-303''

Vipul: 1 revision

2008-05-11T19:43:33Z

1 revision

← Older revision	Revision as of 19:43, 11 May 2008
(No difference)

Vipul at 20:24, 17 December 2007

2007-12-17T20:24:42Z

New page

{{topospace property}}

==Definition==

{{fillin}}

==Relation with other properties==

===Stronger properties===

* [[Metrizable space]]

===Weaker properties===

* [[Paracompact Hausdorff space]]
* [[Monotonically normal space]]
* [[Hereditarily collectionwise normal space]]
* [[Hereditarily normal space]]
* [[Normal space]]

==References==

* ''Paracompactness and elastic spaces'' by Hisahiro Tamano and J. E. Vaughan, ''Proc. Am. Math. Soc., Vol. 28. No. 1 (Apr 1971) pp. 299-303''