Elastic space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[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=== | ||
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! | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
|- | |- | ||
| [[Stronger than:: | | [[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:: | | [[Stronger than::monotonically normal space]] || || || || {{intermediate notions short|monotonically normal space|elastic space}} | ||
|- | |- | ||
| [[Stronger than:: | | [[Stronger than::hereditarily collectionwise normal space]] || || || || {{intermediate notions short|hereditarily collectionwise normal space|elastic space}} | ||
|- | |- | ||
| [[Stronger than:: | | [[Stronger than::hereditarily normal space]] || every subspace is [[normal space|normal]] || || || {{intermediate notions short|hereditarily normal space|elastic space}} | ||
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| [[Stronger than:: | | [[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}} | ||
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| [[Stronger than:: | | [[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}} | ||
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Latest revision as of 23:12, 24 January 2012
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
Definition
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Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Metrizable space | underlying topology of a metric space | metrizable implies elastic | elastic not implies metrizable | Protometrizable space|FULL LIST, MORE INFO |
| Manifold | (via metrizable) | Metrizable space, Protometrizable space|FULL LIST, MORE INFO | ||
| Sub-Euclidean space | ||||
| Closed sub-Euclidean space | Metrizable space, Protometrizable space|FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| paracompact Hausdorff space | paracompact and Hausdorff | elastic implies paracompact Hausdorff | paracompact Hausdorff not implies elastic | |FULL LIST, MORE INFO |
| monotonically normal space | |FULL LIST, MORE INFO | |||
| hereditarily collectionwise normal space | Monotonically normal space|FULL LIST, MORE INFO | |||
| hereditarily normal space | every subspace is normal | Hereditarily collectionwise normal space, Monotonically normal space|FULL LIST, MORE INFO | ||
| collectionwise normal space | , and any discrete collection of closed subsets can be separated by disjoint open subsets | Hereditarily collectionwise normal space, Monotonically normal space|FULL LIST, MORE INFO | ||
| normal space | , and any two disjoint closed subsets are separated by disjoint open subsets | Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space|FULL LIST, MORE INFO |
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