Elastic space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! property !! quick description !! 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}} | |||
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| [[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=== | ||
{| class="wikitable" border="1" | |||
! property !! quick description !! 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::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'' | ||
Revision as of 22:38, 24 October 2009
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 | quick description | 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 | quick description | 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