Elastic space

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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 T_1, 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 T_1, 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