Paracompact Hausdorff space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Compact Hausdorff space]] | * [[Weaker than::Compact Hausdorff space]] | ||
* [[Locally compact paracompact Hausdorff space]] | * [[Weaker than::Locally compact paracompact Hausdorff space]] | ||
* [[Polyhedron]] | * [[Weaker than::Polyhedron]] | ||
* [[CW-space]]: {{proofat|[[CW implies paracompact Hausdorff]]}} | * [[Weaker than::CW-space]]: {{proofat|[[CW implies paracompact Hausdorff]]}} | ||
* [[Metrizable space]] | * [[Weaker than::Metrizable space]] | ||
* [[Manifold]] | * [[Weaker than::Manifold]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Binormal space]] | * [[Stronger than::Binormal space]] | ||
* [[Normal space]] | * [[Stronger than::Normal space]]: {{proofat|[[Paracompact Hausdorff implies normal]]}} | ||
Revision as of 20:45, 16 July 2009
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
This article describes a property of topological spaces obtained as a conjunction of the following two properties: paracompactness and Hausdorffness
Definition
A topological space is termed paracompact Hausdorff if it satisfies the following equivalent conditions:
- It is paracompact and Hausdorff
- Given any open cover of the space, there is a partition of unity subordinate to that open cover; in other words, there is a partition of unity such that the support of each function is contained in some set of that open cover
- It is regular and every open cover has a locally finite open refinement
- It is regular and every open cover has a locally finite closed refinement
- It is regular and every open cover has a locally finite refinement
- It is regular and every open cover has a countably locally finite open refinement
The second definition is the one used in algebraic topology.
Relation with other properties
Stronger properties
- Compact Hausdorff space
- Locally compact paracompact Hausdorff space
- Polyhedron
- CW-space: For full proof, refer: CW implies paracompact Hausdorff
- Metrizable space
- Manifold
Weaker properties
- Binormal space
- Normal space: For full proof, refer: Paracompact Hausdorff implies normal