Paracompact Hausdorff space: Difference between revisions
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* [[Compact Hausdorff space]] | * [[Compact Hausdorff space]] | ||
* [[Locally compact paracompact Hausdorff space]] | |||
* [[Polyhedron]] | * [[Polyhedron]] | ||
* [[CW-space]]: {{proofat|[[CW implies paracompact Hausdorff]]}} | * [[CW-space]]: {{proofat|[[CW implies paracompact Hausdorff]]}} | ||
Revision as of 20:24, 15 December 2007
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
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
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
- Manifold