Paracompact Hausdorff space: Difference between revisions
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| [[Stronger than::Normal space]] || any two disjoint closed subsets are separated by disjoint open subsets || [[paracompact Hausdorff implies normal]] || [[normal not implies paracompact]] || {{intermediate notions short|normal space|paracompact Hausdorff space}} | | [[Stronger than::Normal space]] || any two disjoint closed subsets are separated by disjoint open subsets || [[paracompact Hausdorff implies normal]] || [[normal not implies paracompact]] || {{intermediate notions short|normal space|paracompact Hausdorff space}} | ||
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| [[Stronger than::Collectionwise normal space]] || any discrete collection of closed subsets can be separated by pairwise disjoint open subsets || [[paracompact Hausdorff implies collectionwise normal]] || [[collectionwise normal not implies paracompact Hausdorff]] || {{intermediate notions short|collectionwise normal space|paracompact Hausdorff space]}} | |||
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Revision as of 00:20, 25 October 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
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Compact Hausdorff space | compact and Hausdorff | compact implies paracompact | paracompact Hausdorff not implies compact | |FULL LIST, MORE INFO |
| Locally compact paracompact Hausdorff space | paracompact Hausdorff and locally compact | |FULL LIST, MORE INFO | ||
| Polyhedron | the underlying topological space of a simplicial complex | CW-space|FULL LIST, MORE INFO | ||
| CW-space | the underlying topological space of a CW-complex | CW implies paracompact Hausdorff | |FULL LIST, MORE INFO | |
| Metrizable space | the underlying topological space of a metric space | metrizable implies paracompact Hausdorff | Elastic space|FULL LIST, MORE INFO | |
| Manifold |
Weaker properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Binormal space | product with the unit interval is normal | paracompact Hausdorff implies binormal | ||
| Normal space | any two disjoint closed subsets are separated by disjoint open subsets | paracompact Hausdorff implies normal | normal not implies paracompact | Binormal space|FULL LIST, MORE INFO |
| Collectionwise normal space | any discrete collection of closed subsets can be separated by pairwise disjoint open subsets | paracompact Hausdorff implies collectionwise normal | collectionwise normal not implies paracompact Hausdorff | The part "]" of the query was not understood.</br>Results might not be as expected.|FULL LIST, MORE INFOThe part "]" of the query was not understood.</br>Results might not be as expected. |