Paracompact Hausdorff space: Difference between revisions
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A [[topological space]] is termed '''paracompact Hausdorff''' if it satisfies the following equivalent conditions: | A [[topological space]] is termed '''paracompact Hausdorff''' if it satisfies the following equivalent conditions: | ||
# It is [[paracompact space|paracompact]] and [[Hausdorff space|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 space|regular]] and every open cover has a [[locally finite collection of subsets|locally finite]] open [[refinement]] | |||
# It is [[regular space|regular]] and every open cover has a locally finite closed refinement | |||
# It is [[regular space|regular]] and every open cover has a locally finite refinement | |||
# It is [[regular space|regular]] and every open cover has a [[countable locally finite collection of subsets|countably locally finite]] open refinement | |||
The second definition is the one used in algebraic topology. | The second definition is the one used in algebraic topology. | ||
Revision as of 19:53, 23 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 |