paracompact topological space
A topological space is said to be paracompact if every open cover of has a locally finite open refinement.
In more detail, if is any family of open subsets of such that
then there exists another family of open sets such that
and any specific is in for only finitely many .
Some properties:
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Any metric or metrizable space is paracompact (A. H. Stone).
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Given an open cover of a paracompact space , there exists a (continuous) partition of unity on subordinate to that cover.
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A paracompact , Hausdorff space is regular.
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A compact or pseudometric space is paracompact.
Title | paracompact topological space |
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Canonical name | ParacompactTopologicalSpace |
Date of creation | 2013-03-22 12:12:47 |
Last modified on | 2013-03-22 12:12:47 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 9 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 54-00 |
Classification | msc 55-00 |
Synonym | paracompact space |
Related topic | ExampleOfParacompactTopologicalSpaces |
Defines | paracompact |
Defines | paracompactness |