paracompact topological space
A topological space X is said to be paracompact if every open cover of X has a locally finite
open refinement.
In more detail, if (Ui)i∈I is any family of open subsets of X such that
∪i∈IUi=X, |
then there exists another family (Vi)i∈I of open sets such that
∪i∈IVi=X |
Vi⊂Ui for all i∈I |
and any specific x∈X is in Vi for only finitely many i.
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 X, there exists a (continuous) partition of unity
on X 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 |
---|---|
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 |