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 ${({U}_{i})}_{i\in I}$ is any family of open subsets of $X$ such that
$${\cup}_{i\in I}{U}_{i}=X,$$ 
then there exists another family ${({V}_{i})}_{i\in I}$ of open sets such that
$${\cup}_{i\in I}{V}_{i}=X$$ 
$${V}_{i}\subset {U}_{i}\text{for all}i\in I$$ 
and any specific $x\in X$ is in ${V}_{i}$ 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  20130322 12:12:47 
Last modified on  20130322 12:12:47 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  9 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 5400 
Classification  msc 5500 
Synonym  paracompact space 
Related topic  ExampleOfParacompactTopologicalSpaces 
Defines  paracompact 
Defines  paracompactness 