PlanetMath: paracompact topological space

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paracompact topological space

(Definition)

In more detail, if $(U_i)_{i\in I}$ is any family of open subsets of such that

$\displaystyle \cup_{i\in I}U_i = X\;,$

then there exists another family $(V_i)_{i\in I}$ of open sets such that

$\displaystyle \cup_{i\in I}V_i = X$

$\displaystyle V_i\subset U_i$ for all $\displaystyle i\in I$

and any specific $x\in X$ is in

for only finitely many

Any metric or metrizable space is paracompact (A. H. Stone).
Given an open cover of a paracompact space , there exists a (continuous) partition of unity on subordinate to that cover.
A paracompact , Hausdorff space is regular.
A compact or pseudometric space is paracompact.

"paracompact topological space" is owned by mathcam. [ full author list (3) | owner history (3) ]

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Other names:

paracompact space

Also defines:

paracompact, paracompactness

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AMS MSC:	54-00 (General topology :: General reference works )
	55-00 (Algebraic topology :: General reference works )

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