54-00-ParacompactTopologicalSpace.tex

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\pmtitle{paracompact topological space}
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A topological space $X$ is said to be \emph{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:
\begin{itemize}
\item Any metric or metrizable space is paracompact (A. H. Stone). 
\item Given an open cover of a paracompact space $X$, there exists a (continuous) partition of unity on $X$ subordinate to that cover.
\item A paracompact , Hausdorff space is regular.
\item A compact or pseudometric space is paracompact.
\end{itemize}
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