| Version 4 |
Version 3 |
| \PMlinkescapeword{normal} |
\PMlinkescapeword{normal} |
| \PMlinkescapeword{continuous} |
\PMlinkescapeword{continuous} |
| A topological space $X$ is said to be \emph{paracompact} if every open cover of $X$ has a locally finite open refinement. |
A topological space $X$ is said to be \emph{paracompact} if every open cover of $X$ has a locally finite open refinement. |
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| In more detail, if $(U_i)_{i\in I}$ is any family of open subsets of $X$ such that $$\cup_{i\in I}U_i = X\;,$$ |
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 |
then there exists another family $(V_i)_{i\in I}$ of open sets such that |
| $$\cup_{i\in I}V_i = X$$ |
$$\cup_{i\in I}V_i = X$$ |
| $$V_i\subset U_i\text{ for all }i\in I$$ |
$$V_i\subset U_i\text{ for all }i\in I$$ |
| and any specific $x\in X$ is in $V_i$ for only finitely many $i$. |
and any specific $x\in X$ is in $V_i$ for only finitely many $i$. |
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| Some properties: |
Any metric or metrizable space is paracompact (A. H. Stone). Also, given an open cover of a paracompact space $X$, there exists a (continuous) partition of unity on $X$ subordinate to that cover. |
| \begin{itemize} |
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| \item Any metric or metrizable space is paracompact (A. H. Stone). |
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| \item 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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| \item A paracompact , Hausdorff space is regular. |
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| \item A compact or pseudometric space is paracompact. |
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| \end{itemize} |
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