# 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:

Title paracompact topological space ParacompactTopologicalSpace 2013-03-22 12:12:47 2013-03-22 12:12:47 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 54-00 msc 55-00 paracompact space ExampleOfParacompactTopologicalSpaces paracompact paracompactness