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:

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