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.