Paracompact space

Remarks

Example: $\R$ is paracompact. Given an open cover $\mathcal{U}$, each compact interval $[n, n + 1]$ is covered by finitely many members of $\mathcal{U}$. Intersect these members with $(n - 1, n + 2)$; the resulting sets, over all $n \in \Z$, form a locally finite open refinement.

Non-example: the long line is not paracompact, although it is Hausdorff and locally homeomorphic to $\R$.

Every compact space is paracompact, since a finite subcover is a locally finite refinement. Some authors include the Hausdorff condition in the definition of a paracompact space.