Paracompact topological space

Let $X$ be a topological space.

An open cover $\{U_i\}_{i\in I}$ of $X$ is locally finite if every point $x\in X$ has a neighborhood $V_x$ that meets only finitely many sets $U_i$.

The space $X$ is paracompact if every open cover of $X$ admits a locally finite open refinement (i.e., there is a locally finite open cover $\{V_j\}_{j\in J}$ such that each $V_j\subset U_{i(j)}$ for some $i(j)$).

Paracompactness is one of the key hypotheses that ensures the existence of partitions of unity subordinate to open covers and underlies many global constructions in differential geometry.

Examples

1. Metric spaces.
   Every metric space is paracompact. In particular, all smooth manifolds modeled on $\mathbb R^n$ with their usual topology are paracompact when they satisfy the standard countability hypotheses.

2. Compact Hausdorff spaces.
   Every compact Hausdorff space is paracompact: open covers admit finite subcovers, hence are automatically locally finite.

3. CW complexes.
   CW complexes are paracompact, which is one reason the classifying space technology behaves well for bundles over CW-type bases.