Paracompact topological space

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $i(j)$ ).

Partitions of unity

Together with the Hausdorff condition, paracompactness ensures the existence of continuous partitions of unity subordinate to open covers (and smooth ones on smooth manifolds) and underlies many global constructions in differential geometry.

Examples

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.

Compact Hausdorff spaces. Every compact Hausdorff space is paracompact: each open cover admits a finite subcover, and that subcover is a locally finite open refinement.

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