Paracompact topological space

@lee2013smooth page 36.

Refinement of an open cover $\mathcal{U}=\{U_{\alpha }\}$: is another open cover $\mathcal{V}=\{V_{\beta }\}$ such that every $V_{\beta }$ is contained in at least one $U_{\alpha }$.

Paracompact space: It is a topological space in which every open cover admits a locally finite refinement. That is the refinement $\mathcal{V}$ is such that for every $x\in M$ there exists a neighbourhood that intersects only a finite number of $V_{\beta }$. It can be characterised in terms of partition of unity.

Remark. A closed subspace of a paracompact space is paracompact (here).