@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.
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Author of the notes: Antonio J. Pan-Collantes
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