Compact sets admit relatively compact open neighborhoods and finite chart pieces

Write $M$ for the ambient Hausdorff manifold. We first record the compactness consequences used below. If $C$ is compact in an open subspace $D\subset M$, every ambient open cover restricts to an open cover in $D$, so $C$ is compact in $M$. Any compact $C\subset M$ is closed: for $x\mathrm{\notin }C$, the open sets disjoint from some open neighborhood of $x$ cover $C$ by Hausdorff separation; take finitely many of these sets covering $C$ and intersect the corresponding finitely many neighborhoods of $x$. This gives a neighborhood of $x$ disjoint from $C$. A closed subset of a compact space is compact, since adjoining its open complement to an open cover gives a cover of the compact space and hence a finite subcover.

Let $p\in O$ with $O\subset M$ open, and choose a smooth chart $(U,\phi )$ containing $p$. The open set $D=U\cap O$ is a topological manifold: it is Hausdorff and second countable as an open subspace, and restricting charts gives local Euclidean coordinates. Apply F2 to $D$ at $p$ to obtain an open $T\subset D$ containing $p$ with $C={\bar{T}}^D$ compact in $D$. Since $D$ is open in $M$, $T$ is open in $M$. By step 1.1, $C$ is compact and closed in $M$, so the ambient closure $\bar{T}$ is contained in $C\subset D\subset O\cap U$ and is a closed subset of compact $C$, hence compact. Thus the closure lies in the prescribed open set and in the original smooth chart domain.

Apply step 2.1 with $O=B$. The family of all neighborhoods $T$ obtained there covers $Q$, so choose finitely many $T_1,\dots ,T_r$ covering $Q$ and put $W={\bigcup }_{i=1}^r{T}_i$. It is open and contains $Q$. The closure of a finite union is the union of the closures, so $\bar{W}={\bigcup }_{i=1}^r\overline{T_i}\subset B$. This union is compact: restrict any open cover to each compact member, take finitely many covering sets for each, and combine these finite lists.

Repeat step 3.1 with the open set $W$ in place of $B$ to obtain an open $V$ containing $Q$ with compact $\bar{V}\subset W$. For a compact $K\subset B$, use step 2.1 and compactness to obtain finitely many open $T_j$ covering $K$ such that each compact $\overline{T_j}$ lies in a smooth chart domain $U_j$. Set $K_j=K\cap \overline{T_j}$. Each $K_j$ is closed in compact $K$, hence compact by step 1.1, and is contained in $U_j$; these sets cover $K$ because the $T_j$ do. If $Q$ is empty take $V=W=\mathrm{\varnothing }$, and if $K$ is empty use the empty family of pieces.