Existence of a open set between a compact and an open set

Question

Let $M$ be a compact manifold, $K\subset M$ compact, $U\subset M$ open.

Does in this case always exist a open set $V\subset M$ such that $K\subset V\subset\bar{V}\subset U$ ?

Answer 1

Since a manifold is locally Euclidean, it inherits all local properties of Euclidean space. In particular, since $\Bbb R^n$ is locally compact, so is the manifold $M$.

If $K$ is compact subspace of $M$ and $U$ is an open neighborhood of $K$, then each point $x\in K$ has an open neighborhood $V_x$ such that its closure $\overline{V_x}$ is compact and is contained in $U$ (this follows from $M$ being a locally compact Hausdorff space). By compactness, finitely many such sets $V_1,\dots V_n$ cover $K$. Let $V=\bigcup_{i=1}^n V_i$. Then $$K ⊆ V ⊆ \overline V = \bigcup_{i=1}^n\overline{V_i} ⊆ U$$ Moreover, $\overline V$ is compact.