Let $X$ be a locally compact Hausdorff space. Let $K \subset X$ be a compact subspace, and $U \subset X$ an open set, such that $K \subset U$.
Can we find an open set $V \subset X$ such that $K \subset V \subset U$ and $\overline V$ is compact?
I know that for all $x \in U$ there is an open set $V_x \ni x$ with compact closure contained in $U$. But the closure of $\bigcup_{x \in K} V_x$ doesn't have to be compact.