Let $W$ a neighborhood of $p\in M$. Why is there $V\subset M$ s.t. $V\subset \overline{V}\subset W$?

Question

Let $M$ be a manifold and let $W$ a neighborhood of $p$. Why is there an open $V$ s.t. $p\in V\subset \overline{V}\subset W\subset M$ where $V$ is compact ? It's in a proof of a theorem of my course, but I don't understant why.

Answer 1

The property you describe is the property of being a locally compact Hausdorff space. Manifolds are always locally compact Hausdorff. Indeed, they are Hausdorff by definition, and to see that they are locally compact, you can prove that euclidian spaces are locally compact and since manifolds are locally euclidian, they are locally compact.