If $M$ a manifold and $p\in M$, why is there a neighborhood V s.t. $V\subset \overline V\subset W$ with $\overline V$ compact? If $M$ a manifold and $p\in M$. Let $W\subset M$ a neighborhood of $p$. Why is there a neighborhood $V$ of $p$ s.t. $V\subset \overline V\subset W$ with $\overline V$ compact ? I would say (but with no certitude) that if $W$ is a neighborhhod of $p$, we can take a chart $(U,\varphi)$ where $U\supset W$. Since $\varphi:U\to\mathbb R^n$, and that $\varphi(W)\ni \varphi(p)$ is open (since $W$ open), we can consider $\varepsilon>0$ small enough to have $B(\varphi(p),\varepsilon)\subset \varphi(W)$, and thus, if we set $$V=\varphi^{-1}(B(\varphi(p),\varepsilon))$$ we have that $$p\in V\subset \overline{V}\subset W\subset M.$$ Is it correct ?

It is almost correct except that you should only have $U\subset W$ as $W$ can be big. Then it suffices to find a $V$ in $U$. In this case you might want to find $\epsilon$ so that $$\overline{B(\varphi(p),\epsilon)} \subset \varphi (U)$$ instead of just $B(\varphi(p),\epsilon) \subset \varphi (U)$.