Let X be a compact topological Hausdorff space and for $x$ in $X$ let $N$ be an open neighborhood.
It is said that because $X$ is locally compact, there exist neighborhoods $M\subseteq C\subseteq N$ of $x$ with $C$ closed and $M$ open.
I see the existence of $C:$ Locally compact means that for each $x\in X$ each neighborhood contains a compact neighbourhood of $x$. Here, $C$ is closed since $X$ is Hausdorff. Ok!
But why does there exist this $M$? I do not see that.