We have to prove that
Let $X$ is locally compact and $T_2$. Then for all $x\in X$ and an open set $U$ containing $x$, there is an open set $V$ such that $x\in V\subset \overline{V}\subset U$ with $\overline{V}$ compact.
Definition of Locally compactness: (Munkres) $X$ is said to locally compact if every point of $X$ has a compact neighbourhood i.e. there exists a compact set $C$ and an open set $V$ such that $x\in V\subset C$.
I have tried the problem in the following manner-
As $X$ is locally compact, we have $x\in V\subset C$ where $C$ is compact and $V$ is open. So, we have $$x\in U\cap V\subset C$$. As $X$ is $T_2$, $C$ is $T_2$ as well. Hence, $C$ is compact and $T_2$, then $C$ is normal. As $U\cap V$ is open in $X$ and it is contained in $C$, $U\cap V $ is open is $C$ as well.
Then by an well known property of normal spaces we have $x\in W\subset \overline{W}^C\subset U\cap V$ where $W$ is open set in $C$ and $\overline{W}^C $ denotes the closure of $W$ in $C$ (i.e. $\overline{W}^C=\overline{W}\cap C$). But now the problem is this $W$ may not be open in $X$.
Can anyone help me to complete the proof? Thanks for help in advance.
[Edit: There is a question here https://math.stackexchange.com/questions/2101393/a-topological-space-is-locally-compact-then-here-is-an-open-base-at-each-point-h . But the thing is that in that question it only askes for $x\in V\subset U$ with $\overline{V}$ compact, but I need another condition to be satisfied which is $\overline{V}\subset U$.
Best Answer
Clearly we may assume that $C=\operatorname{cl}V$. Let $K=C\setminus V=\operatorname{cl}V\setminus V$. If $K=\varnothing$, then $V=C$, and $W$ is open in $X$, so suppose that $K\ne\varnothing$.
For each $y\in K$ there are disjoint open sets $W_y\subseteq V$ and $G_y$ such that $x\in W_y$ and $y\in G_y$. $K$ is compact, so there is a finite $F\subseteq K$ such that $K\subseteq\bigcup_{y\in F}G_y$. Let $W=\bigcap_{y\in F}W_y$ and $G=\bigcup_{y\in F}G_y$; then $x\in W\subseteq C\setminus G$, and $C\setminus G$ is closed in $X$, so $x\in W\subseteq\operatorname{cl}W\subseteq C\setminus G\subseteq V$.