Open (closed) sets of a locally compact space.

Question

Let $X$ a locally compact space. How do I show that if $A$ is a open (closed) set in $X$ then $A$ is locally compact? Thank you very much.

Answer 1

Let $A$ be a closed set in $X$. Let $a \in A$. Now, since $X$ is locally compact, we can find a compact neighborhood $V$ of $a$ in $X$. $V \cap A$ is closed in $V$, and closed subsets of compact sets are compact.