Question

Show that a compact metric space X is locally compact.

Answer 1

Let $X$ be a compact metric space, $x\in X$ a point and $x\in U$ an open neighbourhood of $x$. As $U$ is open, there is a certain $r>0$ such that the open ball $B(x,r)\subseteq U$. Now by considering the closed ball $F:=B_{closed}(x,r/2)$ we have $x\in F\subseteq B(x,r)\subseteq U$. Finally, $F$ is closed and thus is compact, because $X$ is a compact metric space.