Proof. (1)$\to$(2): Let $U\suf X$ be open then there exists a countable dense subset $S\suf U$. For each $x\in S$, let $r_{x}>0$ such that $B(x,r)\suf U$, then $U=\cups{}{x\in S}B(x,r_{x})$ is the countable union of open balls.
(2)$\to$(3): For any $x\in X$ and $r>0$, $B(x,r)=\cups{}{n\in\N^+}\cl{B(x,r-1/n)}$
(3)$\to$(1):For any $x\in X$ and $r>0$, $\cl{B(x,r)}$ is closed.
$\square$