Proposition 14.2.5. Let $X$ be a separable metric space, then the Borel $\sigma$-algebra on $X$ is generated by the following families of sets:
Open sets of $X$.
$\bracs{B(x, r)|x \in X, r > 0}$.
$\bracsn{\ol{B(x, r)}|x \in X, r > 0}$.
Proof. (1) $\subset$ (2): Let $U \subset X$ be open. By Definition 4.5.5, there exists a countable dense subset $S \subset U$. For each $x \in S$, let $r_{x} > 0$ such that $B(x, r) \subset U$, then $U = \bigcup_{x \in S}B(x, r_{x})$ is a countable union of open balls.
(2) $\subset$ (3): For any $x \in X$ and $r > 0$, $B(x, r) = \bigcup_{n \in \natp}\overline{B(x, r - 1/n)}$ is a countable union of closed balls.
(3) $\subset$ (1): For each $x \in X$ and $r > 0$, $\overline{B(x, r)}$ is closed.$\square$