Borel Sigma-Algebra Generated by Closed Sets

Theorem
Let $\mathcal B \left({S, \tau}\right)$ be a Borel $\sigma$-algebra generated by the set of open sets in $S$.

Then $\mathcal B \left({S, \tau}\right)$ is equivalently generated by the set of closed sets in $S$.

Proof
By definition, a closed set is the relative complement of an open set.

The result follows from Sigma-Algebra Generated by Complements of Generators.

Also see

 * Characterization of Euclidean Borel $\sigma$-Algebra:Open equals Closed