Borel Sigma-Algebra Generated by Closed Sets

Theorem

Let $\map \BB {S, \tau}$ be a Borel $\sigma$-algebra generated by the set of open sets in $S$.

Then $\map \BB {S, \tau}$ 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.

$\blacksquare$

Also see

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

Sources

• 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S 1.2$