Borel Sigma-Algebra Generated by Closed Sets
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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
Sources
- 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S 1.2$