Characterization of Euclidean Borel Sigma-Algebra/Open equals Closed

Theorem

Let $\OO^n$ and $\CC^n$ be the collections of open and closed subsets of the Euclidean space $\struct {\R^n, \tau}$, respectively.

Then:

$\map \sigma {\OO^n} = \map \sigma {\CC^n}$

where $\sigma$ denotes generated $\sigma$-algebra.

Proof

Recall that a closed set is by definition the relative complement of an open set.

Hence Sigma-Algebra Generated by Complements of Generators applies to yield:

$\map \sigma {\OO^n} = \map \sigma {\CC^n}$

$\blacksquare$