Characterization of Euclidean Borel Sigma-Algebra/Open equals Closed

Theorem
Let $\mathcal{O}^n$, $\mathcal{C}^n$ and $\mathcal{K}^n$ be the collections of open and closed subsets of the Euclidean space $\left({\R^n, \tau}\right)$, respectively.

Then:


 * $\sigma \left({\mathcal{O}^n}\right) = \sigma \left({\mathcal{C}^n}\right)$

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:


 * $\sigma \left({\mathcal{O}^n}\right) = \sigma \left({\mathcal{C}^n}\right)$