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:


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