Definition:Borel Sigma-Algebra

Definition
The Borel sigma-algebra (or Borel $$\sigma$$-algebra) of a topological space $$\left({X, \mathcal T}\right)$$ is the smallest $\sigma$-algebra which contains all open subsets of $$X$$.

More precisely, the Borel $$\sigma$$-algebra of a topological space $$X$$ is the only $$\sigma$$-algebra $$\mathcal B$$ over $$X$$ such that:


 * 1) $$\mathcal T \subseteq \mathcal B$$.
 * 2) If $$\mathcal A$$ is any $$\sigma$$-algebra over $$X$$ such that $$\mathcal T \subseteq \mathcal A$$, then $$\mathcal B \subseteq \mathcal A$$.

This definition makes sense because the smallest $\sigma$-algebra containing a given collection of sets is well-defined.