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.