Definition:Borel Sigma-Algebra

The Borel sigma-algebra (or Borel $$\sigma$$-algebra) of a topological space $$(X, \mathcal{T})$$ 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.