Borel sigma-algebra

A Borel sigma-algebra on a topological space $X$ is the sigma-algebra $\mathcal B(X)$ generated by the open sets of $X$, i.e. the smallest sigma-algebra that contains every open subset of $X$.

Equipping $X$ with $\mathcal B(X)$ turns it into a measurable space in a way that is compatible with topology: many naturally occurring functions (especially continuous ones) become measurable with respect to Borel sigma-algebras.

Examples:

• On $\mathbb R$ with its usual topology, $\mathcal B(\mathbb R)$ is generated by open intervals such as $(a,b)$.
• If $X$ has the discrete topology (all sets are open), then $\mathcal B(X)=\mathcal P(X)$, the power set .
• If $X$ has the trivial topology $\{\varnothing, X\}$, then $\mathcal B(X)=\{\varnothing, X\}$.