Borel set

A Borel set in a topological space belongs to the smallest sigma-algebra containing every open set.

For a topological space, ${\Sigma }_1^0$ is the class of open sets and ${\Pi }_{\alpha }^0$ consists of complements of ${\Sigma }_{\alpha }^0$ sets. For countable $\alpha >1$, ${\Sigma }_{\alpha }^0$ consists of countable unions of sets in ${\bigcup }_{\beta <\alpha }{\Pi }_{\beta }^0$. These countable levels contain every Borel set. Continuous inverse images preserve each level, by induction using openness, complementation and countable unions.

The Borel sigma-algebra of a topological space $X$ is the sigma-algebra generated by its open sets. Its members are exactly the Borel sets.

Every subset of a discrete space is open, so its Borel sigma-algebra is the full power set.