Borel set

In mathematics, a Borel set is a set that belongs to the $\sigma -$algebra (see the CZ entry sigma algebra) generated by the open sets of a topological space.

Formal definition

Let $(X,O)$ be a topological space, i.e. $X$ is a set and $O$ are the open sets of $X$ (or, equivalently, the topology of $X$). Then $A\subset X$ is a Borel set of $X$ if $A\in \sigma (O)$, where $\sigma (O)$ denotes the sigma algebra generated by $O$.

Note in the above that by definition $\sigma (O)$ is simply the smallest $\sigma -$algebra containing the sets in $O$ or, equivalently, the intersection of all $\sigma -$algebras containing $O$.

See also

Topological space

Sigma algebra

Measure theory

Probability theory