# Borel set

Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
Let ${\displaystyle (X,O)}$ be a topological space, i.e. ${\displaystyle X}$ is a set and ${\displaystyle O}$ are the open sets of ${\displaystyle X}$ (or, equivalently, the topology of ${\displaystyle X}$). Then ${\displaystyle A\subset X}$ is a Borel set of ${\displaystyle X}$ if ${\displaystyle A\in \sigma (O)}$, where ${\displaystyle \sigma (O)}$ denotes the σ-algebra generated by ${\displaystyle O}$.
The σ-algebra generated by ${\displaystyle O}$ is simply the smallest σ-algebra containing the sets in ${\displaystyle O}$ or, equivalently, the intersection of all σ-algebras containing ${\displaystyle O}$.