Borel σ-algebra

The Borel σ-algebra of a topological space \((X,\mathcal{O})\) is the σ-algebra generated by the open sets \(\mathcal{O}\), i.e. \(\sigma(\mathcal{O})\).

For example, taking the topological space \((\mathbb{R},\mathcal{O})\) (where \(\mathcal{O}\) is the standard topology on \(\mathbb{R}\) defined with \(|.|\) as the norm), the corresponding Borel σ-algebra \(\sigma(\mathcal{O})\) is such that for \(a,b \in \mathbb{R}\cup \{-\infty,\infty\}\) and \(a

• \((a,b) \in \sigma(\mathcal{O})\)
• \(\bigcap_{n\geq 1} (a-\frac{1}{n},b) = [a,b) \in \sigma(\mathcal{O})\) (Similary \((a,b] \in \sigma(\mathcal{O})\))
• \(\bigcap_{n\geq 1} (a-\frac{1}{n},a+\frac{1}{n}) = \{a\} \in \sigma(\mathcal{O})\)