Definition:Borel Sigma-Algebra

Definition
The Borel or topological sigma-algebra (or $\sigma$-algebra) $\mathcal B \left({X, \tau}\right)$ of a topological space $\left({X, \tau}\right)$ is the $\sigma$-algebra generated by $\tau$.

Sometimes, the name Borel sigma-algebra is reserved for $\mathcal B \left({\R^n, \tau}\right)$, where $\tau$ is the Euclidean topology.

Borel Sets
The elements of $\mathcal B \left({X, \tau}\right)$ are called the Borel (measurable) sets of $\left({X, \tau}\right)$.

Alternative Notation
When the set $X$ or the topology $\tau$ are clear from the context, one may encounter $\mathcal B \left({\tau}\right), \mathcal B \left({X}\right)$ or even just $\mathcal B$.

Also, some authors write $\mathcal{B}^n$ for $\mathcal B \left({\R^n, \tau}\right)$.