Definition:Borel Sigma-Algebra

Topological Spaces
Let $\left({S, \tau}\right)$ be a topological space

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

That is, it is the $\sigma$-algebra generated by the set of open sets in $S$.

Metric Spaces
Let $\left({X,\Vert \cdot \Vert}\right)$ be a metric space.

The Borel sigma-algebra (or $\sigma$-algebra) on $\left({X,\Vert \cdot \Vert}\right)$ is the $\sigma$-algebra generated by the open sets in $\mathcal P(X)$.

By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on topological spaces.

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

Also known as
When the set $S$ or the topology $\tau$ are clear from the context, one may encounter $\mathcal B \left({\tau}\right), \mathcal B \left({S}\right)$ or even just $\mathcal B$.

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