\begin{definition}[Definition:Borel Sigma-Algebra/Metric Space]
Let $\struct {S, \norm {\, \cdot \,} }$ be a metric space.
The '''Borel sigma-algebra''' (or '''$\sigma$-algebra''') on $\struct {S, \norm {\, \cdot \,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset S$.
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 a topological space.
\end{definition}