Definition:Borel Sigma-Algebra/Metric Space

Definition
Let $\struct {X, \norm {\,\cdot\,} }$ be a metric space.

The Borel sigma-algebra (or $\sigma$-algebra) on $\struct {X, \norm {\,\cdot\,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset X$.

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.