Borel Sigma-Algebra of a Metric Space

Let $(X,d)$ be a metric space. The Borel $\sigma$-algebra of $(X,d)$, denoted $\mathcal{B}(X,d)$, is the $\sigma$-algebra generated by the family of all open subsets of $(X,d)$. Its members are called the Borel subsets of $(X,d)$.

When the metric is clear from the context we write $\mathcal{B}(X)$ for $\mathcal{B}(X,d)$.