Let $(X,d)$ be a [metric space](/page/Metric%20Space). The Borel sigma-algebra $\mathcal B(X)$ is generated by the open balls $B(x,r)$ with $x \in X$ and $r>0$. If $X$ is separable, then $\mathcal B(X)$ is generated by the balls $B(x,r)$ with $x$ in a fixed countable [dense subset](/page/Dense%20Subset) of $X$ and $r \in \mathbb Q$ with $r>0$.