Let $X$ be a metric space, and $\mathscr{B}$ the $\sigma$-algebra generated by open sets of $X$. Can we find a countable dense subsets of the metric space $(\mathscr{B},d)$ with the metric $d(A,B)=m(A\Delta B)$ where $m$ is any Borel probability measure on the measurable space $(X,\mathscr{B})$?