Countable discrete spaces are standard borel

On S define $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ otherwise. Symmetry and separation are immediate, and if $x\mathrm{\ne }z$, at least one of $x\mathrm{\ne }y$ or $y\mathrm{\ne }z$ holds, giving $d(x,z)=1\le d(x,y)+d(y,z)$. Balls of radius one half are singletons. Every Cauchy sequence is eventually constant, by applying the Cauchy condition with tolerance one half; hence d is complete. On N, for instance, $d(2,5)=1$ and $B_d(2,1\mathrm{/}2)=\{2\}$.

The countable set S itself is dense. Every subset is a union of singleton opens, so the Borel sigma-algebra is exactly its full power set. By [F1] S is Polish and the identity gives the presentation in [F2]. For empty S there are no Cauchy sequences, the empty metric is complete, and the empty set is a countable dense subset; for a singleton the only sequence is constant.