Standard-Borel Real Codings and Determining Classes: Examples

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.

For $n\ge 1$, [F1] supplies the metric. A d-infinity Cauchy sequence is Cauchy in each coordinate since $\mathrm{\mid }{x}_i-y_i\mathrm{\mid }\le d_{\mathrm{\infty }}(x,y)$. The coordinate limits exist by [F2]. For a fixed tolerance take the maximum of the finitely many coordinate convergence thresholds; beyond it all coordinate errors are below that tolerance, so the vectors converge in d-infinity. This metric induces the usual Euclidean topology: $d_{\mathrm{\infty }}\le d_2\le \sqrt{n}{d}_{\mathrm{\infty }}$ follows by bounding each squared coordinate by the maximum squared.

Induction using [F3]–[F4] makes ${\mathbb{Q}}^n$ countable. Given a vector x and positive epsilon, [F5] gives a rational in each of its finitely many coordinate intervals of radius epsilon; the resulting vector q satisfies $d_{\mathrm{\infty }}(x,q)<\epsilon$. Hence Q to the nth power is dense. For instance in dimension two, $d_{\mathrm{\infty }}((0,1),(1\mathrm{/}3,4\mathrm{/}3))=1\mathrm{/}3$.

Steps 1.1–1.2 and [F6] show R to the nth power is Polish. The identity is the presentation of [F7]. For n=0 there is just the empty tuple, with zero metric and itself as a finite dense set; it is complete and Polish. No maximum over an empty index set is used.

Apply [F1] with its AC hypothesis [F3]. It gives a Polish topology on B whose Borel sigma-algebra is the trace of that of P. The identity map from the trace measurable space to this presentation is therefore bimeasurable, proving the general assertion, including B empty.

For the instance, R is complete by [F4] and separable by [F2]–[F5], hence Polish. Each singleton rational is closed in R (a point outside it has a ball avoiding it), so Q and every subset of Q are Borel by [F2] and countable unions. Thus $\mathcal{B}(\mathbb{R}){\mathrm{\mid }}_{\mathbb{Q}}=\mathcal{P}(\mathbb{Q})$. The discrete metric $d(q,r)={\mathbf{1}}_{\{q\mathrm{\ne }r\}}$ is complete because a Cauchy sequence is eventually constant; Q itself is countable dense for this topology. Its Borel sets are again all subsets. For example the preimage of $\{1\mathrm{/}2,2\mathrm{/}3\}$ under the identity is exactly $\{1\mathrm{/}2,2\mathrm{/}3\}$ in both measurable structures. This gives the claimed explicit Polish presentation without requiring the inherited metric to be complete.