Uncountable splitting in a Polish space

A countable dense set with positive rational radii gives an enumerated metric basis $(U_n)$: inside any ball around a point, choose a dense centre sufficiently near the point and a rational radius large enough to contain the point but small enough that its ball stays inside the original ball. For each n with nonempty countable $A\cap U_n$, A1 selects an enumeration of that intersection; an injection into $\mathbb{N}$ gives such a surjection by filling unused indices with the value at the least occupied index. Pairing n and enumeration indices shows that $M=\bigcup \{A\cap U_n:A\cap U_n\text{ is countable}\}$ is countable; empty terms contribute nothing. If there are no nonempty terms then M is empty.

The set $A\setminus M$ is uncountable: otherwise an enumeration of it and one of M, interleaved, would enumerate A. In particular it contains distinct x,y. Every basis neighbourhood of either meets A uncountably, by the definition of M. Take disjoint balls around x,y with radii less than $\min (d(x,y)\mathrm{/}3,ϵ\mathrm{/}3)$, and refine each at its centre to a basis neighbourhood. The refinements are disjoint, each has diameter less than $ϵ$, and both have uncountable intersection with A. This proves the statement for every positive bound. QED.