Lusin set ($|A\cap M|\le {\aleph }_0\text{ for every meagre }M$)

A Lusin set is an uncountable subset $A$ of the real numbers such that $A\cap M$ is countable for every meagre set $M$. Every meagre subset has a meagre $F_{\sigma }$ cover, so testing those covers suffices. The Lusin set construction under CH gives an example. A Lusin set has no uncountable meagre subset of itself.

Under the Continuum hypothesis, enumerate the meagre $F_{\sigma }$ covers by ${\omega }_1$. At stage $\alpha$ choose a new real outside all earlier covers and the previous choices. The excluded union is meagre, so the Baire category theorem makes the choice possible. Each cover then contains only countably many selected points. Arbitrary meagre sets need not themselves have only continuum many possible choices; the coded covers are what is enumerated.