Assuming the axiom of choice, we can prove that there are subsets of $\Bbb{R}$ (e.g., the Bernstein sets) that does not have the Baire property. What about a random ($\leftarrow$the everyday meaning is used here, i.e., not a mathematical term) Polish space?
If a Polish space is a countable space with the discrete topology, then there's obviously no such a subset.
If a Polish space is uncountable, then using the Cantor–Bendixson theorem, we can apply the canonical method in the case of $\Bbb{R}$ to the uncountable part got from the deconstruction, which is also a Polish space, and there exists such a subset.
The kind of Polish spaces where there is a subset without the Baire property must be uncountable or countable but with a topology other than the discrete one. What should the topology be like?
