Where to find Kuratowski's Theorem?

Question

I am looking for a Theorem by Kuratowski stating something like:

For any two Polish Spaces $A,B$ there are meager sets $M_1 \subseteq A$, $M_2 \subseteq B$ and a homeomorphism $f: A \setminus M_1 \rightarrow B \setminus M_2$.

Does anybody know the title of the original paper and where to find it - or another source?

Thanks.

Answer 1

I’ve not been able to track down the original source, but this paper by Stefan Geschke (who sometimes posts here) states the theorem as follows:

For any perfect Polish spaces $X$ and $Y$ there are meagre Borel sets $A\subseteq X$ and $B\subseteq Y$ such that $X\setminus A$ and $Y\setminus B$ are homeomorphic.

The reference is to K. Kuratowski, Topology, vol. 1, 1966, no page number given.