Let $X$ and $Y$ be uncountable Polish spaces and let $h\colon X\to Y$ be a Borel isomorphism. Suppose that $A\subset X$ has the [Baire property](https://en.wikipedia.org/wiki/Property_of_Baire). Must $h[A]$ have the Baire property in $Y$ too?



Do Borel isomorphisms of Polish spaces preserve the Baire property?
No, it need not.


Take for example $X=Y=\mathbb{R}$, let $A=[0,1]$, let $B$ be some nowhere dense perfect subset of $[2,3]$, and consider a Borel isomorphism $h:\mathbb{R}\cong \mathbb{R}$ swapping $A$ and $B$ and fixing everything else. 


Then for each $U\subseteq A$ the preimage $h^{-1}(U)$ is meager in $\mathbb{R}$, hence has the property of Baire. Taking some $U\subseteq A$ which doesn't have the property of Baire we get that $h$ does not preserve the Baire-ness of $h^{-1}(U)$.


