Consider the invariants appearing in the Cichoń's diagram: $add(\mathcal I)$, $cov(\mathcal I)$, $non(\mathcal I)$, $cof(\mathcal I)$, where $\mathcal I$ is either the ideal of null sets for the Lebesgue measure or the ideal of meager sets.

By theorem 17.41 of Kechris' book *Classical Descriptive Set Theory,* if $X$ is any Polish space and $\mu$ is any continuous Borel measure on $X$ (continuous = points are null sets), there exists a Borel isomorphism $f:X\longrightarrow [0,1]$ through which $\mu$ corresponds to the Lebesgue measure.

This result implies that the cardinal invariants corresponding to the ideal of null sets are the same for every Polish space and every continuous Borel measure on it.

I would like to know if there is an analogous result for the ideal of meager sets. Specifically: Are the four invariants the same for the ideal of meager sets of any perfect Polish space?

I suspect that the answer should be "yes", but I have not seen any result in this direction.

Of course, the answer is positive when restricted to the "usual" Polish spaces: $[0,1]$, $\mathbb R$, $^\omega\omega$, $^\omega2$, etc.