Classification of Polish spaces up to a $\sigma$-homeomorphism

A function $$f:X\to Y$$ between topological spaces is called

$$\bullet$$ $$\sigma$$-continuous if there exists a countable cover $$\mathcal C$$ of $$X$$ such that for every $$C\in\mathcal C$$ the restriction $$f{\restriction}_C$$ is continuous;

$$\bullet$$ a $$\sigma$$-homeomorphism if $$f$$ is bijective and the maps $$f$$ and $$f^{-1}$$ are $$\sigma$$-continuous.

Two topological spaces $$X,Y$$ are called $$\sigma$$-homeomorphic if there exists a $$\sigma$$-homeomorphism $$h:X\to Y$$.

By Theorem $$\Delta^0_3$$ from this MO-post, any two countable-dimensional uncountable Polish spaces are $$\sigma$$-homeomorphic. We recall that a topological space is countable-dimensional if it is the countable union of zero-dimensional spaces.

By a famous result of Pol, there exists a totally disconnected Polish space $$P$$ which is not countable-dimensional. Since the countable-dimensionality is preserved by $$\sigma$$-homeomorphisms, Pol's space $$P$$ is not $$\sigma$$-homeomorphic to the Cantor cube $$2^\omega$$. By the same reason, the Hilbert cube $$[0,1]^\omega$$ is not $$\sigma$$-homeomorphic to the Cantor cube. Since $$[0,1]^\omega$$ is not a countable union of totally disconnected spaces, the Hilbert cube $$[0,1]^\omega$$ is not $$\sigma$$-homeomorphic to Pol's space $$P$$.

Therefore, we have three spaces: $$2^\omega$$, $$[0,1]^\omega$$, $$P$$ which are not pairwise $$\sigma$$-homeomorphic.

Problem. Are there infinitely (countably, continuum) many uncountable Polish spaces which are not pairwise $$\sigma$$-homeomorphic?

Remark. Repeating the argument of the proof of Theorem $$\Delta^0_n$$ in this MO-post, it is possible to show that two Polish space $$X,Y$$ are $$\sigma$$-homeomorphic if and only if there exist countable partitions $$\{X_n\}_{n\in\omega}$$ and $$\{Y_n\}_{n\in\omega}$$ of the spaces $$X,Y$$ such that $$X_n$$ and $$Y_n$$ are homeomorphic Polish spaces for every $$n\in\omega$$. This characterization should imply that the maximal number of pairwise non-$$\sigma$$-homeomorphic Polish spaces belongs to the set $$\omega\cup\{\omega,\aleph_1,\mathfrak c\}$$.

• @Anonymous Any countable-dimensional uncountable spaces are $\sigma$-homeomorphic, is particular $[0,1]^n$ and $[0,1]^m$ are $\sigma$-homeomorphic. Jul 23, 2022 at 12:44
• Oh, right. In order to avoid trivialities (like finite spaces of different cardinalities) you probably are interested in spaces of cardinality $\frak c$. Jul 23, 2022 at 13:07
• @Anonymous Thank you for this comment. Indeed, I am interested in uncountable Polish spaces. Jul 23, 2022 at 13:28