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](https://mathoverflow.net/a/426983/61536), 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](https://www.jstor.org/stable/2043785), 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?*
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**Remark.** Repeating the argument of the proof of Theorem $\Delta^0\_n$ in [this MO-post](https://mathoverflow.net/a/426983/61536), 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 $\omega\cup\{\omega,\aleph\_1,\mathfrak c\}$.



Classification of Polish spaces up to a $\sigma$-homeomorphism
There are indeed continuum many. See:


Kihara, T., & Pauly, A. (2022). Point Degree Spectra of Represented Spaces. Forum of Mathematics, Sigma, 10, E31. doi:10.1017/fms.2022.7


