Strongly zero-dimensinal spaces that are homeomorphic to other spaces under certain conditions

Question

According to a theorem from van Engelen: Homogeneous zero-dimensional absolute Borel sets (1986):

Theorem: If $X$ is a separable metrizable zero-dimensional absolute $F_{\sigma\delta}$ that is nowhere $G_{\delta\sigma}$ and of the first category (in itself), then $X \simeq \mathbb{Q}^\omega$ .

My question is: do similar conditions as im the Theorem apply for any other spaces that are strongly zero-dimensional?

If not with similar assumptions, the conditions can be even very different from those in the theorem. But I am interested whether for any other strongly zero-dimensional space $Y$, there exists $X$ such that $X \simeq Y$ under some assumptions.

I am thinking that maybe Cantor set or irrationals could have similar property, but not very sure. Also I am not 100% sure if Cantor set and irrationals are strongly zero-dimensinal or just zero-dimensional.

Answer 1

In the following I will mean by space a (non-empty) separable metrisable space.

For a space zero-dimensional and strongly zero-dimensional are equivalent, as is well known. So you can forget about that distinction in the remaining part.

Some classical theorems:

(Brouwer (1910)) A space $X$ is homeomorphic to the Cantor set $C$ iff it is zero-dimensional, has no isolated points (aka as "crowded" or dii, "dense-in-itself"), and is compact (i.e. absolutely closed).

(Sierpiński (1920)) A space $X$ is homeomorphic to the rationals $\Bbb Q$ iff it is countable and crowded.

(Aleksandrov and Urysohn (1928)) A space $X$ is homeomorphic to $C\setminus \{p\}$ (where $C$ is the Cantor set and $p$ any of its points) iff $X$ is zero-dimensional, locally compact, non-compact and crowded.

(Aleksandrov and Urysohn (1928)) A space $X$ is homeomorphic to $\Bbb P$ (the space of irrationals) iff $X$ is zero-dimensional, completely metrisable (which means it is an absolute $G_\delta$) and nowhere compact (which means that no point of $X$ has a compact neighbourhood; neighbourhood meant in the broad way).

(Aleksandrov and Urysohn (1928)) A space $X$ is homeomorphic to $C \times \Bbb Q$ iff $X$ is zero-dimensional, $\sigma$-compact (so an absolute $F_\sigma$), which is nowhere compact and nowhere countable.

Then Aleksandrov and Urysohn asked (in their 1928 paper) for other characterisation theorems like this, but for higher Borel classes. Similar I think to what you want. So it's an old problem area.

If you want more info on some solutions to this see van Engelen's thesis which has proofs for all of the above too and his own characterisation of $\Bbb Q^\omega$ that you referenced. I won't repeat that info here. But it gets really technical and quite deep later on. (I've followed a one semester class/seminar on descriptive set theory by van Engelen during my studies).