Among separable metrizable spaces:


Cantor set is the unique compact zero-dimensional space without isolated points.


$\mathbb Q$ is the unique countable space without isolated points 


$\mathbb R \setminus \mathbb Q$ is the unique zero-dimensional, $G\_\delta$-space with no compact neighborhood.


$\mathbb Q ^\omega$ is the unique zero- dimensional, first category $F\_{\sigma\delta}$-space with the property that no nonempty clopen subset is a $G\_{\delta\sigma}$-space.


**Question.** Is there a simple classification of zero-dimensional $G\_{\delta\sigma}$-spaces which have no compact neighborhoods? 
The simplest examples would be $\mathbb Q$, $\mathbb R\setminus \mathbb Q$, and $\mathbb Q\times (\mathbb R\setminus \mathbb Q)$. Are there many others?


Is there a nice characterization of $\mathbb Q\times (\mathbb R\setminus \mathbb Q)$?



A classification of $G_{\delta\sigma}$ zero-dimensional spaces?
[This paper](https://www.ams.org/journals/tran/1981-264-01/S0002-9947-1981-0597877-9/S0002-9947-1981-0597877-9.pdf) by Van Mill from 1981 gives a characterisation of $\Bbb Q \times \Bbb P$ (where $\Bbb P$ is a common notation for the irrationals) in Thm 5.3:



> 
> If $X$ is separable metrisable and zero-dimensional, $\sigma$-complete and nowhere complete and nowhere $\sigma$-compact then $X \simeq \Bbb Q \times \Bbb P$.
> 
> 
> 


Where by complete I mean topologically complete (i.e. in this context: completely metrisable) and a nowhere-$P$ space is one where no non-empty open subset has property $P$, so $\Bbb P$ and $\Bbb Q$ are nowhere locally compact, e.g.) 


I think $\Bbb Q \times C$ (with $C$ the Cantor set) is another example for your list.


A lot of information can be found in van Engelen's PhD-thesis from 1985: *homogeneous zero-dimensional absolute Borel sets*, where he shows there are are $\omega\_1$ many homeomorphism types of subsets of $C$ that are both $F\_{\sigma \delta}$ and $G\_{\delta\sigma}$, also separately written up [here](https://www.ams.org/journals/tran/1985-290-01/S0002-9947-1985-0787953-3/S0002-9947-1985-0787953-3.pdf). In his thesis he also gave the first characterisation of $\Bbb Q^\omega$ (now relegated to the appendix of it). Look up Van Engelen's work from around that time for related results.


