Following a paper of Van Douwen called "Characterizations of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$", where he is proving some properties for specific spaces, I would like to know whether other spaces have the same property.
My question: Is every clopen subspace of $\mathbb{Q}^{\mathbb{N}}$ homeomorphic to $\mathbb{Q}^{\mathbb{N}}$?
The author claims this is true for $\mathbb{Q}$, but I am not sure, whether it still holds for $\mathbb{Q}^{\mathbb{N}}$ = $\mathbb{Q}^\omega$. I think yes, but cannot tell a proof.
Thank you for your insights.
Edit: For $\mathbb{Q}$, I assume the standard topology inherited from $\mathbb{R}$. For $\mathbb{Q}^{\mathbb{N}}$, I primarly assume the product topology, but also interested in the case with the topology inherited from $l^2$ (induced by the norm on $l^2$).
For $\Bbb Q$ this is true because it's a theorem by Sierpiński that any countable metric space without isolated points is homeomorphic to $\Bbb Q$. Any open subset of $\Bbb Q$ obeys that property.
In a comment on an earlier question on this space I noted that for $\Bbb Q^\omega$ (in the product topology!) there also is a more technical characterisation of that space along these lines (the irrationals $\Bbb P$ and the Cantor set also have such characterisations): If $X$ is a separable metrisable zero-dimensional absolute $F_{\sigma\delta}$ that is nowhere $G_{\delta\sigma}$ and of the first category (in itself), then $X \simeq \Bbb Q^\omega$. So to prove your hunch it would suffice that any open subset of $\Bbb Q^\omega$ also obeys the aforementioned properties. Separable metrisable zero-dimensional are immediate. The others I would have to think a bit more about. I won't rule out that your hunch is correct.
Your space (with inherited topology from $\ell_2$) is called (incomplete) Erdös space. It has dimension $1$ and not $0$. Many papers have been written about characterisations of it. Look for them online..