I am studying properties of clopen subsets of $\mathbb{Q}^{\omega}$ (subsets of the space of all rational sequences, endowed with the product topology) and for verifying some properties, I need the clopen subsets to be $F_{\sigma\delta}$ (and nowhere $G_{\delta\sigma}$).
I have only basic understanding of this property and definitions ($F_{\sigma\delta}$ is defined as the intersection of countably many unions of countable closed sets, $G_{\delta\sigma}$ is kind of a dual notion).
The question: Could you please help me prove that any clopen subset of $\mathbb{Q}^{\omega}$ is $F_{\sigma\delta}$ and nowhere $G_{\delta\sigma}$?
My thoughts are to first check the $F_{\sigma}$ (= being union of countable closed sets) and then check $F_{\sigma\delta}$.
The $F_{\sigma}$ seems trivial to me - any clopen set of rational sequences should be "composed" from countable closed sets, since we are in a rational world, right?
With the second part (checking this forms countable intersection to get $F_{\sigma\delta}$) seems much harder to me.
Any thoughts?