I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).



> 
> It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
> written as an infinite disjoint union of non-empty basic clopen
> subsets; hence, $U = \mathbb{N} \times \mathbb{Q}^\infty \simeq \mathbb{Q}^\infty$.
> 
> 
> 


In particular, I don't see **how the $U$ can be written as the said disjoint union and how that implies that $U \simeq \mathbb{Q}^\infty$**.


Can anyone please help with this or provide any reading source? I would be really grateful.


**Definition:**
$\mathbb{Q}^\infty$ is defined as a set of all rational sequences, endowed with the standard product topology.


**Source:** Engelen - [Countable Product of zero-dimensional absolute $F\_{\sigma \delta}$ spaces](https://ur.booksc.me/book/8624681/d91b4b), Lemma 2.1 (a).



Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?
We can first express the open set $U$ as a countable union of basic clopen subsets $A\_n=\{(q\_k)\_{k\in\mathbb{N}};i\_{n,k}<q\_k<j\_{n,k} \text{ for $k=1,\dots,n$}\}$, where $i\_{n,k}$ and $j\_{n,k}$ are irrational numbers. Some $A\_n$ may be empty.


To answer the question it is enough to express $A\_n\setminus\bigcup\_{i=1}^{n-1}A\_i$ as a finite union of disjoint basic clopens. To do it, we can work in $\mathbb{Q}^n$ instead of $\mathbb{Q}^\infty$ due to the construction of the sets $A\_n$.


Writing $\mathbb{Q}^n=\mathbb{Q}\_1\times\dots\times\mathbb{Q}\_n$, notice that for each $k\leq n$, the irrationals $i\_{1,k},\dots,i\_{n,k},j\_{1,k},\dots,j\_{n,k}$ give a partition of $\mathbb{Q}\_k$ into finitely many clopen subsets. Taking a product of these partitions, we obtain a partition of $\mathbb{Q}^n$ into disjoint clopens such that any of the $A\_i$, with $i\leq n$, is a finite union of these clopens. So $A\_n\setminus\bigcup\_{i=1}^{n-1}A\_i$ is also a finite union of the clopens.


If we pick the $A\_n$ so that no finite union of them covers $U$, we also get infinitely many non empty clopens.


This implies that $U\equiv\mathbb{Q}^\infty$ because any basic clopen set like the ones above is homeomorphic to $\mathbb{Q}^\infty$, which can be deduced using that any clopen subset of $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}$ by [Sierpinski's theorem](https://math.wvu.edu/%7Ekciesiel/Other/ElectronicReprints/141.SierpinskiThmOnQReprint.pdf).


