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][1], Lemma 2.1 (a).


  [1]: https://ur.booksc.me/book/8624681/d91b4b