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, Lemma 2.1 (a).

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, Lemma 2.1 (a).