Is $\mathbb{Q}^{\mathbb{N}}$ strongly zero-dimensional? Why/ why not? I think I am struggling to check the definition for this space and also failing to get any intuition, since it is just too "large" and abstract.

**Why I am asking**

If $\mathbb{Q}^{\mathbb{N}}$ is strongly zero-dimensional, it means that $\beta \mathbb{Q}^{\mathbb{N}}$ (the Stone-Čech compactification) is **totally disconnected**, that is why it interests me. (I assume Hausdorff compactification, hence Tychonoff spaces).

**Definitions**

$\mathbb{Q}^{\mathbb{N}}$ = $\mathbb{Q}^{\omega}$ = the space of all rational sequences.

Strongly zero-dimensional space = A Hausdorff topological space $X$ is called strongly zero-dimensional whenever for every closed subset $A$ of $X$ and every open subset $U$ of $X$ such that $A⊆U$, there exists a clopen subset $V$ of $X$ such that $A⊆V⊆U$.

## Best Answer

It is because zero-dimensional and strongly zero-dimensional are equivalent for separable metrisable spaces and $\Bbb Q^\omega$ is separable metrisable and obviously zero-dimensional as a product of zero-dimensional spaces.