Baire space is zero dimensional

Question

Here is the basis of Baire's topology with $s\in A^{<\mathbb{N}}$ and $\mathcal{B}=\{N_{s}:s\in A^{<\mathbb{N}}\}$ I was trying to show that elements of $\mathcal{B}$ is both open and closed. I assumed that complement of Ns is not open but I am confused about choosing initial segments etc. Can you please help me

Answer 1

Hint. Let $s\in A^{<\mathbb{N}}$ be a finite sequence over $A$, of length $n$. Can you prove that $$A^\mathbb{N} \setminus N_s=\bigcup_{}\{N_t \mid t\neq s, \text{ length}(t)=n\}?$$