I'm trying to understand the proof of the theorem:
The Baire space $\mathcal{N}$ is unique up to homeomorphism, non-empty zero-dimentional Polish space for which all compact subsets have empty interior.
It is mentioned in the proof that "since a compact set in $X$ (the space satisfies all above conditions to be homeomorphic to $\mathcal{N}$)have empty interior, it follows that the closure of $U$(a open set in $X$) is not totally bounded."
I think I may miss something obvious, and I just don't know how to start.
The closure of $U$ is not compact, since it has non-empty interior. It is complete, however, since it’s a closed subset of a complete space. Therefore it cannot be totally bounded, since a set is compact iff it is complete and totally bounded.