The interior of a countable set is always empty?

Question

By reading my lectures notes, I stumbled on one justification of a fact that seemed pretty non intuitive for me. It's obvious that $\mathring{\mathbb{Z}} = \emptyset$, but the justification given by my teacher is that simply because $\mathbb{Z}$ is a countable set. So I wonder, what would the proof of this look like?

Answer 1

If by set you mean "subset of $\Bbb R$" then any non-empty open set contains a nontrivial open interval $(a,b)$ and that is an uncountable set. So any set with a non-empty interior is uncountable.