Good afternoon all. This question appeared on my real analysis midterm. I got it wrong (very wrong!) and the prof isn't releasing solutions. Out of curiosity, I'd like to know how to attack the question, which I'll reproduce in full here:
Prove that any countable subset of $\mathbb{R}$ has empty interior. Is the converse true? Explain.
Here're a few ideas I had: Any countable subset $A$ of $\mathbb{R}$ is equivalent to $\mathbb{N}$ (or to some subset of $\mathbb{N}$). We can express the interior of $A$ as the union of all open sets contained in $A$. So if we can show that this union is empty, we'll be done.
As for the converse, if $A$ has empty interior, then the union of all open sets contained in $A$ must be empty. What's the best way to proceed from here?