You're confusing what my claim is. I never claimed that closed sets cannot be countable, but rather that there are nonclosed countable sets and so the statement as I remember it that "countable sets are examples" from the question means that closed cannot be a valid answer given that not all countable sets are examples of closed sets. My request for the wording, to see whether I've forgotten a clause, still stands.
N \subset R is indeed closed (and noncompact) but jagluski's argument is correct but obfuscates a fairly trivial argument.
David