# Is the closure of a discrete subspace of $[0,1]$ necessarily countable?

Question is in the title.

According to this, every discrete subspace of $[0,1]$ must be countable. But what about its closure?

• In fact, every closed nowhere dense subset of $[0,1]$ is the closure of a discrete set. – Henno Brandsma Jun 25 '17 at 21:27