# set theory – Compactness in closed metric space

Is it true that if I have a compact metric space $$(M,d)$$, where $$M$$ is compact and has distance metric $$d$$, then for all sets $$Ssubseteq M$$ the closure of $$S$$, $$overline{S}$$, is compact?

If not, what additional assumptions would I need to ensure that $$overline{S}$$ is compact?