Compact set in metric space

Let $$H$$ be a complete metric space, we equip $$H$$ with the induced metric norm. Let $$S$$ be a set of $$H$$.

Assume that for any $$\epsilon>0$$, we can find compact set $$K_\epsilon$$, such that $$S$$ is contained in a $$\epsilon$$ neighborhood of $$K_\epsilon$$.

Q Can we say $$\bar S$$, i.e. the closure of $$S$$, is compact?

• What notion of "dimension" are you using? What do you mean by "the induced metric norm"? – Eric Wofsey Sep 25 '18 at 4:11

Answer assuming completeness: the hypothesis is very confusing but the result is true in any complete metric space. Since compact sets are totally bounded, the hypothesis tells you that $$S$$ is totally bounded. Hence its closure is compact.