# Is it true that every totally bounded set in a metric space is compact?

Every compact set is totally bounded, but can we say that every totally bounded set is compact?

I'm a beginner in metric space. My thinking is that a totally bounded set behaves like a finite set and in some sense it is small. So it is very much like a compact set.

• closed balls are compact... (In $\mathbb{R}^n$ at least) – Yanko Dec 8 '18 at 13:41
• @ArjunBanerjee It doesn't really matter, we have $B(x_0,r/2) \subset B[x_0,r] \subset B(x_0,2r)$ anyway. Of course, here I used $B[x_0,r]$ to denote a closed ball of radius $r$. – BigbearZzz Dec 8 '18 at 13:48
Consider the set $$(0,1)$$ in the metric space $$\Bbb R$$, it is totally bounded but not compact. However, it is well known that totally boundedness & completeness is equivalent to compactness. You can read more about that here.