Is there any metric space which is compact, not totally bounded but complete? Thanks for your help.
2026-03-29 10:18:21.1774779501
Compact, not totally bounded but complete
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No. If $(X,d)$ is a compact metric space, then, for $\epsilon>0$, the open covering of $X$ by all $\epsilon$-balls $$\bigcup_{x\in X} B(x,\epsilon)$$ must have a finite subcoveringe which means that $X$ is totally bounded.