Totally Bounbdedness

Definition

Def Totally Bounbdedness A set in a metric space is called totally bounded if for any there exist finitely many points such that

Proposition

Prop Totally bounded sets must be bounded, but bounded sets may not be totally bounded.

Proposition

Prop Bounded sets in are totally bounded. Proof Let be a bounded set in . Then there is such that For any , we take an integer and divide into small cubes of equal size with length . Let , denote the center of these cubes. Then for any there is such that for all . Therefore

Thrm A metric space is compact iff it is complete and totally bounded. Proof Assume is compact. Let be any Cauchy sequence. Since is sequentially compact, has a convergent subsequence. According to theorem, is convergent. Thus is complete. To see the totally boundedness of , note that for any we have By the compactness of , we can find such that Therefore is totally bounded. Conversely, assume is complete and totally bounded. We show is sequentially compact.