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Totally Bounded & Completeness

  1. Nov 15, 2011 #1
    1. The problem statement, all variables and given/known data

    I haven't been able to find any theorems stating the relationship between a totally bounded space and a complete metric space, i.e., whether totally boundedness implies completeness. (I know that completeness implies totally boundedness though). Is it true that totally boundedness implies completeness?

    Any help is much appreciated.

    2. Relevant equations

    A metric space is totally bounded if give any ε>0, there exist finitely many points [itex] x_1, ... ,x_n[/itex] [itex]\in[/itex] M s.t. A [itex]\subset[/itex] [itex]\bigcup_{i=1}^{n}[/itex] [itex]B_{\epsilon} (x_i) [/itex] where n is finite.

    A metric space M is said to be complete if every Cauchy sequence in M converges to a point in M.
  2. jcsd
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