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Compactness iff precompact

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data

    Let S be a subset of C. Prove that S is precompact if and only
    if S(closure) is compact.

    2. Relevant equations

    I have already showed if S(closure) compact, then S is precompact
    how can I show if S is precompact, then S(closure) is compact?

    3. The attempt at a solution
  2. jcsd
  3. Oct 11, 2011 #2
    What is your definition ofS being precompact? The one I knew is that the closure of S is compact.
  4. Oct 11, 2011 #3
    I think the OP means totally bounded. I've seen this terminology used before...

    Anyway, you are working in [itex]\mathbb{C}[/itex]. So try to prove that the closure of a totally bounded set is (totally) bounded. Then use Heine-Borel.
  5. Oct 11, 2011 #4
    A set S is precompact if every ε>0 then S can be covered by finitely many discs of radius ε .
  6. Oct 11, 2011 #5
    We didn't cover totally boundedness. I think we should use definition of precompactess.
  7. Oct 11, 2011 #6
    The definition you gave is the same as that of totally-bounded. I mean, totally.
  8. Oct 17, 2011 #7
    totally bounded

    how Can I show that the closure of a totally bounded set is (totally) bounded?

    solution Tried:

    Assume S is totally bounded. then for very ε>0 there are finitely many discs (O=Union of finitely many discs) that covers S let x be a limit points of S that is in S closure. but not in S.
    hence x is in O (how can I show this?)
    So x is in O for all x in S closure.
    Hence S closure is totally bounded.

    Am I on the right track?
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