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Homework Help: Closed, bounded but not compact

  1. Sep 26, 2010 #1
    1. The problem statement, all variables and given/known data

    let |e-x-e-y| be a metric, x,y over R.
    let X=[0,infinity) be a metric space.
    prove that X is closed, bounded but not compact.

    2. Relevant equations



    3. The attempt at a solution

    there is no problem for me to show that X is closed and bounded. but how do I prove it's not compact?
    I assume it must be done with the use of Cauchy sequence. if xn is Cauchy but it's not convergent then X is not complete and then it's not compact. but how do I right it down in algebraical form?

    thanks in advance.
     
  2. jcsd
  3. Sep 26, 2010 #2

    LCKurtz

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    What about trying to find a sequence of points each of which has its own open cover so that no finite subcover exists?
     
  4. Sep 27, 2010 #3
    i'm afraid i don't know how to do it. can you show me please?

    and is the way with Cauchy sequence correct?
     
  5. Sep 27, 2010 #4

    hunt_mat

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    You can use the form of compactness with regard to subsequences. For every sequence x_{n} in the metric space then there is a convergent subsequence.
     
  6. Sep 27, 2010 #5

    HallsofIvy

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    Normally one would prove that [itex][0, \infty)[/itex], with the usual metric, is not compact, directly from the definition, "every open cover contains a finite subcover", by looking at the "open cover" [itex]U_n= [0, n)[/itex]. Are those sets open in this metric?
     
    Last edited by a moderator: Sep 28, 2010
  7. Sep 27, 2010 #6

    LCKurtz

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    I think Halls means [0,n).
     
  8. Sep 28, 2010 #7

    HallsofIvy

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    Yes, I did. Thanks. I will edit it.
     
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