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Example: intersection of compact sets which is NOT compact

  1. Apr 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Let S = {(a,b) : 0 < a < b < 1 } Union {R} be a base for a topology. Find subsets M_1 and M_2 which are compact in this topology but whose intersection is not compact.

    2. Relevant equations



    3. The attempt at a solution
    I'm not even sure what it means for an element of S to be compact, so I haven't been able to make any attempt at a solution.
     
  2. jcsd
  3. Apr 6, 2013 #2

    Bacle2

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    I assume you're working on the real line, right?

    Do you know the meaning of the terms 'open set', 'cover' and ' finite subcover'?
    I don't mean to be condescending; just to know.
     
  4. Apr 6, 2013 #3
    I assume it is the real line, and so the topological space will be (R,S).

    Yes I do know what open set/cover/finite sub cover mean
     
  5. Apr 6, 2013 #4

    Bacle2

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    The most general definition is that a subset S is compact iff (def.) every cover of S by open sets has a finite subcover. There are more specialized results, e.g., for R^n, compactness is equivalent to being closed and bounded,and, for metric spaces you have, e.g., every sequence has a convergent subsequence, but the first one covers all cases.
     
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