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Closed sets

  1. Sep 17, 2007 #1
    1. The problem statement, all variables and given/known data
    Given A and B are closed sets in R does it follow that A+B is closed? (A+B={a+b|a in A and b in B})


    2. Relevant equations
    A set X is closed iff all of its limiting points are in X.


    3. The attempt at a solution
    I don't think this is true. I've tried constructing convergent sequences A and B and having the limit of the sum not being contained in A+B. But then A and B can't be closed.
     
  2. jcsd
  3. Sep 17, 2007 #2

    Dick

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    Ok, let A=N where N={1,2,3...}. Let B={-n+1/(n+1)} for n in N. I claim 0 is in the closure of A+B. But is not in A+B. Can you prove me wrong? Whew, that took a while.
     
  4. Sep 17, 2007 #3
    Gah. That's nice and clean. Thanks.
     
  5. Sep 17, 2007 #4

    Dick

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    No problem. Though it did hurt. I figure if you are having problems, it's not going to be easy for me, either.
     
  6. Sep 17, 2007 #5

    StatusX

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    Incidentally, if A and B are both bounded, A+B is closed. I'm not sure about the case when only one of them is bounded though.
     
  7. Sep 18, 2007 #6

    Dick

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    Well, if the C sequence is convergent and A is bounded then the A sequence has a convergent subsequence (compact). Doesn't that imply the corresponding B subsequence is convergent and seal everyone's fate?
     
  8. Sep 18, 2007 #7

    StatusX

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    Not bad. My idea for the case where both were compact was to send AxB under +:RxR->R, the product and images of compact sets being compact.
     
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