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Question about the set (a,a]

  1. Sep 22, 2004 #1
    question about the set [a,a)

    If S=[a,a) Can such a set exist? It implies that a is in S and not in S, which doesn't make sense, but it seems a problem I'm trying to do requires it to be considered empty.

    The question is:

    Let In = [an,bn) where

    In+1 < In for all natural numbers n. [< denotes subset]

    Give an example of those In for which the intersection of In (for all n) is empty.

    I can't see any other way to construct an empty set.
     
    Last edited: Sep 22, 2004
  2. jcsd
  3. Sep 22, 2004 #2

    Hurkyl

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    (a, a] would be the set of all x such that a < x <= a.

    However, I think the question intends you to limit yourself to intervals whose endpoints are distinct.


    Recall that when you take a nested intersection of nonempty closed sets, you never get the empty set. Since you're searching for behavior that is not demonstrated by closed sets, I would suggest focusing at the open end of your intervals, where they resemble closed sets the least.
     
  4. Sep 22, 2004 #3
    Hi. Thanks for the response. Based on your definition [a,a) is indeed empty. Although I'm not sure why you think the question requires the endpoints to be distinct.

    If every set In+1 is a subset of In, then the only way the intersection of In for all n is empty is if one of those sets is empty. To my eyes (which admittedly are stupid at times), the only set of the form [a,b) that is empty is when a=b (or b<a)
     
    Last edited: Sep 22, 2004
  5. Sep 22, 2004 #4
    ****

    I just learned that the set [a,a) is not a proper subset of itself. So if In=[a,a), In+1 can't satisfy In+1 < In.

    Now I have no idea what's going on. If no In can be empty, how can the intersection of all In be empty in any case! HELP
     
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