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Analysis Question

  1. Sep 23, 2006 #1
    Guys I would appreciate any help on this. I've been trying to find an example of a collection of closed intervals of R that is uncountable. I proved that if I take a collection of open intervals of R and bijectively map them to Z, then the collection is countable, and I would assume the same with a collection of closed intervals, but clearly there must be an example where that doesnt happen and I don't understand why my logic on the collection of open sets cannot be extended to the collection of closed sets. Thanks for any help.
  2. jcsd
  3. Sep 23, 2006 #2


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    The collection { [0,a] | a[itex]\in[/itex]A}, where A is a subset of (0,[itex]\infty[/itex]), can be put in a bijection with A.
  4. Sep 23, 2006 #3

    I forgot to mention the closed sets have to be disjoint.
  5. Sep 23, 2006 #4


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    Well then you could always take points as your closed intervals. It is not possible to form an uncountable set of disjoint closed intervals, each of finite length.
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