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Homework Help: Uncountable family of disjoint closed sets

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data
    Determine whether the following statements are true or false
    a) Every pairwise disjoint family of open subsets of ℝ is countable.
    b) Every pairwise disjoint family of closed subsets of ℝ is countable.

    2. Relevant equations
    part (a) is true. we can find 1-1 correspondence with rational numbers

    But part (b) I know it is false. I need a counter example. Could you help me with that?


    3. The attempt at a solution
     
  2. jcsd
  3. Oct 12, 2011 #2

    Dick

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    You are probably thinking too hard. Think of sets consisting of a single element. Those are closed, yes?
     
  4. Oct 12, 2011 #3
    Let me clarify myself.
    let X be a collection of disjoint closed sets. Define X := { {x} such that x in ℝ }
    {x}_1 is the one of the disjoint closed set.
    {x}_2 is another disjoint closed set.
    and so fourth
    {x}_i is the another disjoint closed set
    Since ℝ is uncountable X must be uncountable.

    Is this what you mean?
     
  5. Oct 12, 2011 #4

    Deveno

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    the way you are listing the {x}_i, makes it look as if X is countable.

    but in fact, |X| = |U(x in R){x}| = |R|, because we have a bijection from X to R:

    {x}<---> x
     
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