1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor

    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Uncountable family of disjoint closed sets
  1. Uncountable sets (Replies: 5)

Loading...