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Clopen subsets of the reals?

  1. Nov 27, 2012 #1
    Prove that the only subset of ℝ with the absolute value metric that are both open and closed are ℝ and ∅.

    I know I'm supposed to prove by contradiction, but i'm having trouble:

    Suppose there exists a clopen subset A of ℝ, where A≠ℝ, A≠∅. Let [x,y] be a closed interval in ℝ, where x is in A and y is in A' (complement of A). Now, let b=sup{z[itex]\in[/itex][x,y]|z[itex]\in[/itex]A}. Then I know b[itex]\in[/itex]A or b[itex]\in[/itex]A'.

    I know that b is an upper bound for A implies b is a lower bound for A'. I'm just not sure how to arrive at a contradiction. I'm still not grasping the intuition behind it, can anyone explain intuitively what this means?

    Thanks.
     
  2. jcsd
  3. Nov 27, 2012 #2

    micromass

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    That's not true is it? Take A=[a,b], then b is an upper bound of A. But [itex]A^\prime=(-\infty,a)\cup (b,+\infty)[/itex] and b is certainly not a lower bound of this.

    Anyway, by definition you know that b is the supremum of [itex][x,y]\cap A[/itex]. But the set [itex][x,y]\cap A[/itex] is closed (what is your definition of closed anyway?), what does that tel you about b?
     
  4. Nov 27, 2012 #3
    Oh okay, I see my mistake.
    Closed means a set contains its limit points. So if that intersection is closed, then b is in A?
     
  5. Nov 27, 2012 #4

    micromass

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    OK, so b is an element of A. Can you make a similar argument to conclude that b is an element of A'?
     
  6. Nov 27, 2012 #5
    Okay, b is an element of A because it is the intersection and A is closed. Why would it necessarily have to be in A'?
     
  7. Nov 27, 2012 #6
    Unless, A' is clopen too right? So A' will have to contain all of its limit points as well, and b is a boundary point for A'...? Am I thinking about this correctly?
     
  8. Nov 27, 2012 #7

    micromass

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    Yes. Why is A' clopen too? (you just need that A' is closed by the way)
     
  9. Nov 27, 2012 #8
    A' is clopen because A is both opened and closed. Thanks for your help!
     
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