1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: 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
    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
    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
    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!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook