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In topology: homeomorphism v. monotone function

  1. Feb 2, 2008 #1
    1. Let [tex]f:\mathbb{R}\rightarrow\mathbb{R}[/tex] be a bijection. Prove that f is a homeomorphism iff f is a monotone function.



    I think I have it one way (if f is monotone, it is a homeomorphism), but I'm stuck on the other way (if f is a homeomorphism, then it is monotone). I tried to prove this using contradiction.

    Assume it is a homeomorphism but not monotone. Then there exists a,b,c in R s.t. a<b and a<c but f(a)<f(b) and f(a)>f(c). I think the statement I want to eventually contradict is the following: [tex]U\subset\mathbb{R}[/tex] is open iff [tex]f(U)\subset\mathbb{R}[/tex] is open.

    Could someone give me a small hint? Thanks. :)
     
  2. jcsd
  3. Feb 3, 2008 #2

    morphism

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    Use the intermediate value theorem (draw a sketch!).
     
  4. Feb 3, 2008 #3

    HallsofIvy

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    Can you prove that if f is not monotone, then f(x)= f(y) for some [itex]x\ne y[/itex]?
     
  5. Feb 3, 2008 #4
    Thanks! I got it very quickly using both of your suggestions. :D
     
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