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Show that a homeomorphism preserves uniform continuity

  1. May 2, 2012 #1
    1. The problem statement, all variables and given/known data

    (X,d1),(Y,d2) and (Z,d3) are metric spaces, Y is compact,
    g(y) is a continuous function that maps Y->Z with a continuous inverse
    If f(x) is a function that maps X->Y, and h(x) maps X->Z such that h(x)=g(f(x))
    Show that if h is uniformly continuous, f is uniformly continuous

    2. Relevant equations

    if h is continuous, f is continuous

    3. The attempt at a solution

    I proved that f is continuous when h is continuous.
    I also know that g is a homeomorphism, which preserves pretty much everything.
    and that f(x)=g^(-1)(h(x)) so I just need to show that the homeomorphism preserves the uniform continuity.
     
  2. jcsd
  3. May 2, 2012 #2

    micromass

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    What do you know about continuous functions with compact domain?
     
  4. May 2, 2012 #3
    THey have a compact range. But I don't know that f's domain is compact, only that it's range is...
     
  5. May 2, 2012 #4

    micromass

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    g has a compact domain.
     
  6. May 3, 2012 #5
    I know Y is compact, and so g(Y) is compact.
    h(X) c g(Y), so h is bounded. Also, because the range of f is compact, f is bounded.
    I qualitatively know that uniform continuity means that the function is not allowed to get infinitely steep, and that a continuous, bounded function obviously can't be infinitely steep, therefore it should be uniformly continuous, but I'm not sure how to say it mathematically.
     
  7. May 3, 2012 #6

    micromass

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    Can you show that a continuous function on a compact domain is uniform continuous?
     
  8. May 3, 2012 #7
    Ooooh.....
    Thanks. I've got it now.
     
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