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: 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
    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
    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
    Can you show that a continuous function on a compact domain is uniform continuous?
  8. May 3, 2012 #7
    Thanks. I've got it now.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook