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Homework Help: Compact Hausdorff space.

  1. May 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X,\tau)[/tex] be a compact Hausdorff space,
    and let [tex]f : X \to X[/tex] be continuous, but not surjective. Prove that
    there is a nonempty proper subset [tex]S \subset X[/tex] such that [tex]f(S) =
    S[/tex]. [Hint: Consider the subspaces [tex]S_n := f^{\circ n}(X)[/tex] where
    [tex]f^{\circ n} := f \circ \cdots \circ f[/tex] ([tex]n[/tex] times)].

    2. Relevant equations

    3. The attempt at a solution

    If such [tex]S[/tex] exists then [tex]f^{\circ n}(S) = S[/tex]. How should I use this in the proof? I don't have any clue where to start.
  2. jcsd
  3. May 9, 2010 #2
    What can you say about the sets S_n? For example, are they nested? What do you know about the continuous image of a compact set?
  4. Apr 22, 2012 #3
    Sorry for digging up an old thread, but I am stuck on the same problem.

    I let S = lim S_n so we have f(S) = f(lim S_n) = lim f(S_n) = lim S_{n+1} = S. Obvisouly S is non-empty since each f(S_n) is not empty.

    I am not sure if I got it right. We know that each S_n is closed and compact since X is a compact Hausdorff space and f is continuous, but I didnt use this property at all in my solution.

    Any help would be appreciated.
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