(adsbygoogle = window.adsbygoogle || []).push({}); 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Compact Hausdorff space.

**Physics Forums | Science Articles, Homework Help, Discussion**