Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$.

(Hint: Set $F_1 = f(X), F_{n+1} = f(F_n), ... $)

So, using Cantor's intersection theorem: If ($F_n$) is a decreasing sequence of closed sets then the intersection is non-empty. I also know that f is a continuous map so I could use that (topology definition or metric space def).

I'm not actually sure if each $F_n$ is closed, or how to show this. Also, if they are closed what would this non-empty intersection even show? The closest I can come to is that using the sequential continuity property - one could possibly show that $X, f(X), f(f(X)), ... $ converges to $A$ implies that $F_n$ converges to $f(A)$. Again, just seeing what I have really. Any help?

  • $\begingroup$ Hint: the image of a compact set under a continuous map is compact. $\endgroup$ – Callus Apr 9 '14 at 10:19
  • $\begingroup$ @Callus Hm, thanks but I'm still unsure of the next step. $\endgroup$ – McT Apr 9 '14 at 10:41

Set $F_1=X$. Then $F_1$ is compact.

Set $F_2=f(F_1)=f(f(X))$. Then $F_2$ is compact because continuous image of a compact set is compact and also $F_2\subset F_1=X$.

$F_3=f(F_2)$ and $F_3= f(F_2)\subset f(F_1)=F_2$

By induction prove that there exists a decreasing sequence ($F_n$) of compact sets. Then $\bigcap_{n} F_n=A\neq \emptyset$.

Then $A=\bigcap_{n+1} F_{n+1}=\bigcap_{n} f(F_n)=f(\bigcap_{n}F_n)=f(A)$.

  • $\begingroup$ How do you know that the intersection contains only one point? $\endgroup$ – Martin Sleziak Apr 9 '14 at 12:38
  • $\begingroup$ @MartinSleziak sorry. i got confused(i'm sleepy).i corrected it $\endgroup$ – Haha Apr 9 '14 at 12:44
  • $\begingroup$ Ohhh. This is quite an interesting result actually! $\endgroup$ – McT Apr 9 '14 at 13:24
  • $\begingroup$ Why can we conclude $\cap_{n}f(F_{n})=f(\cap_{n}F_{n})$? This is false in general. $\endgroup$ – T. Eskin Apr 10 '14 at 4:42
  • $\begingroup$ @ThomasE. $f$ is continuous $\endgroup$ – Haha Apr 10 '14 at 9:52

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