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Suppose $X$ is a compact connected metric connected space and for every $\epsilon >0$ , there exists a continuous surjective function $f : X \rightarrow [0,1]$, that for all $y \in [0,1]$, the diameter of the set $f^{-1} (y)$ is less than $\epsilon$.

Prove that $X$ has the fixed point property; that is, for all continuous $g : X \rightarrow X$, there exists $x_0 \in X$ that $g(x_0) = x_0$.

If we know that for a continuous surjective function $f : X \rightarrow [0,1]$ the diameter of set $f^{-1} (y)$ is zero, then we know that $X$ and $[0,1]$ are homeomorphic, so the statement is proven. But I have no idea for that.

Hints are appreciated.

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  • $\begingroup$ You may be able to prove that $X$ is simply connected. Choose two paths on $X$ with the same start and end-point, they will also be paths on $[0,1]$ which is simply connected, and so are homotopic. By Brouwers fixed point theorem any continuous function will then have a fixed point. $\endgroup$ – Damien Feb 15 '18 at 20:06
  • $\begingroup$ @DanielBeale: Simply connected and compact does not imply the fixed point property. $\endgroup$ – Moishe Kohan Feb 16 '18 at 0:41
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    $\begingroup$ It's a very neat problem, but you should not be trying to prove the existence of a homeomorphism $X\to [0,1]$. Instead, show that there is a sequence of points $x_n\in X$ such that $\lim_{n\to\infty} d(g(x_n), x_n)=0$. This even works if you replace $[0,1]$ with a closed $N$-disk. $\endgroup$ – Moishe Kohan Feb 16 '18 at 3:03
  • $\begingroup$ @MoisheCohen I was thinking of compact sets in $\mathbb{R}^N$. Is it not true that the Lefschetz fixed point theorem works for simply connected compact spaces? One can take a convex subset of $Y \subset X$ and then create homotopies to the continuous functions in $Y$. $\endgroup$ – Damien Feb 16 '18 at 15:13
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    $\begingroup$ @DanielBeale: Think of the antipodal map of the 2-sphere to itself. How many fixed points does it have? $\endgroup$ – Moishe Kohan Feb 16 '18 at 15:15
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Here is one solution which works even when you replace the closed unit interval with the closed unit $N$-ball $B$. I will describe steps of the proof and leave you to fill in the details. (Since the problem smells like a homework.)

(0). Prove the following lemma akin to the uniform continuity of continuous functions on compact metric spaces (and proven in a similar fashion):

Lemma. Let $f: (X,d_X)\to (Y,d_Y)$ be a continuous map between two compact metric spaces such that for every $y\in Y$, $diam(f^{-1}(y))\le \epsilon$. Then there exists $\sigma>0$ (depending on $f$ and on $\epsilon$) such that $$ \forall x_1, x_2\in X, ~~~d_Y(f(x_1), f(x_2))<\sigma \Rightarrow d_X(x_1, x_2)< 2\epsilon. $$

Proof. Suppose the claim fails. Then for every natural number $n$ there exists a pair of points $x_n, x'_n\in X$ such that $d_X(x_n, x'_n)\ge 2\epsilon$ while $$ d_Y(f(x_n), f(x'_n))<\frac{1}{n}.$$ In view of compactness of $X$, after passing to a subsequence, we can assume that $$ \lim_{n\to\infty} x_n=x, \lim_{n\to\infty} x'_n=x', $$ with $d_X(x,x')\ge 2\epsilon$, while $$ \lim_{n\to\infty} f(x_n)= \lim_{n\to\infty} f(x'_n)=y\in Y, $$ and $f(x)=f(x')=y$. Thus, the preimage of $y$ has diameter $\ge 2\epsilon$. A contradiction. qed

(1). Given $f$ and $g$ as in your problem ($f$ is surjective and $diam(f^{-1}(y))<\epsilon$ for all $y\in B$), construct a piecewise-linear continuous map $h: B\to B$ such that the following diagram is "almost commutative": $$ \require{AMScd} \begin{CD} X @>{g}>> X\\ @VVfV @VVfV \\ B @>{h}>> B \end{CD} $$ meaning that $d(f\circ g, h\circ f)<\delta(\epsilon)$, where $$ \lim_{\epsilon\to 0}\delta(\epsilon)=0. $$

I will explain how to do this in the case $B=[0,1]$, the extension to the case when $B$ is higher-dimensional is quite straightforward. In order to construct such $h$ first pick a finite subset $0=y_0< y_1 < y_2 < ... <y_n\in [0,1]$ (with $|y_i- y_{i-1}|$ sufficiently small for all $i$) such that $$ \forall i\in \{1,...,n\}, ~~diam(f^{-1}([y_{i-1}, y_i])) < 2\epsilon $$ Then define $h$ on the finite subset $\{y_0,...,y_n\}$ so that $$ \forall y_i, \exists x_i\in f^{-1}(y_i), h(y_i)=fg(x_i). $$ Then extend $h$ to the rest of $[0,1]$ linearly on each interval $[y_{i-1},y_i]$.

(2). Use the fact that $B$ has the fixed point property to show that $g$ "almost" has a fixed point, i.e.: For every $\epsilon>0$ there is a point $x\in X$ such that $d(g(x), x)\le \eta(\epsilon)$, where $$ \lim_{\epsilon\to 0}\eta(\epsilon)=0. $$

To prove this, take $y\in B$ such that $h(y)=y$ and think of its preimage $f^{-1}(y)$.

(3). Conclude that there is a sequence $(x_n)$ in $X$ such that $$ \lim_{n\to\infty} d(g(x_n), x_n)=0. $$

(4). Use compactness of $X$ to show that $g$ has a fixed point in $X$.

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  • $\begingroup$ thank you very much.I'm sorry to tell I don't know the proof of the Lemma,although I know the proof of the statement u mentioned. $\endgroup$ – user115608 Feb 17 '18 at 19:19
  • $\begingroup$ @user115608: Well, try to prove it (by analogy with the uniform continuity proof) since you do not know a proof of the lemma. Math is not about knowing some proofs but about coming up with new proofs. I also did not know how to prove this lemma before I proved it. $\endgroup$ – Moishe Kohan Feb 17 '18 at 19:55
  • $\begingroup$ I did,I tried.But I don't know if the diameter of $f^{-1}(y)$ is small, how can I use it to prove that the diameter of the preimage of a small interval is small. $\endgroup$ – user115608 Feb 18 '18 at 3:48
  • $\begingroup$ @user115608: How would you prove uniform continuity of a continuous function on a compact metric space (say, on an interval)? $\endgroup$ – Moishe Kohan Feb 18 '18 at 13:44
  • $\begingroup$ @Mouse Cohen would you please see this:math.stackexchange.com/questions/2653248/… $\endgroup$ – user115608 Feb 19 '18 at 6:15
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consider $h: X \rightarrow X$ continuous. I want to prove there's $x \in X$ that $h(x)=x$. for $\epsilon > 0$ consider $foh -f : X \rightarrow [0,1]$.because $f$ is surjective, there is $x \in X$ that $(foh -f) (x) \leq 0$ and there is $y \in X$ that $(foh -f) (y)\geq 0$. $foh -f $ is continuous and $X$ is connected, so there exists $t \in X$ that $(foh -f)(t) =0$. then $t , h(t) \in f^{-1}(f(t))$. so $d(t , h(t)) < \epsilon$.so we can have a sequence $t_n$ that $d(t_n , h(t_n)) < 1/n$. because $X$ is compact we can find $s \in X$ that $s = h(s)$.

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