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Let $M$ be a compact set and $(M,d)$ be a metric space, define function $f:M\to M$ such that for all $\,p,q\in M$ $$d(f(p),f(q))\ge d(p,q)$$ Prove $f$ is surjective.

I observed that compactness of $M$ is crucial, supposing $M$ being only closed or bounded would have counterexamples of $f$.

Also I found another problem which might be related somehow, I couldn't help this but guessing we have to prove $f$ is isometric.

@TommasoSeneci Thank you very much for the efforts, but as we see problem is not completely solved, so any clarification of showing $Y$(discussed in the below answer) is closed or other ways maybe, would be so appreciated!

Community
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Amir Naseri
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  • No, this one is much easier than proving $f$ is an isometry. What make it a little tricky is: we do not assume $f$ is continuous. – GEdgar Apr 02 '16 at 15:44
  • @GEdgar I was just able to realize $f^{-1}$ is continuous when its domain is $f(M)$, have you found a solution? If so, would you please explain more? – Amir Naseri Apr 02 '16 at 16:01
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    It seems I was wrong when I said this one is easier than proving isometry. – GEdgar Apr 05 '16 at 14:49

2 Answers2

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Solution by "ifk" at http://www.artofproblemsolving.com/community/c7h360582p1972843 . There, $(X,d)$ is the metric space, and we are to prove $f$ is both surjective and isometric.

Let $x\in X$. Define $x_{0} = x, \ x_{n+1} = f(x_{n})$. By compactness $\{x_{n}\}_{n}$ has convergent (in particular Cauchy) subsequence $\{x_{n_{k}}\}_{k}$. W.l.o.g. $\{n_{k+1} - n_{k}\}_{k}$ is increasing.

We will show that the sequence $\{x_{n_{k+1} - n_{k}}\}_{k}$ converges to $x$. Let $\varepsilon > 0$. Since $\{x_{n_{k}}\}_{k}$ is Cauchy sequence, then for sufficiently large $k$ we've got $d(x_{n_{k}}, x_{n_{k+1}}) < \varepsilon$. Now $d(x_{n_{k}}, x_{n_{k+1}}) \ge d(x_{n_{k+1} - n_{k}}, x)$.

Since $\{x_{n}\}_{n\ge 1} \subset f(X)$ it follows that $f(X)$ is dense in $X$.

Now suppose that $d(f(x), f(y)) > d(x,y)$ for some $x,y\in X$. Define $\{x_{n}\}_{n}$ as before and similarly $y_{0} = y, \ y_{n+1} = f(y_{n})$ By the preceding argument there exist sequence $\{n_{j}\}_{j}\subset \mathbb{N}$ s.t. $x_{n_{j}}\to x, \ y_{n_{j}}\to y$ as $j\to \infty$. Then $$ d(f(x), f(y)) = d(x_{1},y_{1}) \le d(x_{n_{j}},y_{n_{j}})\to d(x,y) $$ so $d(x,y) < d(f(x),f(y) \le d(x,y)$, contradiction.

Therefore $f$ is isometry, in particular continuous mapping. Hence $f(X)$ is compact, in particular complete, in particular $f(X) = \text{cl}(f(X)) = X$. Q.E.D.

GEdgar
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  • So many thanks to you and "ifk" from "Art of problem solving" community! – Amir Naseri Apr 05 '16 at 15:09
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    When given a mathematical puzzle (at the undergraduate level or below), "Art of Problem Solving" is often a good place to find a solution. – GEdgar Apr 05 '16 at 15:13
  • @GEdgar We know that we can find a sequence $x_{n_j} \to x$ and a sequence $y_{m_j} \to y$, but why can we find a single sequence that works for both $x$ and $y$? – Emolga Jun 03 '16 at 15:31
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    @Leullame ... we have $x_n$ and $y_n$. We take a subsequence (a subset of the indices $n$) so that one converges. Then a further subsequence (a subset of the first subset of indices) so that the other converges. – GEdgar Jun 03 '16 at 20:36
  • @GEdgar Let's say $x_{n_k} $ converges to $x$. Then we have a sequence $y_{n_{k_{j+1}}-n_{k_j}}$ converging to $y$, but it seems $x_{n_{k_{j+1}}-n_{k_j}}$ does not converge to $x$, since it is not a subsequence of $x_{n_k}$. So how do we keep a sub-sequence of $x_{n_k}$ while doing the trick of $n_{k_{j+1}}-n_{k_j}$? – Emolga Jun 06 '16 at 18:45
  • A friend suggested an alternative explanation that I do understand: we have a convergent sub-sequence $(x_{n_k},y_{n_k})$ in the space $X^2$ which is Cauchy, thus $\forall\epsilon>0 \forall k\gg 0: d(x,x_{n_{k+1}-n_k})<d(x_{n_k},x_{n_{k+1}})<d((x_{n_k},y_{n_k}),(x_{n_{k+1}},y_{n_{k+1}}))<\epsilon$ and the same for $y$, and we are done. – Emolga Jun 06 '16 at 18:45
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Firstly we see that $f$ is injective since $x \neq y \Rightarrow d(f(x),f(y)) \geq d(x,y) > 0$ so $f(x) \neq f(y)$. Moreover the inverse, namely $g := f^{-1}$, is continuous on its domain, denoted by $Y := f(X)$, because $x,y \in Y \Rightarrow d(g(x),g(y)) \leq d(x,y)$ the function $g$ is non-expansive. So we have a continuos map $g : Y \rightarrow X$ and we assume not surjectiveness for $f$, i.e. $\exists x_0\in X\setminus Y$. $Y=f(X)=g^{-1}(X)\subset X$ is closed because it is the preimage through the continuous function $g$ of the compact set $X$, and compact set are closed in Hausdorff topologies such as metric induced ones. So since it cannot exists any sequence converging to $x_0$ we conclude $\exists \varepsilon >0 \; B_{\varepsilon}(x_0)\subset X\setminus Y$. Thus $\min_{Y} d(x_0,y)=d(x_0,\tilde{y}) > \varepsilon$. We are done because consider the sequence $\{ f^n(x_0) \}_n \subset Y$ by compactness $\exists f^{n_k}(x_0)\rightarrow \tilde{x}$ and then it is Cauchy, so for $\delta = \frac{\varepsilon}{2} \; \exists \, n_k \; d(f^{n_k}(x_0),f^{n_{k+1}}(x_0))<\delta$ which implies $ d(x_0,f^{n_k}(x_0)) \leq d(f^{n_{k}}(x_0),f^{n_{k+1}}(x_0)) < \delta = \frac{\varepsilon}{2}$.

Tommaso Seneci
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  • Great, thank you! I understood the whole solution except as @GEdgar mentioned, how can we conclude from surjectiveness and continuity of $g$ that $Y$ is closed? – Amir Naseri Apr 02 '16 at 17:49
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    @Tommaso Seneci I have faced a wrong conclusion, $Y=f(X)$ is closed as a subset of $Y$, but not $X$ generally, so we can not say it's compact, am I right? – Amir Naseri Apr 04 '16 at 08:39
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    @GEdgar Excuse me sir, if you see this comment and the one I wrote above, would you please clarify how could we be sure $Y$ is closed and compact in $X$? – Amir Naseri Apr 04 '16 at 11:00
  • edited:) I underlined surjectivess just to say that the image of $g$ is the whole $X$ – Tommaso Seneci Apr 04 '16 at 12:40
  • I now do not understand. Note: this is not true... if $X$ compact metric, $Y \subseteq Y$, $g : Y \to X$ continuous and surjective, then $Y$ is closed in $X$. So we need to use more than that to show $Y$ is closed in $X$ in this case. – GEdgar Apr 04 '16 at 12:49
  • We have all these ingredient? $g$ is surjective by definition, indeed it is $f^{-1}$ which domain is the whole $X$ by DEFINITION, and it is also continuous on its domain, so where the problem? – Tommaso Seneci Apr 04 '16 at 13:26
  • @TommasoSeneci For example $g:[0,1)\cup[2,3]\to[0,2]$ such that for ($x\lt1:g(x)=x$) (o.w $g(x)=x-1$) is continuous and surjective and $[0,2]$ is closed, but is the domain closed, too? – Amir Naseri Apr 04 '16 at 13:36
  • @GEdgar Do you probably have a link to a theorem or a proof which shows if $Y\subseteq X$ and $g:Y\to X$ continuous and surjective, then $Y$ is closed in $X$ when $X$ is closed? – Amir Naseri Apr 04 '16 at 13:47
  • An example. $X$ is the unit circle in the complex plane, $Y$ the subset obtained by removing the one point $-1$, and $g : Y \to X$ defined by $g(z) = z^2$. Then $g$ is continuous and surjective, but $Y$ is not closed. So for this problem we have to somehow use the fact that $g$ is contractive, or that $f$ is expansive. – GEdgar Apr 04 '16 at 13:53
  • I would be so thankful if you completed the solution sometime and make me aware of it. – Amir Naseri Apr 04 '16 at 14:10
  • @GEdgar be aware that if you work in the complex plane with the sphere you're using the induced topology from $\mathbb{R}^2$ and then $\mathbb{S}^1\setminus{{-1}}$ is still closed. It is a very general fact that the preimage of a closed set through a continuous function is still closed, you can find it in every calculus or general topology book. – Tommaso Seneci Apr 04 '16 at 17:25
  • @TommasoSeneci ... I disagree. Writing $\mathbb{S}^1$ for the set of complex numbers of modulus $1$: Although $\mathbb{S}^1\setminus{{-1}}$ is closed in $\mathbb{S}^1\setminus{{-1}}$, it is not closed in $\mathbb{S}^1$ – GEdgar Apr 04 '16 at 17:32
  • Now I see what you pointed out, if the following function $d: f(X) \rightarrow \mathbb{R}$ ; $ x \rightarrow d(x,y)$, for $y\in X$, is continuous then it's done (I do not have enough hypothesis to claim it). Indeed if $y_n\in f(X) \rightarrow \tilde{y}\in X$ then $\exists x_n ; f(x_n)=y_n$ thus $\inf_{x\in X} d(f(x),\tilde{y})=0$ and by compactness argument on $X$ we could have the minimum, hence $d(f(\tilde{x}),\tilde{y})=0$. @GEdgar what fo you think? – Tommaso Seneci Apr 04 '16 at 19:48