Let $(X, d)$ be a metric space. If $f \colon X \to X$ satisfies the condition
$$ d \big( f(x), f(y) \big) = d(x, y) $$
for all $x, y \in X$, then $f$ is called an isometry of $X$. Show that if $f$ is an isometry and $X$ is compact, then $f$ is bijective and hence a homeomorphism. [Hint: If $a \not\in f(X)$, choose $\varepsilon$ so that the $\varepsilon$-neighborhood of $a$ is disjoint from $f(X)$. Set $x_1 = a$, and $x_{n+1} = f\left( x_n \right)$ in general. Show that $d \left( x_n, x_m \right) \geq \varepsilon$ for $n \neq m$.]
My attempt:
Approach(1): let $f(x)=f(x’)$, where $x,x’\in X$. Then $d(f(x),f(x’))=d(x,x’)=0$ by isometry property. Which implies $x=x’$. Thus $f$ is injective. Now we show $f$ is continuous. Let $x\in X$. Given $\epsilon \gt 0$. Take $\delta=\epsilon$. If $\forall y\in X$ with $d(x,y)\lt \delta$ , then $d(f(x),f(y))=d(x,y)\lt \epsilon$. Thus $f$ is continuous, $\forall x\in X$.
Now we show $f$ is surjective. Assume towards contradiction, $f(X)\neq X$. So $\exists a\in X$ such that $a\in X-f(X)$. By theorem 26.5, $f(X)$ is compact. By theorem 26.3, $f(X)$ is closed in $X$. So $X-f(X)\in \mathcal{T}_X$. $\exists \epsilon \gt 0$ such that $B_d(a,\epsilon) \subseteq X- f(X)$. Define $x_1 =a$ and $x_{n+1}=f(x_n)$. It’s easy to check $x_{n+1}\in f(X)$, $\forall n\in \Bbb{N}$ by induction. Since $B_d(a,\epsilon) \cap f(X)=\emptyset$, $x_{n+1}\notin B_d(a,\epsilon), \forall n\in \Bbb{N}$. Which implies $d(a,x_{n+1})\geq \epsilon, \forall n\in \Bbb{N}$. Claim: $\forall k\in \Bbb{N}$, $d(x_k, x_n)\geq \epsilon$ $\forall n\in \Bbb{N} \setminus \{k\}$. Proof: Base case, $k=1$. $d(x_1, x_n)=d(a,x_n)\geq \epsilon$, $\forall n\in \Bbb{N} \setminus \{1\}$. Inductive step: Assume $d(x_l,x_n)\geq \epsilon$ $\forall n\in \Bbb{N}\setminus \{l\}$; $l\in \Bbb{N}$. Then $d(x_{l+1}, x_n)=d(f(x_l), f(x_{n-1}))=d(x_l, x_{n-1})\geq \epsilon, \forall n\in \Bbb{N}\setminus \{l+1\}$, last inequality follows from inductive hypothesis. By PMI, $d(x_k,x_n)\geq \epsilon, \forall \in \Bbb{N}\setminus \{k\}$ holds $\forall k\in \Bbb{N}$. So $\forall n,m\in \Bbb{N}$ with $n\neq m$, we have $d(x_n,x_m)\geq \epsilon \gt 0$. Which implies $x_n \neq x_m$, if $n\neq m$. By theorem 28.2, $X$ is sequentially compact. So $\{x_n\}$ have convergent subsequence. Let $\{x_{n_i}\}$ is subsequence of $\{x_n\}$ such that $ \{x_{n_i}\} \to x_0 \in X$. For $\epsilon /2$, $\exists N \in \Bbb{N}$ such that $d(x_{n_i},x_0) \lt \epsilon$, $\forall i\geq N$. Then $d(x_{n_N},x_{n_{N+1}})\leq d(x_{n_N},x_0)+d(x_0,x_{n_{N+1}})\lt \epsilon$. So $d(x_{n_N},x_{n_{N+1}})\lt \epsilon$; $n_N \lt n_{N+1}$. Thus we reach contradiction. Hence $f$ is subjective.
Now we show $f^{-1}$ is continuous. Since $f$ is bijective, $f$ is invertible. So $\exists !g:X\to X$ such that $f\circ g= id_X =g\circ f$. Given $\epsilon \gt 0$. Take $\delta =\epsilon$. Let $x,y \in X$ with $d(x,y)\lt \delta$. Since $x,y\in X$, $\exists i,j \in X$ such that $f(i)=x$ and $f(j)=y$. Then $d(g(x),g(y))=d(g(f(i)),g(f(j)))=d(i,j)=d(f(i),g(j))=d(x,y)\lt \epsilon$. Thus $g=f^{-1}$ is uniformly continuous. Which implies $g$ is continuous. Hence $f$ is bijective and homeomorphism. Is this proof correct?
Approach(2): In proof of $f$ is surjective, we can also contradict definition of limit point compact in place of sequential compact. By theorem 28.1, $X$ is limit point compact. $\forall k\in \Bbb{N}$, $d(x_k,x_n)\geq \epsilon, \forall \in \Bbb{N}\setminus \{k\}$. So $\forall n,m\in \Bbb{N}$ with $n\neq m$, we have $d(x_n,x_m)\geq \epsilon \gt 0$. Which implies $x_n \neq x_m$, if $n\neq m$. So $E=\{x_n |n\in \Bbb{N}\}$ is an infinite subset of $X$. By definition of limit point compact, $\exists x\in X$ such that $x\in E’$. $\forall V\in \mathcal{N}_x, V\cap E\neq \emptyset$. Let $V=B_d(x,\epsilon /2)\in \mathcal{N}_x$. By theorem 17.9, $|B_d(x,\epsilon /2)\cap E|=\infty$. So $\exists p,q \in B_d(x,\epsilon /2)\cap E$ such that $p\neq q$. Then $d(p,q)\leq d(p,x)+d(x,q)\lt \epsilon$ and $p,q \in E$. So $d(p,q)\lt \epsilon, p\neq q$. Thus we reach contradiction. Rest of the proof is similar to approach(1). Is this proof correct?
Approach(3): $f$ is surjective by above proof. By exercise 2 section 21(link: Exercise 2, Section 21 of Munkres’ Topology), $f$ is imbedding. Hence $f$ is bijective and homeomorphism.
This post is little bit different then Prob. 6, Sec. 28, in Munkres' TOPOLOGY, 2nd ed: An isometric self-map of a compact metric space is homeomorphism post.
Best Answer
First I will give an alternative proof:
Note that by Theorem 26.6, you don't actually need to directly check that $f^{-1}$ is continuous, it is enough to show that $f$ is a continuous bijection.
The function $f$ is continuous because it is an isometry. Indeed, for any $\epsilon > 0$, choose $\delta = \epsilon$. Then if $d(x,y) < \delta$, we have $d(f(x),f(y)) = d(x,y) < \delta = \epsilon$.
An isometry is immediately injective. As for subjectivity, assume to the contrary that there is an $a\in X$ such that $a\not\in f(X)$. Then there is an $\epsilon$-neighborhood around $a$ disjoint from $f(X)$. Define the sequence $(x_n)$ as in the hint. Since $x_k\in f(X)$ for all $k>1$ we have $d(x_1,x_k) \geq \epsilon$. Fix some positive integer $n$ and let $k = m - n + 1$ and apply the fact that $f$ is an isometry $n-1$ times. The above then becomes $d(x_n, x_m) \geq \epsilon$ whenever $n\neq m$.
Finally, since $X$ is a compact metric space, it is sequentially compact. Thus the sequence $(x)_n$ has a convergent subsequence. However, as shown above the distance between any two points in this subsequence must be larger than or equal to $\epsilon$, thus it cannot converge. This contradiction shows $f$ must be surjective.
All in all, $f$ is a continuous bijection, and per the first paragraph, $f$ is a homemorphism.
Now I will comment on your proof(s):
A general comment is that you overuse symbols, which makes the text difficult to read. This looks more like a proof sketch which was copied, rather than a completed proof. Even if completely logically correct, proofs written in this style are difficult to parse. Write in complete sentences. I looked at some of your other questions and this is a recurring situation.
A second general comment is that you do not always need to specify logical quantifiers if the context is obvious. In particular, your repeated use of the "for all" symbol.
All in all, your proof needs a lot of work. It's a mess to read. I'll try and cover it in detail, but keep in mind these comments are also comments on your general style, and do not just apply to these specific instances. I will also not suggest alternative strategies in this proof, since I covered that above. Thus I will only focus on clarifying the proof using your proof strategy.
You do not need to specify where $x$ and $x'$ come from, it is obvious from context.
This is better expressed as "since $f$ is an isometry". It's also not a correct English sentence. It would at least have to be "the isometry property".
This is a mess. You simply have to show that if two distinct points are delta-close, their images are epsilon-close. You instead fix your $x$, and check every $y$, which is equivalent, but a lot more messy. The penultimate sentence is also malformed, since it reads: "If for all y in X with the distance from x to y being less than delta, then ...". Again, there is that overuse of the "for all" symbol at the end.
The context is already clear here. You do not need to specify that $a\in X - f(X)$. Simply saying that $a \not\in f(X)$ is enough.
It's generally better to write "is open" rather than use set membership.
Do not start a sentence with mathematical symbols. Also, instead of using symbols here you should simply write "there is an epsilon-ball around $a$ disjoint from $f(X)$. It's the exact same statement as what you wrote, but is much easier to parse.
First, you do not need to specify where the $n$s live. It is clear from context. Second, this is not shown by induction. Every point $x_k$ lives in $X$, so its image will live in $f(X)$. You can use induction, but it is overkill.
Please, oh please stop cramming "∀n∈N" in everywhere. The reader understood it the first time, and would have understood it the zeroth time. This is the last time I will comment on it.
Generally inserting claims in the middle of proofs is bad style. Not always, but generally. It is bad style in this case. What you wrote is also much clearly expressed as "$d(x_k, x_n) \geq \epsilon$ whenever $k\neq n$. Again, logically equivalent but easier to read.
Explicitly labelling base cases and inductive steps is not something that is done.
I know what you're trying to do here, but writing $l\in\mathbb{N}$ makes it seem like you've shown it for all $l$ already, which you haven't. Also, you reference $n-1$ which might be 0, which is an issue. This needs to be fixed.
It does, but you never use this, so why include it?
Spelling error. have -> has
This should be "Let ... be a subsequence".
You explicitly choose epsilons here, which is fine, but a bit overkill. You already know different points in the sequence are "far" apart, so its impossible for any subsequence to converge, for the subsequence will consist of point that are "far" apart.
This is mostly fine, if a bit wordy. It is certainly uniformly continuous, but you only need continuity, so settle for that. You do not need to mention uniform continuity.
Your proof is logically correct, but a mess in every other way. Now on to approach 2.
I'm assuming here you're using the same strategy as in the first approach to show this? It's not clear exactly.
This seems correct, yes. Again, horrible to parse. You need to define your symbols.
As for approach 3, it is correct, although you still need to show surjectivity, injectivity, and continuity of $f$. You only really save on showing that the inverse is continuous.