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I don't even know how to begin to answer this question?

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2 Answers

up vote 12 down vote accepted

The trick is to make $0$ and $1$ infinitely far apart (even though neither of them belongs to the interval). Consider the map $f\colon(0,1)\to\mathbb R$, $x\mapsto \frac1x$ and let $d(x,y)=|f(x)-f(y)|$. Then $d(\epsilon,1-\epsilon) $ becomes arbitrarily big.

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Think of any homeomorphism from $(0,1)$ to $\mathbb{R}$

One diffeomorphism from $\mathbb{R}$ to $(0,1)$ would be $$ x\mapsto \frac{1}{2}\cdot \frac{x}{1+ |x|}+\frac{1}{2}$$

So the diffeomorphism from $(0,1)$ $$\frac{2y-1}{y}=2-\frac{1}{y}$$ for $x\in (0,\frac{1}{2})$ and $$\frac{1-2y}{2y-2}$$ for $x\in [\frac{1}{2},1)$

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I suspect if he is having trouble with this, he won't know what a homeomorphism is. –  Thomas Andrews Mar 6 '13 at 21:31
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@ThomasAndrews I expect the OP to post if he don't unterstand a solution, I still don't think it is a reason to downvote the answer –  Dominic Michaelis Mar 6 '13 at 21:47
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I don't mean to be rude. When I saw the answer, I thought, "well, that can't be possibly helpful to anybody who asked this question." The fact that you've edited after giving a quick and un-helpful answer makes me wonder why you posted your original unhelpful answer. Were you just trying to post quickly, with the intent of posting a helpful answer later, or did my criticism spur you to change your answer? In either event, my criticism for your original answer stands, and down votes are not just for wrong answers. –  Thomas Andrews Mar 6 '13 at 22:06
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And I don't suspect "diffeomorphism" is in the vocabulary of a person asking a basic metric space question, either. You don't even need a diffeomorphism. You are shooting an ant with a cannon. –  Thomas Andrews Mar 6 '13 at 22:10
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True, but anybody seeking this answer is likewise not going to know what a diffeo- or homeo- is, either. (You and I and the other answerer assumed something about the posters question - you don't even need continuity as the question stands, but I think it is right to assume that was the intent of the problem.) –  Thomas Andrews Mar 6 '13 at 22:46
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