# I have to think of a metric that makes (0,1) an unbounded interval

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

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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
@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