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Are there any unbounded metrics on $(0,1)$? I've tried to use $d_1(x,y)=\tan(\frac{\pi d(x,y)}{2})$, where $d(x,y)=|x-y|$, but I couldn't prove the triangle inequality, so I don't really know whether it is a metric or not.

easternland
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    $(0, 1)$ is homeomorphic to $\mathbb{R}$, so pick any such homeomorphism and transfer the usual metric on $\mathbb{R}$ to it. – Qiaochu Yuan Nov 28 '20 at 08:09
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    In a similar vein, $d_2(x,y) = | \tan(\frac{\pi x}2) - \tan(\frac{\pi y}2) |$ is exactly such a metric as in @QiaochuYuan's suggestion (although it uses a homeomorphism $x\mapsto\tan(\frac{\pi x}2)$ from $(0,1)$ to $(0,\infty)$ rather than to $\Bbb R$), which might be what you (the OP) were thinking. Another such example would be $d_3(x,y) = |\log x - \log y|$. – Greg Martin Nov 28 '20 at 08:52
  • Note that the transfer function being homeomorphic has no bearing on the prolem. Any bijection between $(0,1)$ and and unbounded set of real numbers does the trick. – Ingix Nov 28 '20 at 10:08

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