Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are uniformly continuous. And $d_1$ and $d_2$ are strongly equivalent if there exist constants $\alpha,\beta>0$ such that $\alpha d_1(x,y)\leq d_2(x,y)\leq\beta d_1(x,y)$ for all $x,y\in X$.

Now if $d_1$ and $d_2$ are strongly equivalent, then they are uniformly equivalent and they have the same bounded sets. My question is, is the converse true? That is, if $d_1$ and $d_2$ are uniformly equivalent and have the same bounded sets, then are they strongly equivalent?

If not, is there an example of metrics which are uniformly equivalent and have the same bounded sets but are not strongly equivalent?


Try $[0,1]$ with $d_1(x,y) = |x-y|$ and $d_2(x,y) = |x^2 - y^2|$.

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  • $\begingroup$ Is there an example where the metrics are not bounded? $\endgroup$ – Keshav Srinivasan Dec 23 '18 at 17:12
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    $\begingroup$ Try $[0,\infty)$ with $d_2(x,y) = |f(x) - f(y)|$ where $f(x) = x^2/(1+x)$. $\endgroup$ – Robert Israel Dec 23 '18 at 20:27

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