Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)\le K<\infty$ for all $x,y\in X$. Is there a standard way to convert $\rho$ into another metric $\widetilde{d}$ on $X$ with the property that $\widetilde{d}(x,y)\to\infty$ if and only if $d(x,y)\to 1$? One way would be to find some function $f$ such that $\widetilde{d}(x,y)=f(d(x,y))$ satisfies the given conditions, but it is not obvious that this is always possible.

The following properties are additionally useful, but not necessary:

• $\widetilde{d}$ preserves the topology of $(X,d)$
• $d(x,y)>d(x',y')\implies \widetilde{d}(x,y)>\widetilde{d}(x',y')$

Also, if this is possible when $(X,d)$ satisfies certain extra assumptions but not in general, answers in this direction are welcome.