What properties of functions preserve the topology of standard metric, such that $f(|x-y|)=d_2(x,y)$?
One example is $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ where $f(d) = \frac{d}{1+d}$, but can that statement be made more general to allow $f$ to be any continuous function? Or any differentiable function? Or any bijective function over $[0. \infty)$? Or otherwise, what set of known conditions?
Two metrics are equivalent iff they have the same convergent sequences with the same limits. Hence if $f(|x-y|)$ is a metric then it is equivalent to the usual metric iff $\lim_{t \to 0} f(t)=0$ and, conversely, $f(t) \to 0$ implies $t \to 0$.
$f(|x-y|)$ is a metric if it satisfies the following conditions:
$f(x) \geq 0$ for all $x$,
$f(x)=0$ iff $x=0$
$f$ is non-decreasing
$f(x+y) \leq f(x)+f(y)$ for all $x,y$.