Bounded metric transform
= Bounded metric transform
{title2=$d_b(x,y)=d(x,y)/(1+d(x,y))$}
The increasing function $\phi(s)=s/(1+s)$ is subadditive on nonnegative arguments: each summand $b/(1+b)$ and $c/(1+c)$ is at least its numerator divided by $1+b+c$. Thus $\phi\circ d$ is a <metric> whenever $d$ is. It is bounded and induces the same topology because $\phi$ and its inverse near zero preserve convergence to zero.