Arctangent pullback metric
= Arctangent pullback metric
{title2=$d(x,y)=|\arctan x-\arctan y|$}
This <metric> identifies $\mathbb R$ isometrically with the open interval $(-\pi/2,\pi/2)$ in the usual real distance. It induces the usual topology but is incomplete: the integer sequence has transformed limit $\pi/2$ outside that interval. This illustrates that completeness depends on the <metric>, not solely on its topology.