Equivalent metrics

Two metrics are equivalent when they induce the same topology. Global two-sided constant bounds imply equivalence, but equivalence alone need not supply such bounds.

On $\mathbb{R}$, the usual metric and $|\arctan x-\arctan y|$ induce the same topology, but their ratio along $(0,n)$ tends to zero. Thus no positive global lower comparison constant exists.

Topological equivalence does not determine uniform convergence. For the usual and arctangent metrics on $\mathbb{R}$, maps that change the value $n$ to $2n$ at a single domain point have arctangent sup-error tending to zero while their usual sup-error diverges.