 Equivalence of 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.

The equivalence of metrics is a concept in metric spaces that refers to the idea that two different metrics define the same topology on a set. In more formal terms, two metrics \( d_1 \) and \( d_2 \) on a set \( X \) are said to be equivalent if they induce the same notions of convergence, continuity, and open sets.