 Strong equivalence of two metrics implies topological equivalence, but not vice versa.An intuitive reason why topological equivalence does not imply strong equivalence is that bounded sets under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.When the two metrics formula_33 are those induced by norms formula_34 respectively, then strong equivalence is equivalent to the condition that, for all formula_11,

In finite dimensional spaces, all metrics induced by the p-norm, including the euclidean metric, the taxicab metric, and the Chebyshev distance, are strongly equivalent.