Definition:Uniform Equivalence

Metric Spaces
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Then the mapping $f: A_1 \to A_2$ is a uniform equivalence of $M_1$ with $M_2$ iff $f$ is a bijection such that $f$ and $f^{-1}$ are both uniformly continuous.

Metrics
Let $A$ be a set on which there are two metrics imposed: $d_1$ and $d_2$.

Then $d_1$ and $d_2$ are uniformly equivalent iff the identity mapping of $A$ is uniformly $\left({d_1, d_2}\right)$-continuous and also uniformly $\left({d_2, d_1}\right)$-continuous.