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.