Definition:Uniform Equivalence

Definition

Metric Spaces

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Then the mapping $f: A_1 \to A_2$ is a uniform equivalence of $M_1$ with $M_2$ if and only if $f$ 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 if and only if the identity mapping of $A$ is uniformly $\tuple {d_1, d_2}$-continuous and also uniformly $\tuple {d_2, d_1}$-continuous.