Definition:Uniform Equivalence/Metric Spaces
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Definition
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.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.8$: Compactness and Uniform Continuity: Definition $5.8.5$