Definition:Uniform Equivalence/Metrics
< Definition:Uniform Equivalence(Redirected from Definition:Uniformly Equivalent Metrics)
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Definition
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.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.8$: Compactness and Uniform Continuity: Definition $5.8.4$