Definition:Metrizable Uniformity
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Definition
Let $M = \struct {A, d}$ be a metric space.
Let $\UU$ be the uniformity on $X$ defined as:
- $\UU := \set {u_\epsilon: \epsilon \in \R_{>0} }$
where:
- $\R_{>0}$ is the set of strictly positive real numbers
- $u_\epsilon$ is defined as:
- $u_\epsilon := \set {\paren {x, y}: \map d {x, y} < \epsilon}$
Then $\UU$ is defined as metrizable.
Also see
Linguistic Note
The British English spelling for metrizable is metrisable, but it is rarely found.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metric Uniformities