Metric Space Generates a Uniformity
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Theorem
Let $M = \left({A, d}\right)$ be a metric space.
Let $\mathcal U$ be the set of sets defined as:
- $\mathcal U := \left\{{u_\epsilon: \epsilon \in \R_{>0}}\right\}$
where:
- $\R_{>0}$ is the set of strictly positive real numbers
- $u_\epsilon$ is defined as:
- $u_\epsilon := \left\{{\left({x, y}\right): d \left({x, y}\right) < \epsilon}\right\}$
Then $\mathcal U$ is a uniformity on $X$ which generates a uniform space with the same topology as the topology induced by $d$.
Proof
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$: Metric Uniformities
