Definition:Metrizable Uniformity

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Let $\mathcal U$ be the uniformity on $X$ defined as:
 * $\mathcal U := \left\{{u_\epsilon: \epsilon \in \R^*_+}\right\}$

where:
 * $\R^*_+$ 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 defined as metrizable.

Also see

 * See Metric Space Generates a Uniformity for demonstration that $\mathcal U$ is indeed a uniformity.