Definition:Metrizable Uniformity

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

 * Metric Space generates Uniformity