Pseudometric Space generates Uniformity

Theorem
Let $P = \struct {A, d}$ be a pseudometric space.

Let $\UU$ be the set of sets 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:
 * $u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$

Then $\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$.