Pseudometric Space generates Uniformity

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

Let $\mathcal U$ be the set of sets 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 a uniformity on $X$ which generates a uniform space with the same topology as the topology induced by $d$.