# Pseudometric Space generates Uniformity

From ProofWiki

## Theorem

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

Let $\mathcal U$ be the set of sets defined as:

- $\mathcal U := \left\{{u_\epsilon: \epsilon \in \R_{>0}}\right\}$

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

## Sources

- Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(1970)... (previous)... (next): $\text{I}: \ \S 5$: Metric Uniformities