Metric Space generates Uniformity

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Let $M = \struct {A, d}$ be a metric space.

Let $\UU$ be the set of sets defined as:

$\UU := \set {u \in A \times A: \exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u}$


$\R_{>0}$ 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 $d$.


From definition it is clear that a metric space is a pseudometric space.

The result then follows from Pseudometric Space generates Uniformity.