Definition:Pseudometrizable Uniformity

Theorem

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

Let $\UU$ be the uniformity on $A$ 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 defined as pseudometrizable.

Also see

• Pseudometric Space generates Uniformity

Linguistic Note

The British English spelling for pseudometrizable is pseudometrisable, but it is rarely found.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metric Uniformities