Definition:Metrizable Uniformity

From ProofWiki
Jump to navigation Jump to search

Definition

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

Let $\UU$ be the uniformity on $X$ 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:
$u_\epsilon := \set {\paren {x, y}: \map d {x, y} < \epsilon}$


Then $\UU$ is defined as metrizable.


Also see


Linguistic Note

The British English spelling for metrizable is metrisable, but it is rarely found.


Sources