Category:Definitions/Metrizable

This category contains definitions related to Metrizable.
Related results can be found in Category:Metrizable.

Metrizable Topology

$T$ is said to be metrizable if and only if there exists a metric $d$ on $S$ such that:

$\tau$ is the topology induced by $d$ on $S$.

Metrizable Uniformity

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.