Definition:Quasiuniform Space

Definition
Let $\UU$ be a quasiuniformity on a set $S$.

Then a topology $\tau$ can be created from $\UU$ by:
 * $\tau := \set {\map u x: u \in \UU, x \in S}$

where:
 * $\forall x \in S: \map u x := \set {y: \tuple {x, y} \in u}$

The resulting topological space $T = \struct {S, \tau}$ is called a quasiuniform space.

It can be denoted $\struct {\struct {S, \UU}, \tau}$, or just $\struct {S, \UU}$ if it is understood that $\tau$ is the topology created from $\UU$.

Also see

 * Quasiuniformity Induces Topology for a proof that $\tau$ is indeed a topology.