Definition:Quasiuniform Space

Definition
Let $\mathcal U$ be a quasiuniformity on a set $S$.

Then a topology $\tau$ can be created from $\mathcal U$ by:
 * $\tau := \left\{{u \left({x}\right): u \in \mathcal U, x \in S}\right\}$

where:
 * $\forall x \in S: u \left({x}\right) := \left\{{y: \left({x, y}\right) \in u}\right\}$

The resulting topological space $T = \left({S, \tau}\right)$ is called a quasiuniform space.

It can be denoted $\left({\left({S, \mathcal U}\right), \tau}\right)$, or just $\left({S, \mathcal U}\right)$ if it is understood that $\tau$ is the topology created from $\mathcal U$.

Also see

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