Definition:Quasiuniform Space

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

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

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

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

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

Also see

 * Quasiuniformity Yields a Topology for a proof that $\vartheta$ is indeed a topology.