Definition:Quasiuniformity

Definition
Let $S$ be a set.

A quasiuniformity on $S$ is a set of subsets $\mathcal U$ of the cartesian product $S \times S$ satisfying the following quasiuniformity axioms:

That is, a quasiuniformity on $S$ is a filter on the cartesian product $S \times S$ (from $(U1)$ to $(U3)$) which also fulfils the condition:


 * $\forall u \in \mathcal U: \exists v \in \mathcal U$ such that whenever $\left({x, y}\right) \in v$ and $\left({y, z}\right) \in v$, then $\left({x, z}\right) \in u$

which can be seen to be an equivalent statement to $(U4)$.

$u \circ v$ in this context can be seen to be equivalent to composition of relations.

Thus a quasiuniformity on $S$ is a filter on $S \times S$ which also fulfils the condition that every element is the composition of another element with itself.

Also see

 * Definition:Uniformity
 * Definition:Entourage