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:


 * U1: $\forall u \in \mathcal U: \Delta_S \subseteq u$, where $\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \left\{{\left({x, x}\right): x \in S}\right\}$


 * U2: $\forall u, v \in \mathcal U: u \cap v \in \mathcal U$


 * U3: $u \in \mathcal U, u \subseteq v \subseteq S \times S \implies v \in \mathcal U$


 * U4: $\forall u \in \mathcal U: \exists v \in \mathcal U: v \circ v \subseteq u$ where $\circ$ is defined as:
 * $u \circ v := \left\{{\left({x, z}\right): \exists y \in S: \left({x, y}\right) \in v, \left({y, z}\right) \in v}\right\}$

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

 * Uniformity
 * Entourage