\begin{definition}[Definition:Quasiuniformity Axioms]
Let $S$ be a set.
The '''quasiuniformity axioms''' are the conditions on a set of subsets $\UU$ of the cartesian product $S \times S$ which are satisfied for all elements of $\UU$ in order to make $\UU$ a quasiuniformity:
{{begin-axiom}}
{{axiom | n = \text U 1
        | q = \forall u \in \UU
        | m = \Delta_S \subseteq u
}}
{{axiom | n = \text U 2
        | q = \forall u, v \in \UU
        | m = u \cap v \in \UU
}}
{{axiom | n = \text U 3
        | q = \forall u \in \UU
        | m = u \subseteq v \subseteq S \times S \implies v \in \UU
}}
{{axiom | n = \text U 4
        | q = \forall u \in \UU
        | m = \exists v \in \UU: v \circ v \subseteq u
}}
{{end-axiom}}
where:
:$\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$
:$\circ$ is defined as:
::$u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$
\end{definition}