# Definition:Quasiuniformity

## Definition

Let $S$ be a set.

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

 $(\text U 1)$ $:$ $\ds \forall u \in \UU:$ $\ds \Delta_S \subseteq u$ $(\text U 2)$ $:$ $\ds \forall u, v \in \UU:$ $\ds u \cap v \in \UU$ $(\text U 3)$ $:$ $\ds \forall u \in \UU:$ $\ds u \subseteq v \subseteq S \times S \implies v \in \UU$ $(\text U 4)$ $:$ $\ds \forall u \in \UU:$ $\ds \exists v \in \UU: v \circ v \subseteq u$

where:

$\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$
$\circ$ is defined as:
$u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$

That is, a quasiuniformity on $S$ is a filter on the cartesian product $S \times S$ (from $(\text U 1)$ to $(\text U 3)$) which also fulfils the condition:

$\forall u \in \UU: \exists v \in \UU$ such that whenever $\tuple {x, y} \in v$ and $\tuple {y, z} \in v$, then $\tuple {x, z} \in u$

which can be seen to be an equivalent statement to $(\text U 4)$.

$u \circ v$ 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

• Results about uniformities can be found here.