# Definition:Uniformity

## Definition

Let $S$ be a set.

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

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

where:

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

and also:

$(\text U 5): \forall u \in \UU: u^{-1} \in \UU$ where $u^{-1}$ is defined as:
$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$
That is, all elements of $\UU$ are symmetric.

These five axioms are together known as the uniformity axioms.

## Also see

• Results about uniformities can be found here.

## Sources

but beware the error: $u^{-1}$ is defined as:
$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in \UU}$