Definition:Uniformity

Definition
Let $S$ be a set.

A uniformity on $S$ is a set of subsets $\mathcal U$ of the cartesian product $S \times S$ satisfying the 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: $\forall u \in \mathcal U: \exists v \in \mathcal U: v \circ u \subseteq u$ where $\circ$ is defined as:
 * $u \circ v := \left\{{\left({x, z}\right): \exists y \in S: \left({x, y}\right) \in v, \left({y, z}\right) \in v}\right\}$

... and also:
 * U5: $\forall u \in \mathcal U: \exists u^{-1} \in \mathcal U$ where $u^{-1}$ is defined as:
 * $u^{-1} := \left\{{\left({y, x}\right): \left({y, x}\right) \in \mathcal U}\right\}$

These five axioms are together known as the uniformity axioms.