Axiom:Uniformity Axioms

Definition
Let $S$ be a set.

The uniformity axioms are the conditions on a set of subsets $\mathcal U$ of the cartesian product $S \times S$ which are satisfied for all elements of $\mathcal U$ in order to make $\mathcal U$ a uniformity:

where:
 * $\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \left\{{\left({x, x}\right): x \in S}\right\}$


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


 * $u^{-1}$ is defined as:
 * $u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in u}\right\}$
 * That is, all elements of $\mathcal U$ are symmetric.

Also see

 * Definition:Uniformity
 * Definition:Entourage