Axiom:Uniformity Axioms

Definition
Let $S$ be a set.

The uniformity 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 uniformity:

where:
 * $\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$


 * $\circ$ is defined as:
 * $u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in v}$


 * $u^{-1}$ is defined as:
 * $u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$
 * That is, all elements of $\UU$ are symmetric.

Also see

 * Definition:Uniformity
 * Definition:Entourage