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:

$(\text U 1)$	$:$	$\ds \forall u \in \UU:$	$\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 $
$(\text U 5)$	$:$	$\ds \forall u \in \UU:$	$\ds u^{-1} \in \UU $

where:

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

$\circ$ is defined as:

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

$u^{-1}$ is defined as:

$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$

That is, all elements of $\UU$ are symmetric.

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Uniformities

but beware the error: $u^{-1}$ is defined as:

$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in \UU}$

\((\text U 1)\)	$:$	\(\ds \forall u \in \UU:\)	\(\ds \Delta_S \subseteq u \)
\((\text U 2)\)	$:$	\(\ds \forall u, v \in \UU:\)	\(\ds u \cap v \in \UU \)
\((\text U 3)\)	$:$	\(\ds \forall u \in \UU:\)	\(\ds u \subseteq v \subseteq S \times S \implies v \in \UU \)
\((\text U 4)\)	$:$	\(\ds \forall u \in \UU:\)	\(\ds \exists v \in \UU: v \circ v \subseteq u \)
\((\text U 5)\)	$:$	\(\ds \forall u \in \UU:\)	\(\ds u^{-1} \in \UU \)