Axiom:Quasiuniformity Axioms

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Definition

Let $S$ be a set.

The quasiuniformity 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 quasiuniformity:

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

where:

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


Also see

  • Results about uniformities can be found here.


Sources