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This category contains definitions related to Uniformities.
Related results can be found in Category:Uniformities.

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:

\((U1)\)   $:$     \(\displaystyle \forall u \in \mathcal U:\) \(\displaystyle \Delta_S \subseteq u \)             
\((U2)\)   $:$     \(\displaystyle \forall u, v \in \mathcal U:\) \(\displaystyle u \cap v \in \mathcal U \)             
\((U3)\)   $:$     \(\displaystyle u \in \mathcal U:\) \(\displaystyle u \subseteq v \subseteq S \times S \implies v \in \mathcal U \)             
\((U4)\)   $:$     \(\displaystyle \forall u \in \mathcal U:\) \(\displaystyle \exists v \in \mathcal U: v \circ v \subseteq u \)             


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

... and also:

$(U5)$: $\forall u \in \mathcal U: \exists u^{-1} \in \mathcal U$ where $u^{-1}$ is defined as:
$u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in u}\right\}$
That is, all elements of $\mathcal U$ are symmetric.