# Category:Definitions/Uniformities

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:

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

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

... 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.

## Subcategories

This category has only the following subcategory.

## Pages in category "Definitions/Uniformities"

The following 16 pages are in this category, out of 16 total.