Category:Examples of Uniformities
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This category contains examples of Uniformity.
Let $S$ be a set.
A uniformity on $S$ is a set of subsets $\UU$ of the cartesian product $S \times S$ satisfying the quasiuniformity axioms:
\((\text U 1)\) | $:$ | \(\ds \forall u \in \UU:\) | \(\ds \Delta_S \subseteq u \) | ||||||
\((\text U 2)\) | $:$ | \(\ds \forall u, v \in \UU:\) | \(\ds u \cap v \in \UU \) | ||||||
\((\text U 3)\) | $:$ | \(\ds \forall u \in \UU:\) | \(\ds u \subseteq v \subseteq S \times S \implies v \in \UU \) | ||||||
\((\text U 4)\) | $:$ | \(\ds \forall u \in \UU:\) | \(\ds \exists v \in \UU: v \circ v \subseteq u \) |
where:
- $\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$
- $\circ$ is defined as:
- $u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$
and also:
- $(\text U 5): \forall u \in \UU: u^{-1} \in \UU$ where $u^{-1}$ is defined as:
- $u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$
- That is, all elements of $\UU$ are symmetric.
Pages in category "Examples of Uniformities"
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