Definition:Quasiuniformity
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Definition
Let $S$ be a set.
A quasiuniformity on $S$ is a set of subsets $\UU$ of the cartesian product $S \times S$ satisfying the following 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:
- $u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$
That is, a quasiuniformity on $S$ is a filter on the cartesian product $S \times S$ (from $(\text U 1)$ to $(\text U 3)$) which also fulfils the condition:
- $\forall u \in \UU: \exists v \in \UU$ such that whenever $\tuple {x, y} \in v$ and $\tuple {y, z} \in v$, then $\tuple {x, z} \in u$
which can be seen to be an equivalent statement to $(\text U 4)$.
$u \circ v$ in this context can be seen to be equivalent to composition of relations.
Thus a quasiuniformity on $S$ is a filter on $S \times S$ which also fulfils the condition that every element is the composition of another element with itself.
Also see
- Results about uniformities can be found here.
Sources
- 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