Definition:Quasiuniform Space
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Definition
Let $\UU$ be a quasiuniformity on a set $S$.
Then a topology $\tau$ can be created from $\UU$ by:
- $\tau := \set {\map u x: u \in \UU, x \in S}$
where:
- $\forall x \in S: \map u x := \set {y: \tuple {x, y} \in u}$
The resulting topological space $T = \struct {S, \tau}$ is called a quasiuniform space.
It can be denoted $\struct {\struct {S, \UU}, \tau}$, or just $\struct {S, \UU}$ if it is understood that $\tau$ is the topology created from $\UU$.
Also see
- Quasiuniformity Induces Topology for a proof that $\tau$ is indeed a topology.
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