# Quasiuniformity Induces Topology

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## Theorem

Let $\UU$ be a quasiuniformity on a set $S$.

Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$, created from $\UU$ by:

- $\tau := \set {\map u x: u \in \UU, x \in S}$

where:

- $\forall x \in S: \map u x = \set {y: \tuple {x, y} \in u}$

That is, where $\map u x$ is the image of $x$ under $u$, where $u$ is considered as a relation on $S$.

Then $\tau$ is a topology on $S$.

That is, the quasiuniform space $\struct {\struct {S, \UU}, \tau}$ is also the topological space $\struct {S, \tau}$.

## Proof

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## 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