Quasiuniformity Induces Topology

Theorem
Let $\mathcal U$ be a quasiuniformity on a set $X$.

Let $\vartheta \subseteq \mathcal P \left({S}\right)$ be a subset of the power set of $S$, be created from $\mathcal U$ by:
 * $\vartheta := \left\{{u \left({x}\right): u \in \mathcal U, x \in X}\right\}$

where:
 * $\forall x \in X: u \left({x}\right) = \left\{{y: \left({x, y}\right) \in u}\right\}$

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

That is, the quasiuniform space $\left({\left({X, \mathcal U}\right), \vartheta}\right)$ is also the topological space $\left({X, \vartheta}\right)$.