Quantifying completion (Q1578975)

A set \(X\) together with a family \(({\mathcal U}_\varepsilon)_{ \varepsilon \in\mathbb{R}^+}\) of filters on \(X\times X\) is called an approach uniform space if the following conditions are satisfied: (U1) for each \(\varepsilon\in \mathbb{R}^+\) and for each \(U\in {\mathcal U}_\varepsilon\) the diagonal \(\Delta_X\) is contained in \(U\); (U2) \(U^{-1}\in {\mathcal U}_\varepsilon\) for each \(\varepsilon\in\mathbb{R}^+\) and for each \(U\in {\mathcal U}_\varepsilon\); (U3) \({\mathcal U}_{\varepsilon +\varepsilon'} \subset{\mathcal U}_\varepsilon \circ{\mathcal U}_{ \varepsilon'}\) for each \(\varepsilon\), \(\varepsilon' \in\mathbb{R}^+\); (U4) \({\mathcal U}_\varepsilon= \bigcup\{{\mathcal U}_\alpha \mid\alpha> \varepsilon\}\) for each \(\varepsilon\in\mathbb{R}^+\). With morphisms defined properly the category \textbf{AUnif} of approach uniform spaces becomes a topological category which contains the category \textbf{Univ} of uniform spaces both reflectively and coreflectively. It was introduced by the same authors in [ibid. 21, No. 1, 1-18 (1998; Zbl 0890.54024)]. In this paper they continue their investigations by developing a natural theory of completeness for approach uniform spaces.