Constructing the sobrification of an approach space via bicompletion (Q949866)

Approach spaces form the missing link in the topology-uniformity-metric triad (R. Lowen). \textit{B. Banaschewski, R. Lowen} and \textit{C. Van Olmen} [Topology Appl., 153, No.~16, 3059--3070 (2006; Zbl 1114.54007)] have introduced a notion of sobriety for such approach spaces, modelled after the corresponding concept in the category of topological spaces. In the note under review based on the classical Császár-Pervin quasi-uniform space, the authors construct a compatible quasi-uniform gauge structure on every approach space \(X\). Via bicompletion of this gauge space of a \(T_0\) approach space \(X\) the sobrification of \(X\) is constructed, retrieving especially the old result of \textit{G. C. L. Brümmer} and \textit{H. P. A. Künzi} [Math. Proc. Camb. Philos. Soc. 101, 237--247 (1986; Zbl 0618.54024)] on the sobrification of a \(T_0\) topological space.