Complete nearness \(\sigma\)-frames (Q1046762)

A \(\sigma \)-frame \(L\) is a bounded lattice admitting countable joins satisfying the distribution law \(x\wedge (\bigvee Y)=\bigvee \{x\wedge y:y\in Y\}\) for each \(x\in L\) and any countable \(Y\subseteq L.\) \textit{J. Walters} [Commentat. Math. Univ. Carol. 32, No.~1, 189--198 (1991; Zbl 0735.54014)] first considered structures on a \(\sigma \)-frame, in particular those of uniformity and proximity, and in her Ph.D. Thesis complete uniform \(\sigma \)-frames were characterized. This notion was modelled for metric \(\sigma \)-frames by \textit{M. Vojdani Tabatabaee} and \textit{M. Mehdi Ebrahimi} [Appl. Categ. Struct. 11, No. 2, 135--146 (2003; Zbl 1031.06006)]. The author's papers [Czech. Math. J. 56, No.~4, 1229--1241 (2006; Zbl 1164.06312)] and [Quaest. Math. 30, No.~2, 133--145 (2007; Zbl 1138.06005)] introduced the structure of a nearness on a \(\sigma \)-frame as a generalization of that of uniformity. In this paper, the author looks at the corresponding notion of completeness in the setting of nearness \(\sigma \)-frames. He shows that complete strong nearness \(\sigma \)-frames are exactly the nonzero parts of complete separable strong Lindelöf nearness frames. Also the author relates nearness \(\sigma \)-frames and metric \(\sigma \)-frames and shows that every metric \(\sigma \)-frame admits an admissible nearness such that it is complete as a metric \(\sigma \)-frame if and only if it is complete in this admissible nearness.