On strict extensions of nearness spaces (Q1840747)

Let \((X,\mu)\) be a nearness space with underlying \(T_1\)-topology \(\tau_\mu\), and let \(X^*\) be any collection of round Cauchy filters on \(X\) in the sense of \textit{D. Harris} [Structures in topology, Mem. Am. Math. Soc. 115 (1971; Zbl 0239.54020)] containing all neighbourhood filters of \(\tau_\mu\). It is shown that there is a natural nearness structure \(\mu^*\) on \(X^*\) which has the following properties: (i) the underlying topology of \((X^*,\mu^*)\) is \(T_1\); (ii) \((X,\mu)\) can be embedded into \((X^*,\mu^*)\) by identifying points in \(X\) with its neighbourhood filters; (iii) \((X^*,\mu^*)\) is a strict extension of \((X,\mu)\). It turns out that if \(X^*\) is the collection of all round Cauchy filters on \(X\), then \((X^*,\mu^*)\) is isomorphic to the strict completion of \((X,\mu)\) introduced by \textit{H. Herrlich} [General Topol. Appl. 4, 191-212 (1974; Zbl 0288.54004)]. Moreover, if \(X^*\) is the collection of all Morita-generated filters on \((X,\mu)\), then \((X^*,\mu^*)\) is isomorphic to the simple extension defined by \textit{K. Morita} in [Proc. Japan Acad. 27, 65-72 (1951; Zbl 0042.41203); No. 3, 130-137; No. 4, 166-171 (1951; Zbl 0043.16504); 632-636 (1951; Zbl 0045.11702)]. Using this approach, the completeness concepts for nearness spaces introduced by K. Morita, H. Herrlich, and J. W. Carlson are compared. They are shown to be equivalent in regular nearness spaces.