Separation axioms: Representation theorems, compactness, and compactifications (Q2702347)

The chapter under review is primarily based on the author's earlier paper in [Fuzzy Sets Syst. 73, No. 1, 55-87 (1995; Zbl 0867.54007)]. All proofs not given in the chapter under review are found in [op. cit.]. Sections 1, 3, 4, 5, and 7 coincide with the corresponding sections in [op. cit.], so that it is appropriate to restrict this review to the new material added in Sections 2, 6, and 8, and to the original Section 9.NEWLINENEWLINENEWLINEIn Section 2, new material is added in Discussion 2.15 which establishes a categorical relationship between \(L\)-sobriety and classical sobriety, and in Discussion 2.16 which looks at the \(L\)-2-sobrification functor as a fuzzification functor. Some applications to the fuzzy real lines are given. In Section 6, new material includes an alternative description of the equivalence between the categories of compact Hausdorff spaces and compact regular \(L\)-sober spaces. Also, the latter category is compared with the category of compact Hausdorff-separated \(L\)-topological spaces in the sense of \textit{U. Höhle} and \textit{A. P. Šostak} [Axiomatic foundations of fixed-basis fuzzy topology, Handbook, Fuzzy Sets Ser. 3, 123-272 (1999)]. These categories coincide if \(L\) is a complete Boolean algebra. In Section 8, an error of Proposition 8.7(3) in [the author, op. cit.] is corrected by means of reworked Definition 8.6 and Proposition 8.7. New material is added in Discussions 8.14-8.15 in which the author provides a detailed object-level comparison of compact Hausdorff with its -- mentioned above -- categorical equivalent of compact regular \(L\)-sober spaces, including applications to the fuzzy unit interval. Section 9 generalizes the compactifications given in Section 8 and in [the author, op. cit.] to variable-basis fuzzy topology in the sense of [the author, Handbook, Fuzzy Sets Ser. 3, 273-388 (1999; Zbl 0968.54003), see above], giving the first compactification reflectors for variable-basis fuzzy topology.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].