Two decades of fuzzy topology: Basic ideas, notions, and results.

*(English. Russian original)*Zbl 0716.54004
Russ. Math. Surv. 44, No. 6, 125-186 (1989); translation from Usp. Mat. Nauk 44, No. 6(270), 99-147 (1989).

The author’s intention in this expository article is “to make the reader familiar with basic ideas and categories of fuzzy topology, to present more or less systematically basic concepts, constructions and results of this field, and to discuss directions in which it is being developed.” The sections are: 0. Preliminaries: fuzzy sets; 1. Fuzzy topological spaces: basic categories of fuzzy topology; 2. Basic relationships between the category TOP of topological spaces and categories of fuzzy topology; 3. Local structure of fuzzy topological spaces; 4. Structures of convergence in fuzzy spaces; 5. Separation in fuzzy spaces; 6. Normality and complete regularity type properties in fuzzy topology; 7. Compactness in fuzzy topology; 8. Connectedness of fuzzy spaces; 9. Fuzzy metric spaces and metrizability of fuzzy spaces; 10. The fuzzy real line \({\mathcal F}(R)\) and its subspaces; 11. A fuzzy modification of a linearly ordered space; 12. A fuzzy-probabilistic modification of a topological space; 13. The interval-fuzzy line; 14. On hyperspaces of fuzzy sets; 15. Another look at the object of fuzzy topology and some of its categorical aspects; Appendix: some impressions about the role and importance of fuzzy topology.

Although the author does not claim his survey to be comprehensive, it is the reviewer’s feeling that the title of the paper does require that the choice of topics should reflect more than the author’s interests. For instance, there is no excuse for missing fuzzy uniform structures, a quite well developed topic, especially when some constructions of minor importance are treated in details. (As a result, in spite of the heading of Section 9, there is actually nothing about metrizability in it.) Also, a few papers are missing which could have been included, among them: “Compactification theory for fuzzy topological spaces” by B. Hutton [Fuzzy Math. 3, No.1, 35-44 (1983; Zbl 0531.54005)] (necessary at least in Section 15 where Hutton’s point-free theory is treated) and N. N. Morsi [Fuzzy Sets Syst. 23, 393-397 (1987; Zbl 0625.54007)] (one simple condition discussed by the author as necessary for a fuzzy space to be a neighbourhood space (Subsection 4.5) is proved by Morsi to be also sufficient). It also seems that the crediting of priority in the development of some aspects of the theory is occasionally careless. On the other hand, one has to thank the author for not having mentioned a recent proliferation of papers of the “almost-”, “semi-”, “nearly-” type. Further it must be pointed out that the author almost entirely ignores the lattice-theoretic issue of fuzzy topology by claiming that things become “essentially simpler” under Chang’s definition of a fuzzy space, i.e. with the real unit interval [0,1] as a range space for fuzzy sets. (Notable exception is Section 15 whose topic requires lattices other than [0,1].) It is the reviewer’s feeling that this approach could lead to the abandonment of fuzzy topology by further simplifying to the case of the 2-point lattice, i.e. to ordinary topological spaces. One illustration of how lattices other than [0,1] can be advantageous is the Tychonoff-Goguen theorem which is meaningful for a certain class of lattices, excluding [0,1] but including \(\{\) 0,1\(\}\), so that the classical Tychonoff theorem becomes its special case. Note that the author did include the original Goguen’s statement. There is another anniversary type survey paper by S. F. Rodabaugh [Fuzzy Sets Syst. 40, 297-345 (1991)] which provides a heavily lattice-theoretic view of fuzzy topology.

Presumably for the sake of saving space, neither definitions nor results have been distinguished from the text, so that one has to search for results. A penetrating reader, who might not be impressed with the number of results, should note that during the first quarter of those two decades of fuzzy topology only one paper was published (just the 1968 paper of Chang), and the next quarter (1973-77) brought some 20 papers more.

Although the author does not claim his survey to be comprehensive, it is the reviewer’s feeling that the title of the paper does require that the choice of topics should reflect more than the author’s interests. For instance, there is no excuse for missing fuzzy uniform structures, a quite well developed topic, especially when some constructions of minor importance are treated in details. (As a result, in spite of the heading of Section 9, there is actually nothing about metrizability in it.) Also, a few papers are missing which could have been included, among them: “Compactification theory for fuzzy topological spaces” by B. Hutton [Fuzzy Math. 3, No.1, 35-44 (1983; Zbl 0531.54005)] (necessary at least in Section 15 where Hutton’s point-free theory is treated) and N. N. Morsi [Fuzzy Sets Syst. 23, 393-397 (1987; Zbl 0625.54007)] (one simple condition discussed by the author as necessary for a fuzzy space to be a neighbourhood space (Subsection 4.5) is proved by Morsi to be also sufficient). It also seems that the crediting of priority in the development of some aspects of the theory is occasionally careless. On the other hand, one has to thank the author for not having mentioned a recent proliferation of papers of the “almost-”, “semi-”, “nearly-” type. Further it must be pointed out that the author almost entirely ignores the lattice-theoretic issue of fuzzy topology by claiming that things become “essentially simpler” under Chang’s definition of a fuzzy space, i.e. with the real unit interval [0,1] as a range space for fuzzy sets. (Notable exception is Section 15 whose topic requires lattices other than [0,1].) It is the reviewer’s feeling that this approach could lead to the abandonment of fuzzy topology by further simplifying to the case of the 2-point lattice, i.e. to ordinary topological spaces. One illustration of how lattices other than [0,1] can be advantageous is the Tychonoff-Goguen theorem which is meaningful for a certain class of lattices, excluding [0,1] but including \(\{\) 0,1\(\}\), so that the classical Tychonoff theorem becomes its special case. Note that the author did include the original Goguen’s statement. There is another anniversary type survey paper by S. F. Rodabaugh [Fuzzy Sets Syst. 40, 297-345 (1991)] which provides a heavily lattice-theoretic view of fuzzy topology.

Presumably for the sake of saving space, neither definitions nor results have been distinguished from the text, so that one has to search for results. A penetrating reader, who might not be impressed with the number of results, should note that during the first quarter of those two decades of fuzzy topology only one paper was published (just the 1968 paper of Chang), and the next quarter (1973-77) brought some 20 papers more.

Reviewer: T.Kubiak

##### MSC:

54A40 | Fuzzy topology |