Problems of the theory of bitopological spaces. II.

*(English. Russian original)*Zbl 0852.54028
J. Math. Sci., New York 81, No. 2, 2465-2496 (1996); translation from Zap. Nauchn. Semin. POMI 208, 5-67 (1993).

The first part [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 167, 5-62 (1988; Zbl 0685.54019)] of this work covers the period until 1986 inclusive and part of 1987. The second part, which we present here, mainly concerns the papers published in 1987-1990. It also replenishes the gaps concerning earlier papers, and touches some papers published in 1991.

With few exceptions, we do not repeat here the contents of the first part. Thus, it is necessary to get acquainted with the basic notions and results of the theory of bitopological spaces by using other sources, preferably the first part itself. All references are given according to “Bibliography…,” Nos. 1-245 of the first part [ibid., 63-78 (1988; Zbl 0685.54020)] and Nos. 246-478 of the second part.

The structure of this second part of the work “Problems of the theory of bitopological spaces” is, in general, parallel to the first one: it consists of the same five sections, and each of them can be considered as the continuation of the corresponding section of the first part. Thus, in §2 we consider further generalizations of the theory of topological spaces and related structures on the bitopological spaces in the sense of Kelly. In §3 we continue the presentation of the general theory of bitopological spaces. Sections 4 and 5 are devoted to the applications of the theory.

As before, the greater part of the bitopological literature is devoted to bitopological spaces \((X, \tau_1, \tau_2)\) in the sense of Kelly, but general bitopological spaces \((X,\beta)\) begin to gradually attract the attention of a broader section of specialists. One can suppose that this tendency will persist and general bitopological spaces will become customary objects of study. Nevertheless, §2, which is devoted to bitopological spaces in the sense of Kelly, is still a basic one. In general, its structure is similar to that of §2 of the first part: ten subsections continuing the corresponding subsections of §2 of the first part, and the new §11 devoted to fuzzy bitopological spaces. In §3 we continue the construction of proper bitopological theory: new notions are introduced and their interconnections are studied.

With few exceptions, we do not repeat here the contents of the first part. Thus, it is necessary to get acquainted with the basic notions and results of the theory of bitopological spaces by using other sources, preferably the first part itself. All references are given according to “Bibliography…,” Nos. 1-245 of the first part [ibid., 63-78 (1988; Zbl 0685.54020)] and Nos. 246-478 of the second part.

The structure of this second part of the work “Problems of the theory of bitopological spaces” is, in general, parallel to the first one: it consists of the same five sections, and each of them can be considered as the continuation of the corresponding section of the first part. Thus, in §2 we consider further generalizations of the theory of topological spaces and related structures on the bitopological spaces in the sense of Kelly. In §3 we continue the presentation of the general theory of bitopological spaces. Sections 4 and 5 are devoted to the applications of the theory.

As before, the greater part of the bitopological literature is devoted to bitopological spaces \((X, \tau_1, \tau_2)\) in the sense of Kelly, but general bitopological spaces \((X,\beta)\) begin to gradually attract the attention of a broader section of specialists. One can suppose that this tendency will persist and general bitopological spaces will become customary objects of study. Nevertheless, §2, which is devoted to bitopological spaces in the sense of Kelly, is still a basic one. In general, its structure is similar to that of §2 of the first part: ten subsections continuing the corresponding subsections of §2 of the first part, and the new §11 devoted to fuzzy bitopological spaces. In §3 we continue the construction of proper bitopological theory: new notions are introduced and their interconnections are studied.