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**A new approach for fuzzy topology. I.**
*(English)*
Zbl 0718.54017

Noting that the standard theory of fuzzy topological spaces developed by Chang, Lowen, Pu and Liu and many others is in fact the theory of crisp topological structures on families of fuzzy sets, the author proposes an essentially different approach to the concept of a fuzzy topology, called in the paper a bifuzzy topology. Namely, by a bifuzzy topology on a set X the author calls a mapping \({\mathcal T}: I^ X\to I\) such that (1) \({\mathcal T}(X)=1\); (2) \({\mathcal T}(U\cap V)\geq {\mathcal T}(U)\wedge {\mathcal T}(V)\) for any \(U,V\in I^ X\), and (3) \({\mathcal T}(\cup_{\lambda}U_{\lambda})\geq \cap_{\lambda}{\mathcal T}(U_{\lambda})\) for each \(\{U_{\lambda}:\lambda \in \Lambda \}\subset I^ X\). This definition is being motivated from the fuzzy- logical point of view.

The largest part of the paper deals with special bifuzzy topologies (the s.c. fuzzifying topologies) which are, in a known sense, induced by usual (crisp) topologies. The theory of fuzzifying topologies is being developed: in particular, such problems as the neighbourhood structure, the base, derived sets, net- and filter-convergence are being discussed. (Unfortunately, the author is unaware of the fact, that the same concept of a (bi)fuzzy topology was introduced and studied earlier by the reviewer [Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 89-103 (1985; Zbl 0638.54007); Russ. Math. Surv. 44, No.6, 125-186 (1989); translation from Usp. Mat. Nauk 44, No.6(270), 99-147 (1989; Zbl 0716.54004)]. The fuzzy-logical reasoning of (bi)fuzzy topologies and the study of structures similar to fuzzyfying topologies was conducted by Z. Diskin [Topologicheskie Prostranstva Otobrazheniya 1985, 59-70 (1985; Zbl 0618.54005)].

The largest part of the paper deals with special bifuzzy topologies (the s.c. fuzzifying topologies) which are, in a known sense, induced by usual (crisp) topologies. The theory of fuzzifying topologies is being developed: in particular, such problems as the neighbourhood structure, the base, derived sets, net- and filter-convergence are being discussed. (Unfortunately, the author is unaware of the fact, that the same concept of a (bi)fuzzy topology was introduced and studied earlier by the reviewer [Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 89-103 (1985; Zbl 0638.54007); Russ. Math. Surv. 44, No.6, 125-186 (1989); translation from Usp. Mat. Nauk 44, No.6(270), 99-147 (1989; Zbl 0716.54004)]. The fuzzy-logical reasoning of (bi)fuzzy topologies and the study of structures similar to fuzzyfying topologies was conducted by Z. Diskin [Topologicheskie Prostranstva Otobrazheniya 1985, 59-70 (1985; Zbl 0618.54005)].

Reviewer: A.Šostak (Riga)

### Keywords:

fuzzy-valued logics; fuzzy neighborhood; fuzzy net; fuzzy filter; fuzzy topology; bifuzzy topology; fuzzifying topologies
Full Text:
DOI

### References:

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[2] | Wong, C. K., Fuzzy point and local properties of fuzzy topology, J. Math. Anal. Appl., 46, 316-328 (1974) · Zbl 0278.54004 |

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