Study On Some Problems About Fuzzy Topologies

This paper focus mainly on several problems about L—fuzzifying topology, where L is a completely distributive lattice. The problems are listed as below: (1) How to establish the relationship between L—fuzzifying topology and exten-sional L—fuzzy topology? (2) Whether there is a Galois correspondence be-tween L—FYS of L—fuzzifying topological spaces and the category EL—FTOP of extensional L—fuzzy topological spaces; (3) How to constitute the subspace and the quotient space of extensional L—fuzzy topological space? In this pa-per, there are three purposes. First, it gives the method of transformation be-tween L—fuzzifying topology and extensional L—fuzzy topology. Moreover, the paper gives the relationship between the category L—FYS and the category EL—FTOP. Then, it is showed that there is a Galois correspondence between L—FYS and EL—FTOP; Second, the subspace and the quotient space of ex-tensional L—fuzzy topological space are defined and the rationality of the two definitions are also proved. Lastly, it gives the rectification about some incorrect proofs of I—fuzzifying closure operator and the convergence of net in the pub-lished paper.This paper includes five parts. The main contents are organized as follows:The first section is preface. The background, the stage of development about L—fuzzy topology, status about the research of L—fuzzy topological spaces are introduced and the main problems to be solved in this paper are also shown in this part.The second section is the preparation. It introduces some notations and def-initions. Then some elementary knowledge about L—fuzzy topological spaces, L—fuzzifying topological spaces, extensional L—fuzzy topological spaces, cate-gory, closure operator and the convergence of net are recalled simply.The third section is the research about the method of transformation be-tween L—fuzzifying topology and extensional L—fuzzy topology. In this section, first of all, it defines two mappings between L—fuzzifying topology and exten-sional L—fuzzy topology. Then it proves that the two mappings just consti-tute two functors between the category L—FYS and the category EL—FTOP. Finely, it shows that there is a Galois correspondence between L—FYS and EL-FTOP.The fourth section concerns the method for constitution about the subspace and the quotient space of extensional L—fuzzy topological space. First, the au-thor defines the subspace and the quotient space of extensional L—fuzzy topolog-ical space. Second, the author proves that the subspace and the quotient space of extensional L—fuzzy topological space are also extensional L—fuzzy topological spaces.The last section is major in the rectification about some incorrect proofs of L—fuzzifying closure operator and the convergence of net in the published paper. And then the correct proofs about the important conclusions are given.