The Generation And Representation For Fuzzy Topologies

In this dissertation, we try to study the generation and representation for fuzzy topologies. Its contents are divided into three chapters.In chapter one, we discuss the relationship between fuzzifying topologies and fuzzy relations. A saturated fuzzifying topology can be generated by a reflexive fuzzy relation; and conversely, a fuzzy preorder can be generated by a fuzzifying topology. In fact, there exists a one-to-one correspondence between the set of all saturated fuzzifying topologies and that of all fuzzy preorders. It supplies the theoretical support to set up the internal relationship between fuzzy preorders structure and fuzzifying topologies.In chapter two, we apply the idea of nested sets to propose the concepts of nested topologies based on the concepts of minimal set and maximal set. By introducing so-calledβ-equivalence andα-equivalence on complete lattice of nested topologies on any fixed universe X, we find two quotient objects which are both isomorphic to the complete lattice consisting of all (X, M)-topologies on the fixed universe. In conclusion, we establish the representation theorems for (L, M)-topology. These results summarize relative discussions in research literatures and hopefully it would have good uses in general cases.In chapter three, we propose the concept of Para-Quasi-Uniformities for classical topological spaces, and discuss the problem of the generation and Para-Quasi-Uniforming for topology. Considering that the classical topology is a special case of fuzzy topology and the fuzzy topology is the extension of classical topology, we hope the results obtained in this paper can be extended to fuzzy topological spaces and have a good use in relative research field.