The Fuzzifying Derived Operators And Derived Space Categories

As is known to us, the derived operator plays an important role in topology. A topology is not only determined through it, but also important in aspect of the research on topological spaces. In 1980, Hohle introduced in [5] the concept of fuzzy measurable spaces with the idea of giving degrees in [0, 1] to some topological terms rather than 0 and 1. Then the theory of many-valued logic is applied to many research directions and research fields in topology. In 1991, from a logical view , Ying [17] introduced the concept of fuzzifying topological spaces. From the on, many authors worked on the subject . Many concepts and properties, the examples of fuzzifying (topological) interior operator, fuzzifying (topological ) closure operator, separation axioms, compactness, generalized neighborhood systems and subspaces, are generalized to fuzzifying topological spaces. In this paper, the author introduces the concept of fuzzifying (topological) derived operator. Some of comprehensive characterizations about fuzzifying (topological) derived operator are established. The relation between fuzzifying (topological) derived operator and fuzzifying (topological) closure operator is also discussed. In addition, the author construct the category of fuzzifying topological derived spaces and its continuous mappings, denoted I-GDS, and show that I-GDS is isomorphic to I-GCS.This paper is composed of five sections:In the first section, the author introduces the research state and the development of fuzzifying topological spaces, and gives the research results of derived operator in classical topology and L-topology.The second section is a total description about background of derived operator. The author discusses the definition and properties of derived operator in classical topology and L-topology. The main questions we study are given.In the third section, firstly, the author gives the axioms which the fuzzifying (topological) derived operator satisfies. Some of comprehensive characterizationsabout the fuzzifying (topological) derived operator are established. Then the relation between fuzzifying (topological) derived operator and fuzzifying (topological) closure operator is also discussed, and some important results are obtained .The fourth section is the deepen of the results above. The author makes some discussions on the relation between the fuzzifying (topological) derived operator and I-topology and ascertains that the fuzzy topology can be induced by fuzzifying (topological) derived operator. At last, the author utilizes the category tools to construct the category J-GDS and find J-GDS is isomorphic to I-GCS.The fifth section are some questions to be solved in the future.