Neighborhood Structure Of I-fuzzy Topology On Any Fuzzy Subset

This article attempts to study the neighborhood structure and its related questions in I-fuzzy topological space on a fuzzy subset(we call it general I-fuzzy topological space,and the topology on it is called general I-fuzzy topology). First,we establish a kind of relatively quasi-coincident relation between two fuzzy subsets of a fixed fuzzy subset,it can reduced to primitive quasi-coincident relation of Liu Yingming;Next,based on this new relatively quasi-coincident relation,we introduce the neighborhood structure of I-fuzzy topology on general fuzzy subset successfully,called general I-fuzzy quasi-coincident neighborhood system.And we find that general I-fuzzy quasi-coincident neighborhood system and general I-fuzzy topology can be obtained one from the other.The establishment of general I-fuzzy quasi-coincident neighborhood system,not only had its rationality,but also laid the foundation for the research of local questions of I-fuzzy topological space which take the fuzzy subset as the universe of discourse. As the further application of general I-fuzzy quasi-coincident neighborhood system, we study the first countability,the second countability and the separability in general I-fuzzy topological space.And of course,the rationality of general I-fuzzy quasi-coincident neighborhood systems of the paper is supported by the results obtained.Finally,we give the theory of connectedness in general I-fuzzy topological space,and its description of degree.At the same time,we give its equivalent condition too.This paper is organized as follows:The first section is preface.The author introduces background about gener-ation of I-fuzzy quasi-coincident neighborhood systems in topology,status about the research of lattice-valued topological spaces and gives the main problems to be solved in this paper.The second section is the preparation.The author introduces some notations and reviews some fundamental knowledge about I-fuzzy topological spaces and general I-fuzzy topological spaces simply.The third section is the research about neighborhood structure and base of general I-fuzzy topological space.The author using the redefined quasicoincident relation,establishes a kind of general I-fuzzy quasi-coincident neighborhood system on any fuzzy subset as expected.And shows that a general I-fuzzy topology and a general I-fuzzy quasi-coincident neighborhood system on any fuzzy subset can be induced one from the other.At the end of Section 3,on a given fuzzy subset,the author gives the definition of base of general I-fuzzy topology and establishes its two characteristic theorems.The fourth section is the research about general I-fuzzy countability of general I-fuzzy topological space.The author introduces the concepts of general I-fuzzy first countability,general I-fuzzy second countability,general I-fuzzy separability,and given the affirmative description of the mutually inferred relations of general I-fuzzy first countability,general I-fuzzy second countability and general I-fuzzy separability in general I-fuzzy topological space.The fifth section is the research about the connectedness in general I-fuzzy topological space.In this section,the author gives two definitions of connectedhess in general I-fuzzy topological space,and show the relation of them.