Research On Some Convergences In Fuzzy Metric Spaces

The convergence of fuzzy metric space is a widely concerned topic in fuzzy topology.This thesis mainly studies the statistical convergence,p-statistical convergence of sequences in fuzzy metric space and the uniform convergence and α convergence of mapping sequences.In Chapter 2,we mainly study some related characterizations of statistical convergence,statistical cluster point and statistical limit point of sequence in fuzzy metric space.Firstly,we discuss the equivalent characterization of statistical convergence in complete fuzzy metric spaces,and further obtain the consistency of statistical completeness and completeness.Then,we give the definitions of statistical cluster point,statistical limit point and limit point,and explore the inclusion relationship between them.In Chapter 3,we mainly study the relationship and equivalent characterization of the pstatistical convergence and p-statistical Cauchy of sequences in fuzzy metric space,and the relationship between p-statistical completeness and statistical completeness of fuzzy metric spaces.Firstly,we introduce the concept of p-statistical convergence and discuss the uniqueness of its convergence point,and then prove the equivalence between p-statistical convergence and statistical convergence in principal fuzzy metric space.Secondly,we give the concepts of pstatistical Cauchy and p-statistical completeness,and study the implication relationship between p-statistical completeness and statistical completeness.In Chapter 4,we mainly study some related characterizations of uniform convergence andα convergence of mapping sequences in fuzzy metric spaces.Firstly,we explore the judgment methods of uniform convergence of mapping sequences in general fuzzy metric spaces and standard fuzzy metric spaces,then give the concepts of α convergence,almost uniform convergence,and exhaustive of mapping sequences,and discuss the relationship between αconvergence and almost uniform convergence and exhaustive respectively.