Generalization Of Sequence Convergences

Sequence convergences is an important research topic in the field of mathematics.At first,the study of convergence was carried out on real lines,but with the emergence of the concept of topological space,convergence naturally has also been extended.People change |xn-a|<εinto xn∈Ua in the definition of sequence limit,where Ua is any neighborhood of a,and define sequence limit in general topological space.Many properties of topological space can be described by sequences.Generalizing sequences and using them to study problems in topological space has become one of the research topics that people pay attention to.Generalizing sequence convergence has gradually become a hot research topic in the mathematical field.This article discusses the generalization of sequence convergence in different spaces,and the main content is as follows:1.I-convergence in general topological space.We studied the necessary and sufficient conditions for its equivalence to general convergence and proved the uniqueness of the I-limit in T2 space.On this basis,the I-limit point and I-cluster point of the sequence were defined,and the connections and differences between the three were discussed.Finally,it was proved that the I-limits,I-limit points,and I-cluster points of two sequences that are I-equivalent to each other are same.2.Rough convergence and rough I-convergence in finite dimensional normed linear spaces.First,we study the properties of these two limit sets,and find that they are bounded,closed and convex.Secondly,the relationships between rough convergence and general convergence,as well as between rough I-convergence and I-convergence were studied.Among them,when studying rough convergence,a rough Cauchy sequence was defined and it was found that a r-convergent sequence must be a 2r-Cauchy sequence,but a 2r-Cauchy sequence may not necessarily be a r-convergent sequence.Finally,these two convergence concepts are extended to metric space and semi-metric space.3.Rough Weighted I-Convergence in Locally Solid Riesz Spaces.First,we discuss the boundedness,closeness and convexity of the rough weighted I-limit set of the sequence,then we introduce the weighted I-cluster point of the sequence,and discuss the closeness of the weighted I-cluster point set.Then,the relationship between the rough weighted I-limits and weighted I-cluster points of two sequences that are Z-equivalent to each other was studied.It was proved that when {tn}n∈N is a bounded sequence,the rough weighted I-limits and weighted I-cluster points of the two sequences that satisfy xn-yn(?)0 are the same.Finally,the relationship between the rough weighted I-limit set and the weighted I-cluster point set of the sequence was discussed.