The G-connected Property And G-topological Groups

G-connected and G-convergence are significant topics in general topology and analysis.On the one hand,in addition to the general convergence of sequences,there are various kinds of convergence and connectivity which play an important role in pure mathematics,and they play a very important role in other branches of mathematics,especially in computer science,information theory,biological science and dynamic system;On the other hand,G-connected is closely related to the properties of G-convergence,G-compactness and G-continuity,which play an indispensable and key role in the application of mathematics and other fields.Therefore,understanding the properties of G-connected will help us to explore the mysteries of topology and construct special topological space structures.This paper has done the following four aspects of work:Firstly,in the second section of the first chapter,we introduce the necessary symbols and terms in this paper,introduce the definition of the category of group with operation,the category of topological group with operation,G-hull,G-kernel,G-connected related properties,and give the equivalent characterization of G-open set and G-closed set.Secondly,in the first section of the second chapter,based on the research of reference[28],we discuss some properties of G-hull and G-kernel,generalize some results about G-hull,G-kernel,G-open set,G-closed set and G-continuity,and give examples to answer the question about infinite product of G-kernel.Thirdly,in the second section of the second chapter,we study the function of G-connected under G-continuity,generalize the finite product problem of G-connected,and prove the countable product problem of G-connected.Fourthly,in the third section of the second chapter,we introduce the concept of G-topological group,establish the relationship between G-topological group and G-topology under the assumption of keeping the regular method of subsequence,and obtain some results about G-topological group.