Statistically Sequentially Compact Spaces, Statistically Sequential Spaces And G-methods

Sequential convergence is an important research object in topology and analysis.On the one hand,convergence is closely related to continuity,compactness and other related properties.On the other hand,it has played a fundamental role in mathematics and its applications.Statistical convergence as an important generalization of convergence has a great significance.The main results are the following:Firstly,we introduce the statistical definition of sequential compact spaces in general topological spaces,discuss some topological properties on statistically sequential spaces and deepen some results on sequential compact spaces in topological groups and metrizable spaces.Secondly,the properties of statistically sequential spaces are studied.After introducing the concepts of statistically sequentially continuous mappings,statistically sequence-covering mappings and statistically sequentially quotient mappings,the relationship among the new or old mappings and their effects in statistically sequential spaces are discussed.Also a negative answer is given to a problem about the products of statistically sequential spaces.Thirdly,based on general convergence,statistical convergence and other convergences,we try to introduce the notions of G-methods,G-sequential spaces and G-Fr�chet-Urysohn spaces,discuss the characterization,hereditary properties and mapping properties of G-sequential spaces and G-Fr�chet-Urysohn spaces,and establish the mutual relationship between continuity and G-sequential continuity,promote G-methods and some related results in topological Hausdorff group which satisfies the first axiom of countability.