A generalized topological convergence of function's sequences confined by weights

Abstract

In this article, an extended method based on weight functions for pointwise and uniform statistical convergence for function sequences in topological settings is presented. Several fundamental theorems are proven and new definitions are suggested in order to rigorously describe these sorts of generalized convergence. Illustrations and counterexamples that highlight the differences and advantages of the suggested approaches provide further support for the theoretical development. Furthermore, this study has also examined the topological implications of these new types of convergence with the presence of Dini's theorem.