G-methods, G-sequential spaces and G-continuity in topological spaces

G-methods and G-continuity for real functions are induced by changing the definition of the convergence of sequences on the set of real numbers. In this paper we introduce the concepts of G-methods, G-submethods and G-topologies on arbitrary sets and the related notion of G-continuity. We investigate operations on subsets that deal with G-hulls, G-closures, G-kernels and G-interiors, and we study topological spaces that are G-sequential spaces, G-Fréchet spaces or G-topologizable spaces. The G-methods on first-countable topological groups and several convergence methods on topological spaces are extended and studied in a unified way. In particular, several results for G-methods on first-countable topological groups are improved.