G-convergently separation axioms

Abstract

A convergence sequence in a Hausdorff space $X$ has a unique limit. Hence this idea gives us a function which is defined on convergence sequences and has the values in $X$. Replacing this limit function with any function $G$ whose domain is a certain subset of the sequences extends the notion of limit and such a function $G$ is called $G$-method. Then sequential definitions of continuity, compactness and connectedness have been extended to $G$-method setting.

In the paper we intent to study some separation axioms such that $\T_i$ $(i=0,1,2,3,4)$ for $G$-methods in sets or topological spaces; and characterise them in terms of $G$-open and $G$-closed subsets. Then we give some different counterexamples of $G$-methods and evaluate them if these separations axioms are satisfied.