G-Convergences of Submethods

Resumo

Classical topological notions such as openness, closedness, and continuity  can be expressed in sequential  form, particularly in the case of first-countable Hausdorff spaces. Motivated by this idea, a generalized convergence method known as the $G$-method, has  been introduced. Different studies related to the $G$-method have been widely explored in recent literature, including studies on $G$-continuity, $G$-compactness, and other related topological properties. In this study, the theory of the $G$-method defined on a set $X$ is extended by introducing the concept of $G$-submethod $G_Y$, induced on a non-empty subset $Y\subseteq X$. We examine the behaviours of $G$-open and $G$-closed subsets under these submethods. Several characterizations are provided to determine when $G_Y$-closedness and $G_Y$-openness are preserved within submethods. In addition, we also investigate the preservation of topological properties such as $G$-compactness, and separation axioms $G$-$T_0$, $G$-$T_1$, and $G$-Hausdorff for $G$-submethods.