sg-PERFECT FUNCTIONS IN SEMI-GENERALIZED TOPOLOGICAL SPACES: ON DECOMPOSITION, INVARIANCE, AND FURTHER DEVELOPMENTS

Abstract

This paper is a pioneering study of sg-perfect functions, a transformative familyof mappings that combine the structural depth of ideal topologies with the nuanced behaviorof semi-open sets. These functions, based on Levine's semi-open sets and Bhattacharyya's sg-closedness, emerge as a three-layered marvel: sg-continuous, sg-closed, and endowedwith sg-compact. We go through decomposition theorems and hereditary properties to seehow sg-perfect functions, sg-Lindelöf perfect functions protect topological integrity. Beyondtheoretical, this approach highlights several practical pathways: sg-perfect functionsand sg-Lindelöf perfect functions permit the transfer of invariants in ideal-rich environments,providing tools for breaking down complex regions into manageable, sg-closed components.This study prepares the way for future research into sg-connectedness, dynamicdecomposition, and beyond. For mathematicians negotiating the intersection of ideals andopenness, sg-perfect functions operate as a guidepost to undiscovered topological realms.