Submaximality, Connectedness, and δβI-Continuous Mappings in Ideal Spaces

This paper extends the concepts of submaximality, connectedness, and hyperconnectedness to ideal topological spaces by means of δβI-open sets, which constitute a class of generalized open sets. Initially, we establish a collection of equivalent conditions that define the δβI-submaximality of an ideal topological space (T, τ, I). This study then discusses δβI-connectedness, which is a more general form of classical connectedness. We demonstrate that some mappings preserve this form of relationship. Finally, we introduce a generalization of hyperconnectedness based on δβI-open sets and provide characterizations of these spaces. These findings jointly augment the theory of ideal topological spaces by establishing coherent frameworks for submaximality, connectedness, and hyperconnectedness while expanding their applicability to more expansive topological situations where conventional separation axioms and continuity ideas are insufficient.