Ideal-Driven βθ-I-Open Sets and Their Role in Connectedness, Hyperconnectedness, and Extremal Disconnectedness

In this paper, we introduce a novel type of open set, called βθ−I-open sets, constructed by means of an ideal on a topological space. These sets broaden the classical definition of openness. Employing these ideal-based generalized open sets, we investigate essential topological properties—including connectedness, hyperconnectedness, and extremal disconnectedness—and derive results that unify, generalize, and sharpen earlier theories. The inclusion of ideals enhances the structural framework and yields new characterizations, demonstrating the versatility and broad relevance of these sets in both classical and ideal topological settings. Moreover, we analyze various classes of functions that preserve or reflect these properties between ideal topological spaces, offering a refined perspective on connectedness under different mappings.