Two Idealizations of h-Open Sets: Interior Operator and Continuity

Given an ideal topological space (X,τ,I), we consider two opposite ways of idealizing the class of h-open sets. The first, the class of hI-open sets introduced by Açıkgöz and Noiri, pushes the ideal inward through the Janković–Hamlett local-function closure: a set A is hI-open if A⊆int(A∪cl∗(V)) for every nonempty proper open subset V of X. The second, introduced here as the class of Ih-open sets, pushes the ideal outward by tolerating a small defect: a set A is Ih-open if A∖int(A∪V)∈I for every nonempty proper open subset V of X. We show that h-openness implies both hI-openness and Ih-openness, while hI-openness and Ih-openness are mutually independent. We also show that the two idealizations collapse to h-openness at opposite ends of the ideal lattice: hI-openness coincides with h-openness when I=P(X), whereas Ih-openness coincides with h-openness when I={∅}. Replacing the ordinary interior by the α-, pre-, semi- and β-interior operators, we obtain the variants Ihα-, Ihp-, Ihs- and Ihβ-openness, and we describe how they are related. Finally, we study the family τIh of all Ih-open sets and the induced operator intIh. We prove that τIh is a topology whenever X is finite or, more generally, whenever I is closed under arbitrary unions of its members. We also give a counterexample showing that, without such a hypothesis, arbitrary unions of Ih-open sets need not be Ih-open.