On semi-I-regular sets, AB I -sets and decompositions of continuity, R I C-continuity, A I -continuity

Summary We introduce the notion of a semi-I-regular set and investigate some of its properties. We show that it is weaker than the notion of a regular-I-closed set. Additionally, we also introduce the notion of an AB_I -set by using the semi-I-regular set and study some of its properties. We conclude that a subset A of an ideal topological space (X,τ,I) is open if and only if it is an AB_I -set and a pre-I-open set.