Weak structure space (briefly, wss) has master looks, when the whole space is not open, and these classes of subsets are not closed under arbitrary unions and finite intersections, which classify it from typical topology. Our main target of this article is to introduce ψH(.)-operator in hereditary class weak structure space (briefly, Hwss) (X,w,H) and examine a number of its characteristics. Additionally, we clarify some relations that are credible in topological spaces but cannot be realized in generalized ones. As a generalization of w-open sets and w-semiopen sets, certain new kind of sets in a weak structure space via ψH(.)-operator called ψH-semiopen sets are introduced. We prove that the family of ψH-semiopen sets composes a supra-topology on X. In view of hereditary class H0, wT1-axiom is formulated and also some of their features are investigated.