Weak structure space (briefy, wss) has master looks, when the whole space is not
open, and these classes of subsets are not closed under arbitrary unions and fnite
intersections, which classify it from typical topology. Our main target of this article is
to introduce ψH(.)-operator in hereditary class weak structure space (briefy, 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