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 $$\psi _{{\mathcal {H}}}(.)$$ ψ H ( . ) -operator in hereditary class weak structure space (briefly, $${\mathcal {H}}wss$$ H w s s ) $$(X, w, {\mathcal {H}})$$ ( 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 $$\psi _{{\mathcal {H}}}(.)$$ ψ H ( . ) -operator called $$\psi _{{\mathcal {H}}}$$ ψ H -semiopen sets are introduced. We prove that the family of $$\psi _{{\mathcal {H}}}$$ ψ H -semiopen sets composes a supra-topology on X . In view of hereditary class $${\mathcal {H}}_{0}$$ H 0 , $$w T_{1}$$ w T 1 -axiom is formulated and also some of their features are investigated.

中文翻译：

---

在具有遗传类的弱结构空间中的 $$\psi _{{\mathcal{H}}}( . )$$-operator

弱结构空间（简称 wss ）具有主外观，当整个空间不是开放的，并且这些子集类在任意联合和有限交集下不是封闭的，这将其从典型拓扑中分类。我们这篇文章的主要目标是在遗传类弱结构空间中引入 $$\psi _{{\mathcal {H}}}(.)$$ ψ H ( . ) -operator（简而言之，$${\mathcal { H}}wss$$ H wss ) $$(X, w, {\mathcal {H}})$$ ( X , w , H ) 并检查它的一些特征。此外，我们澄清了一些在拓扑空间中可信但在广义空间中无法实现的关系。作为 w -open 集和 w -semiopen 集的推广，弱结构空间中的某些新集合通过 $$\psi _{{\mathcal {H}}}(.)$$ ψ H ( . ) -引入了称为 $$\psi _{{\mathcal {H}}}$$ ψ H -semiopen 集合的运算符。我们证明 $$\psi _{{\mathcal {H}}}$$ ψ H -semiopen 集合的族构成了 X 上的超拓扑。鉴于遗传等级$${\mathcal {H}}_{0}$$H 0 ，$$w T_{1}$$ w T 1 -axiom 被公式化，并且还研究了它们的一些特征。