This Paper introduces a new local function ϕSζ∗ in a Grill topological space (Ẋ, ζ, τ) by using a semi open set, which is a superset of the local function ϕζ. Additionally, two equivalent operators, Ψs and ΓS that define topology τSζ, were devised based on ϕSζ∗, along with the presentation of various characteristics and properties ofϕ∗Sζ and the two equivalent operators, Ψs and ΓS. Additionally, newly created sets with weakly open properties (ϕSζ∗.open,ζ s.α-open, ζ s.pre.open, ζ s.semi.open) were established and their interrelations were proofed and diagrams that illustrate the relationship between them. For speed and accuracy of completing counterexamples, creating a computer program with original codes we

The primary objective of this work pertains to the examination of a new class of grill topological spaces denominated as AE-space. This nomenclature is rooted in the delineation of said spaces upon a non-empty set, upon which a myriad of topological structures is imposed via an arbitrary grill. The construction of these grill topologies precedes the establishment of our nascent topological construct through the intersection of these individualized topologies. In this discourse, we introduce and rigorously define the notions of AE-open sets and AE-closed sets. Furthermore, we establish the result that the AE-space exhibits the fundamental attributes of a topological space, assuming the absence of pairwise disjoint non-empty sets within th

 We use the idea of Grill, this study generalized a new sort of linked space like –connected –hyperconnected and investigated its features, as well as the relationship between it and previously described notation. It also developed new sorts of functions, such as hyperconnected space, and identifying their relationship, by offering numerous instance and attributes that belong to this set. This set will serve as a starting point for further research into the sets many future possibilities. Also, we use some of the theorems and observations previously studied and relate them to the grill and the Alpha group, and benefit from them in order to obtain new results in this research. We applied the concept of Connected to them and obtained