New Local Function in Grill Topological Spaces

This Paper introduces a new local function φSζ*in a Grill topological space (X˙, ζ,τ) by using a semi open set, which is a superset of the local function φζ. Additionally, two equivalent operators, Ψs and ΓSthat define topology τSζ, were devised based on φSζ*, along with the presentation of various characteristics and properties of φSζ* and the two equivalent operators, Ψs and ΓS. Additionally, newly created sets with weakly open properties (φSζ*.open, ζs.α-open, ζs.pre.open, ζs.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 developed will be an exciting progression, assisting in saving time, effort, and errors when computing instances related to guiding principles of the study. This is done by using Wolfram Mathematica program. These generic codes can be utilized to implement more programs related to the topic. A program was developed to execute the necessary programming by creating an algorithm that clarifies how the program operates. Different images of the program interface illustrate the stages of its implementation and application.