On P^* g- Closed Set in topological Spaces

Closed sets play a fundamental role in topological spaces. Notably, a topology on a set can even be characterized by specifying the properties of its closed sets. In 1970, N. Levine introduced the concept of generalized closed sets, defined: A subset S of a topological space X is considered generalized closed if the closure of A is contained in U, cl(A)⊆U whenever A⊆U and U is open set. In this study, we define and explore novel classes of sets termed pre star generalized closed sets (P^* g-closed), pre star generalized open sets (P^* g-open) within the context of topological spaces. The relationship of this set with other closed sets has been proven, and its fundamental properties such as union, intersection, and containment have been established. We have also demonstrated that pre star generalized closed set (P*g- closed) in any given set (let it be X) remains a pre star generalized closed set (P*g-closed) in any of its subset Y⊆X. Moreover, we have proven the fundamental properties of pre star generalized open set (P*g –open). In future studies, we aim to extend this research to introduce a new operator similar to the one currently under investigation, sharing its topological characteristics. Keywords: P-closet, P-open, P^* g-closed, 〖 P〗^* g-open.