Utilizing the Advancement of Kasaj Topological Spaces for Engineering Breakthroughs

The primary aim of this article is to present and explore the properties of Kasaj pre-neighborhoods, Kasaj pre-interior, Kasaj pre-closure, and Kasaj pre-limit points and later on to apply the idea in the various engineering domain. We derive significant results by leveraging the concepts of Kasaj pre-open sets, Kasaj pre-closed sets, and Kasaj pre-derived sets. This study delves into the properties and characterizations of pre-open and pre-closed sets within the framework of Kasaj topological space. Furthermore, we examine the interrelationships between the characterizations of Kasaj pre-open sets, Kasaj pre-closed sets, and Kasaj pre-derived sets.