A further study in generalized topological spaces

Abstract

This thesis investigates the theory of generalized topological spaces (GTS),extending classical topology by relaxing or modifying the axioms that define open sets. Motivated by limitations in standard topological frame works when applied to non-traditional structures in analysis. The thesis defines G - neghbourhood, G - neighbourhood system, G - separated sets, G - Lindeloff spaces, G - derived set, G - dense set, G - locally finite, σG - locally finite, G - paracompact spaces, IG - convergent sequence, IG∗ - convergent sequence and presents fundamental properties, theorems and equivalent characterizations. The work explored the properties of the generalized closed sets, generalized connectedness and general- ized local connectedness. G - locally finite collections and σG - locally collections of sets introduced and used to extend the definition of paracompactness into generalized topological spaces. The work investigate the role of separation axioms in generalized paracompactness. The thesis defines IG limit points and IG - cluster points of a sequence and found some relations between ordinary sequential convergence and IG - convergence in generalized topological spaces. The study advances into the notions of G - regular open sets, G - regular closed sets and discuss about some properties of these type of concepts. Further introduced minimal and maximal G - regular open sets and minimal and maximal G - regular closed sets and investigate the role of these concepts in generalized topological spaces. Overall, the thesis provides a comprehensive and unified frame work that not only generalizes classical topological concepts but also enriches the theoretical landscape of topological spaces.