On Connectedness Via Nano Generalized Open Sets

Abstract

 In this paper, the concept of connectedness is explored through Nano Generalized Open Sets (NGOs) in nano topological spaces (NTS). A new class of spaces, named NG-connected spaces, is introduced and evaluated. Several characterizations of NG-connectedness are established, including formulations in terms of NGOs, NGCs and NGOs. Furthermore, two stronger forms of NG- connectedness, namely NG- hyper connectedness (NGHCD) and NG-ultraconnectedness (NGUCD), are introduced and investigated. It is observed that both NGHCD and NGUCD spaces imply NG-connectedness. Various examples and counter examples are provided to illustrate the relationships among these classes and to show the failure of several conserve implications. The obtained results prolong the study of connectedness from NOS to NGOs and present a broader framework for investigating generalized connectedness in NTS. These findings contribute to the advancement of NT and open new directions for further research including generalized topological structures.