Generalized \(\mathcal{I}_\mathcal{p}\)-Closed Sets in Ideal Topological Spaces

Abstract

The main focus of this study is to introduce a new category of generalized closed sets, referred to as \(\mathcal{I}_\mathcal{p}\)- closed sets, within the framework of ideal topological spaces. By using a few instances, we demonstrate \(\mathcal{I}_\mathcal{p}\)- closed sets and establish some fundamental properties of \(\mathcal{I}_\mathcal{p}\)-closed sets. We also investigate the relationship between \(\mathcal{I}_\mathcal{p}\)-closed sets and other classes of generalized closed sets in ideal topological spaces, such as (\mathcal{I}_\mathcal{g}\)- closed sets, \(\alpha\)\(\mathcal{I}_\mathcal{g}\)-closed sets, and (\mathcal{I}_\mathcal{r}\)\mathcal{g}\)-closed sets. Then, we focus on the topological implications of \(\mathcal{I}_\mathcal{p}\)-closed sets and investigate how they relate to the concepts of \(\mathcal{I}_\mathcal{p}\)-continuous map, (\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. First and foremost, we define the \(\mathcal{I}_\mathcal{p}\)-continuous map, investigate the behavior of \(\mathcal{I}_\mathcal{p}\)- continuous map with respect to \(\mathcal{I}_\mathcal{p}\)-closed sets, and derive several important properties of \(\mathcal{I}_\mathcal{p}\)-continuous map. Further, we studied their relationships with other classes of continuous maps in ideal topological spaces. Nevertheless, we defined the definitions of \(\mathcal{I}_\mathcal{p}\)-irresolute maps and strongly \(\mathcal{I}_\mathcal{p}\)-continuous maps in ideal topological spaces. We explored the connections with the notions of \(\mathcal{I}_\mathcal{p}\)-continuous map, (\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. Our results provide new insights into the study of ideal topological spaces.