On \(e^*\theta \)-Regular and \(e^*\theta \)-Normal Spaces

This chapter explores the intricate world of -regular and -normal spaces, building upon the foundational concepts of functional analysis and topology. It begins by discussing the historical context and definitions of regular and normal spaces, tracing their evolution from the works of Vietoris and Tietze to more recent generalizations by Ayhan and Özkoç. The chapter introduces -regular and -normal spaces, defined through - -open sets, and delves into their characterizations and relationships with other topological properties. Key topics include the definitions and properties of various types of open and closed sets, such as regular open, semiopen, and preopen sets, and their respective closures and interiors. The chapter also presents several lemmas and theorems that elucidate the behavior of these spaces, including conditions for regularity and normality. Additionally, it investigates the continuity and irreducibility of functions between these spaces, providing a comprehensive analysis of their topological properties. The chapter concludes by highlighting the significance of -regular and -normal spaces in the broader context of functional analysis and topology, suggesting avenues for further research.