A Characterization of Continuity of βθ and Pθ Monads in Topolog

Nonstandard topology is a kind of topology which is constructed by means of nonstandard analysis. There are several important concepts in topology which can be defined in terms of preopen sets. Among the most well-known are Bourbaki’s submaximal spaces. In this book, we apply the notions of β-open and preopen sets in topological spaces to present and study two new types of monads which are called βθ-monad and Pθ-monad which play an important role in studying and characterizing several topological concepts like β-closed spaces, β-T2 spaces, p-closed spaces, pre-Urysohn spaces, ..etc. and it can be used in an external characterizing of pre -θ-open sets and βθ-open sets.