Between Preopen and Open Sets in Topological Spaces

The aim of the present paper is to introduce the class of omega(p)-open sets which is strictly placed between the class of all preopen and the class of all open subsets of $X$. Relationships with some other type of sets are given. Furthermore, we introduce a new type of continuity termed the class of omega-almost continuous functions which is lies strictly between the class of all continuous and the class of all almost-continuous functions. New decompositions of continuity are obtained.