On certain types of notions via preopen sets

Abstract

In this paper, we deal with the new class of pre-regular $p$-open sets in which the notion of preopen set is involved. We characterize these sets and study some of their fundamental properties. We also present two other notions called extremally $p$-discreteness and locally $p$-indiscreteness by utilizing the notions of preopen and preclosed sets by which we obtain some equivalence relations for pre-regular $p$-open sets. Moreover, we define the notion of regular $p$-open sets by utilizing the notion of pre-regular $p$-open sets. We investigate some of the main properties of these sets and study their relations to pre-regular $p$-open sets.