The Topology T ⊛

PR⋆ in the Frame of Primal Topological Spaces

Amani Rawshdeh1,∗, Ahmad Al-Omari2

1Department of Mathematics, Al-Balqa Applied University, Alsalt, Jordan

2Department of Mathematics, Faculty of Sciences, Al al-Bayt University, Mafraq, Jordan

Emails: amanirawshdeh@bau.edu.jo; omarimutah1@yahoo.com

Abstract

In this paper, we will use the family of regular⋆-open subsets to present and examine two new operators

(.)⊛

PR⋆ and Cl⊛

PR⋆ . We demonstrate that, in contrast to the operator (.)⊛

PR⋆ , the operator Cl⊛

PR⋆ is a Kura-

towski closure operator. We show that each of these operators lies between two previously defined operators

where for each subset H ⊆ S, H3

P ⊆ H⊛

PR⋆ ⊆ H3

PR and H ⊆ Cl3

P (H) ⊆ Cl⊛

PR⋆ (H) ⊆ Cl3

PR(H).

Furthermore, we show that the topology, denoted by T ⊛

PR⋆ , which is obtained by Cl⊛

PR⋆ is independent from

T and it is finer than Tη⋆ , where Tη⋆ is the family of all union of regular⋆-open subsets of (S, T ). Then

we demonstrate several fundamental results concerning this new structure and present many illustrative exam-

ples that relate to our conclusions. Finally, by using the operator Cl⊛

PR⋆ we introduce a new notion namely,

P−generalized closed sets, and study some of their basic properties.

Keywords: Kuratowski closure operator; primal; primal topological; The operator (.)⊛

PR⋆ ; the operator

Cl⊛

PR⋆ .