A problem being presented by us in [4] was "Is the Theorem true for infinite topological spaces?". In this paper, we will solve the above problem by proving that Rational Squence Topological space is a T1-space, that has πgp-regularity but also non πgp-normality. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2013 Vol. 58 No. 7 pp. 59-65 This paper is available online at http πgp-REGULARITY OF RATIONNAL SEQUENCE TOPOLOGICAL SPACE Le Nguyen Thanh Nhon and Bui Quang Thinh Faculty of Education Tien Giang University Abstract. In this paper we prove that Rational Sequence Topological Space is a T1 -space which is a πgp-regular and not a πgp-normal space. Keywords Regular closed π-closed πgp-closed πgp-normal space πgp-regular space. 1. Introduction In 2012 L. . Nhon and . Thinh introduced the notations of πgp-normal and πgp-regular spaces in the Journal of Advanced Studies in Topology see 4 . From theorem in 4 we have the following theorems Theorem . If X is a T1 -space and has πgp-normality then X has πgp-regularity. Theorem . If X is finite T1 -space and has πgp-regularity then X has πgp-normality. A problem being presented by us in 4 was quot Is the Theorem true for infinite topological spaces quot . In this paper we will solve the above problem by proving that Rational Squence Topological space is a T1 -space that has πgp-regularity but also non πgp-normality. 2. Preliminary Notes Throughout this paper space X always means non empty topological spaces on which no separation axioms are assumed unless explicitly stated. We will denote the set of irrational numbers rational numbers and real numbers by I Q and R .For a set A of space X P A X A A int A denote to the power set of A the complement the closure and interior of A in X respectively. Next we need to recall the following definitions Received September 10 2013. Accepted October 30 2013. Contact Le Nguyen Thanh Nhon e-mail address 59 Le Nguyen Thanh Nhon and Bui Quang Thinh Definition . 2 . Subset A of space X is said to be regular open or an open domain if it is the interior of its own closure or equivalently if it is the interior of some closed set. Set A is said to be a regular closed or closed .
