Continuous and Contra Continuous Functions in Bi-topological Spaces

Abstract

The concept of bi-topological spaces was first introduced by J.C. Kelly [2] in 1963. Many authors such as Levine [3] contributed as he defined the semi-open sets and semi-continuity in bi-topological spaces. Maheshwari and Prasad [5] contributed semi-open sets and semi-continuity to bi-topological spaces. The notion of $\beta$-open sets contributed by Mashhour et. al. [6] and Andrijevic [1] define Semi pre-open sets. In this paper we discuss pre-continuity and semi pre-continuity in bi-topological spaces. LellisThivager et.al. [4] introduces $g^{\ast}$-closed sets topological spaces and initiated the concepts of ultra space by using $(1,2)\alpha$-open sets in bi-topological spaces and proved that each $(1,2)\alpha$-open sets is (1, 2) semi-open and (1, 2) pre-open but the converse of each is not true. R-Devi and S.Sampath Kumar and M. Caldas [7] introduced and studied a class of sets and maps between bi-topological spaces Called supra $\alpha-$open sets and supra $\alpha$-continuous maps respectively.