On -Continuous Functions in Biminimal Structure Spaces

We introduce the notion of -continuous functions and some other forms of continuity in biminimal structure spaces. Some new characterizations and several fundamental properties of -continuous functions are obtained. 1. Introduction Weak continuity due to Levine [1] is one of the most important weak forms of continuity in topological spaces. Rose [2] has introduced the notion of subweakly continuous functions and investigated the relationships between subweak continuity and weak continuity. In [3], Baker obtained several properties of subweak continuity which are analogous to results in [4]. Nj？stad [5] introduced a weak form of open sets called -sets. In [6], the author showed that connectedness is preserved under weakly -continuous surjections. Mashhour et al. [7] have called strongly semicontinuous -continuous and obtained several properties of such functions. In [8], they stated without proofs that -continuity implies -continuity and is independent of almost continuity in the sense of Singal [9]. On the other hand, in 1980 Maheshwari and Thakur [10] defined -irresolute and obtained several properties of -irresolute functions. Levine [11] defined the notions of semiopen sets and semicontinuity in topological spaces. Maheshwari and Prasad [12] extended the notions of semiopen sets and semicontinuity to the bitopological setting. Bose [13] further investigated many properties of semiopen sets and semicontinuity in bitopological spaces. Mashhour et al. [7] introduced the notions of preopen sets and precontinuity in topological spaces. Jeli？ [14] generalized the notions of preopen sets and precontinuity to the setting of bitopological spaces. The purpose of the present paper is to introduce the notion of -continuous functions in biminimal structure spaces and investigate the properties of these functions. 2. Preliminaries Definition 1 (see [15]). Let be a nonempty set and the power set of . A subfamily of is called a minimal structure (briefly m-structure) on if and . By , we denote a nonempty set with an -structure on and it is called an -space. Each member of is said to be -open, and the complement of an -open set is said to be -closed. Definition 2 (see [16]). Let be a nonempty set and an -structure on . For a subset of , the -closure of and the -interior of are defined as follows: (1) ; (2) . Lemma 3 (see [16]). Let be a nonempty set and a minimal structure on . For subset and of , the following properties hold: (1) and . (2)If , then and if , then . (3) , , , and . (4)If , then and . (5) and . (6) and . Definition 4 (see [16]). An -structure on a