On M ( i , j ) A-Continuous Functions in Biminimal Structure Spaces

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 andweak continuity. In [3], Baker obtained several properties of subweak continuity which are analogous to results in [4]. Njåstad [5] introduced a weak form of open sets called α-sets. In [6], the author showed that connectedness is preserved under weakly α-continuous surjections.Mashhour et al. [7] have called strongly semicontinuous α-continuous and obtained several properties of such functions. In [8], they stated without proofs that α-continuity implies θcontinuity and is independent of almost continuity in the sense of Singal [9]. On the other hand, in 1980 Maheshwari and Thakur [10] defined α-irresolute and obtained several properties of α-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. Jelić [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 M A -continuous functions in biminimal structure spaces and investigate the properties of these functions.


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].Njåstad [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.Jelić [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 M (,) A -continuous functions in biminimal structure spaces and investigate the properties of these functions.

Preliminaries
Definition 1 (see [15]).Let  be a nonempty set and P() the power set of .A subfamily   of P() is called a minimal structure (briefly m-structure) on  if 0 ∈   and  ∈   .By (,   ), we denote a nonempty set  with an mstructure   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 m-structure on .For a subset  of , the   -closure of  and the   -interior of  are defined as follows: (1)   Cl() = ∩{ :  ⊆ ,  −  ∈   }; (2)   Int() = ∪{ :  ⊆ ,  ∈   }.Lemma 3 (see [16]).Let  be a nonempty set and   a minimal structure on .For subset  and  of , the following properties hold: ( Definition 4 (see [16]).An -structure   on a nonempty set  is said to have propertyB if the union of any family of subsets belonging to   belongs to   .
Lemma 5 (see [17]).Let  be a nonempty set and   an structure on  satisfying property B. For a subset  of , the following properties hold: (1)  ∈   if and only if   Int() = .
Let (,  1   ,  2  ) be a biminimal structure space and  a subset of .The   -closure of  and the   -interior of  with respect to    are denoted by    Int() and    Cl(), respectively, for  = 1, 2. Also ,  = 1, 2 and  ̸ = .
Definition 7 (see [20]).A subset  of a biminimal structure space (,  ) be a biminimal structure space and {  :  ∈ K} a family of subsets of . (1) ( ) be a biminimal structure space and  a subset of .Then the    --closure of  and the    --interior of  are defined as follows: (1) Lemma 10.Let (,  1   ,  2  ) be a biminimal structure space.For a subset  of , the following properties hold:  (2) This follows from (1) immediately.
-continuous at point  ∈  if for each   -open set  of  containing (), there exists an (, )-  --open set  containing  such that () ⊆ .