Some Results of Upper and Lower M -Asymmetric Irresolute Multifunctions in Bitopological Spaces

In this paper, we aim to introduce and study some basic properties of upper and lower M-asymmetric irresolute multifunctions defined between asymmetric sets in the realm of bitopological spaces with certain minimal structures as a generalization of irresolute functions deal to Crossley and Hildebrand [1] and upper and lower irresolute Multifunctions deal to Popa [2].


Introduction
Topology and its vital role to continuity have received considerable attention by several authors not only in the field of functional analysis but also in other branches of applied science. Continuity and multifunctions, the basic concepts in the theory of classical point set topology which plays an important role not only in the field of functional analysis but also in applied sciences: mathematical economics, control theory, and fuzzy topology have also received considerable attention by many scholars. In this regard, several scholars have generalizations these notions of continuity to (bi-)topological spaces using the weaker forms of open and closed sets the semiopen and semiclosed sets see: [1] [3] [4] [5] [6] [7].
The fundamental idea of semiopen sets and semi-continuous functions in topological spaces was first introduced by Levine [3] and these concepts have been extended to the realm of bitopological spaces by Maheshwari and Prasad [4], How to cite this paper: Matindih, L.K., Moyo, E., Manyika, D.K. and Sinyangwe, T. cept to the settings of bitopological spaces in which he studied how the conserving properties of connectedness, compactness and paracompactness are preserved by multifunctions between bitopological spaces. In 2000, Noiri and Popa [10] introduced and studied the notion of upper and lower M-continuous multifunctions as a generalization of upper (lower) continuous multifunction and M-continuous function deal to Berge [9] and, Popa and Noiri [7] respectively.
They showed that the upper (lower) continuity of multifunctions has properties similar to those of upper (lower) continuous functions and continuous multifunctions between topological spaces. Recently, Matindih and Moyo [11] have extended the ideas in [10] and studied M-asymmetric semiopen sets and semicontinuous multifunctions from which they observed that, such functions have properties similar to those of upper (lower) continuous functions and M-continuous multifunctions between topological spaces, with the only difference that, the semiopen sets in use belonged to two topologies.
In 1972, the notion of irresolute functions was introduced and their fundamental properties were investigated by Crossley and Hildebrand [1]. They discovered that, most irresolute functions are not necessarily continuous and neither are continuous functions necessarily irresolute. As an extension and generalization of this idea, Ewert and Lipski [12], then studied the concept of upper and lower irresolute multivalued mappings and, followed by Popa [2] who looked at some characterizations of upper and lower irresolute multifunctions in topological spaces.
In this present paper, we introduce and study some basic properties of M-asymmetric irresolute multifunctions defined between sets satisfying certain minimal conditions in the framework of bitopological spaces, as a generalization of results deal to Crossley and Hildebrand [1], Popa [2] and, Noiri and Popa [10].
The organization of this paper is as follows. Section 2 presents some necessary preliminaries concerning semiopen sets, m-asymmetric semiopen sets and M-(asymmetric semi)-continuous multifunctions [11]. In section 3, we present and discuss some results of M-asymmetric irresolute multifunctions as a generalized idea for irresolute functions [1] and upper and lower M-continuous multifunctions [10]. Section 4 gives some concluding remarks.

Preliminaries and Basic Properties
This section presents some important properties and notations to be used in this 2) The i j is the union of all i j T T -open subsets of X contained in A. Evidently, provided 3) The i j T T -closed subsets of X containing A. Note that asymmetrically, space and, let A and B be non-void subsets of X.

1) A is said to be
2) The i j A γ γ ∈ Γ be a family of subsets of X. Then, the properties below hold: Remark 2.9. [11] It should generally be noted that, the intersection of any two ij m -semiopen sets may not be ij m -semiopen in a minimal bitopological space
Lemma 2.15. [11] For an ij m -space ( ) , ij X m , , 1, 2 i j = ; i j ≠ and any none-void subset A of X, the properties below holds: Lemma 2.16. [11] For an ij m -space ( ) , ij X m , , 1, 2 i j = ; i j ≠ and any none-void subset A of X, the properties below are true: The converse to this assertion is not necessarily true.
Example 2.17. [11] In this example, it is shown that, the converse to part (2) of Lemma 2.16 is not necessarily true: Define the two minimal structures 1 m Lemma 2.19. [11] For an ij m -space ( ) , ij X m , , 1, 2 i j = ; i j ≠ with ij m satisfying property (B ) and subsets A and F of X, the properties below holds: Lemma 2.20. [11] For any ij m -space ( ) , ij X m , , 1, 2 i j = ; i j ≠ with ij m satisfying property B and any none-void subset A of X, the properties outlined below holds: Lemma 2.21. [11] Let ( ) satisfying the property B and let { } : A γ γ ∈ Γ be an arbitrary collection of subsets of X. Then, If the property B of Make is removed in the previous Lemma, the equality does not necessarily hold, refer to Example 3.23 [11].
Lemma 2.23. [11] For a minimal bitopological space ( ; i j ≠ , and any subset U of X, the properties below holds: Definition 2.24. [10] A multifunction is a point-to-set correspondence between two topological spaces X and Y such that for each point x of X, In the sense of Berge [9], we shall denote and define the upper and lower inverse of a non-void subset G of Y with respect to a multifunction F respectively by: Generally, F − and F + between Y and the power set For any non-void subsets A and G of X and Y respectively, ( ) ( ) topological spaces X and Y is said to be: x of X provided for any semiopen subset G of 2) lower irresolute at a point o x of X provided for any semiopen subset G of 3) upper (resp lower) irresolute provided it is upper (resp lower) irresolute at all points o x of X.

Some Properties of Upper and Lower M-Asymmetric Irresolute Multifunctions
In this section, we introduce and study a special kind of multifunctions for which the inverse of M-asymmetric semiopen sets is M-asymmetric semiopen.
3) upper (resp lower) M-Asymmetric irresolute provided it is upper (resp lower) M-Asymmetric irresolute at each and every point o x of X. Clearly, Then, F is lower M-asymmetric irresolute but not upper M-asymmetric irre- In this part of the section, we discuss some characterizations involving upper M-asymmetric irresolute multifunctions.
(2) ⇒ (3): Suppose (2) holds. Let K be an For if (5) holds, Lemma 2.13 implies 2) Clearly follows from (1). Indeed, for any point o neighbourhood of o x and by the hypothesis, In this part of the section, we discuss some characterizations involving lower M-asymmetric irresolute multifunctions.    Proof. For necessity, let F be lower M-asymmetric irresolute at a point o x of X and G be an Cl Cl Z by Theorem 3.10 and . Thus, F is a lower M-asymmetric irresolute multifunction at a point o x in X.

satisfies property
B , then the following properties are equivalent for a multifunction Proof.