NEAR ω-CONTINUOUS MULTIFUNCTIONS ON BITOPOLOGICAL SPACES

In this paper, we introduce and study basic characterizations, several properties of upper (lower) nearly (i, j)-ωcontinuous multifunctions on bitopological space.


Introduction
It is well known that various types of functions play a significant role in the theory of classical point set topology. A great number of papers dealing with such functions have appeared, and a good number of them have been extended to the setting of multifunctions. This shows that both, functions and multifunctions are important tools for studying other properties of spaces and for constructing new spaces from previously existing ones. Generalized open sets play an important role in General Topology and they are now the research topics of many topologists worldwide. Indeed a significant theme in General Topology and Real analysis concerns the introduction of various modified forms of continuity, separation axioms etc. by utilizing generalized open sets. A generalization of closed sets, the notion of ω-closed sets has been introduced and studied by Hdeib [8]. Several characterizations and properties of ω-closed sets has been provided in [2,4,5,6,8,9]. In this paper, we introduce and study upper (lower) nearly (i, j)-ω-continuous multifunctions on bitopological space.

Preliminaries
Throughout this paper, (X, τ 1 , τ 2 ) and (Y, σ 1 , σ 2 ) denote the bitopological spaces in which no separation axioms are assumed unless explicitly stated. Bitopological spaces and its different properties have been investigated by Triparthy and Sarma ([11], [12], [14]), Sarma and Triparthy [15], Tripathy and Acharjee [13], Acharjee and Tripathy [1], Tripathy and Debnath [16], and others. For a subset A of (X, τ ), i Cl(A) (respectively i Int(A)) denote the closure of A with respect to τ i (respectively the interior of A with respect to τ i ). A point x ∈ X is called a condensation point of A if for each U ∈ τ with x ∈ U , the set U ∩ A is uncountable. The set A is said to be ω-closed [8] if it contains all its condensation points. The complement of an ω-closed set is said to be an ω-open set. It is well known that a subset W of a space (X, τ ) is ω-open if and only if for each x ∈ W , there exists U ∈ τ such that x ∈ U and U \W is countable. The family of all ω-open subsets of a topological space (X, τ ) forms a topology on X finer than τ . The intersection of all ω-closed sets containing A is called the ω-closure [8] of A and is denoted by ω Cl(A). For each x ∈ X, the family of all ω-open sets containing x is denoted by ωO(X, x). The family of all ω-open sets of X is denoted by ωO(X). A Multifunction F : X → Y from a topological space X to a topological space Y is a point to set correspondence and is assumed that F (x) ̸ = ∅ for all x ∈ X. We denote the upper and lower inverse of a subset V of Y by [10] if every cover of A by (i, j)-regular open sets of X has a finite subcover.

Upper (lower) nearly (i, j)-ω-continuous multifunctions
has this property at every point of X.
Proof. The proof is similar to that of Theorem 3.5 and hence, F + (G) = iω Int(F + (G)). It follows from Theorem 3.5 that F is upper nearly (i, j)-ω-continuous. The proof of lower nearly (i, j)ω-continuity can be established similarly.

Conversely, let x ∈ ∪
Proof. The proof is similar to that of Theorem 3.8  Proof. We shall established the first case since the proof of the second can be established similar. Let x ∈ D + n(i,j)ω (F ). Then, there exists a σ i -open set V of Y containing F (x) and having (i, j)-N -closed complement such that U ∩ (X\F + (V )) ̸ = ∅ for every (i, j)-open set U containing x. Then we have x ∈ iω Cl(X\F + (V )). On the other hand, since x ∈ F + (V ) ⊂ iω Cl(F + (V )) and hence x ∈ iω-F r(F + (V )). Conversely, suppose that F is upper nearly (i, j)-ω-continuous at x ∈ X. Then for any σ i -open set V of Y containing F (x) and having (i, j)-N -closed complement, there exists U ∈ τ i -ωO(X) containing x such that F (U ) ⊂ V ; hence x ∈ U ⊂ F + (V ). Therefore, x ∈ U ⊂ iω Int(F + (V )). This contradicts to the fact that x ∈ iω-F r(F + (V )).