Upper and Lower Weakly α - (cid:63) -Continuous Multifunctions

. This paper deals with the concepts of upper and lower weakly α - (cid:63) -continuous multifunc-tions. Moreover, some characterizations of upper and lower weakly α - (cid:63) -continuous multifunctions are investigated. Furthermore, the relationships between almost α - (cid:63) -continuity and weak α - (cid:63) -continuity are discussed.


Introduction
Topology is concerned with all questions directly or indirectly related to continuity. Weaker and stronger forms of open sets play an important role in the researches of generalizations of continuity for functions and multifunctions. In 1965, Njåstad [18] introduced a weak form of open sets called α-sets. Mashhour et al. [17] defined a function to be α-continuous if the inverse image of each open set is an α-set and obtained several characterizations of such functions. Noiri [20] investigated the relationships between α-continuous functions and several known functions, for example, almost continuous functions, η-continuous functions, δ-continuous functions or irresolute functions. In [21], the present author introduced the concept of almost α-continuity in topological spaces as a generalization of α-continuity and almost continuity. Neubrunn [19] introduced the notion of upper (resp. lower) αcontinuous multifunctions. These multifunctions are further investigated by the present authors [24].
In 1996, Popa and Noiri [23] introduced the notion of upper (resp. lower) almost α-continuous multifunctions and investigated several characterizations and some basic properties concerning upper (resp. lower) almost α-continuous multifunctions. Moreover, some characterizations of weakly α-continuous multifunctions were investigated in [11], [22] and [23]. Topological ideals have played an important role in topology. Kuratowski [16] and Vaidyanathswamy [25] introduced and studied the concept of ideals in topological spaces. Every topological space is an ideal topological space and all the results of ideal topological spaces are generalizations of the results established in topological spaces. In 1990, Janković and Hamlett [15] introduced the concept of I -open sets in ideal topological spaces. Abd El-Monsef et al. [1] further investigated I -open sets and I -continuous functions. Later, several authors studied ideal topological spaces giving several convenient definitions. Some authors obtained decompositions of continuity. For instance, Açikgöz et al. [4] studied the concepts of α-I -continuity and α-I -openness in ideal topological spaces and obtained several characterizations of these functions.
Hatir and Noiri [14] introduced the notions of semi-I -open sets, α-I -open sets and β-I -open sets via idealization and using these sets obtained new decompositions of continuity. Moreover, Açikgöz et al. [3] introduced and studied the notions of weakly-I -continuous and weak -I -continuous functions in ideal topological spaces. In [10], the present author introduced and investigated the concepts of upper and lower weakly -continuous multifunctions. In this paper, we introduce the concepts of upper and lower weakly α--continuous multifunctions. In particular, several characterizations of upper and lower weakly α--continuous multifunctions are discussed.

Preliminaries
Throughout the present paper, spaces (X, τ ) and (Y, σ) (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a topological space (X, τ ). The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. An ideal I on a topological space (X, τ ) is a nonempty collection of subsets of X satisfying the following properties: (1) A ∈ I and B ⊆ A imply B ∈ I ; (2) A ∈ I and B ∈ I imply A ∪ B ∈ I . A topological space (X, τ ) with an ideal I on X is called an ideal topological space and is denoted by (X, τ, I ). For an ideal topological space (X, τ, I ) and a subset A of X, A (I ) is defined as follows: In case there is no chance for confusion, A (I ) is simply written as A . In [16], A is called the local function of A with respect to I and τ and Cl (A) = A ∪ A defines a Kuratowski closure operator for a topology τ (I ) finer than τ . A subset A is said to be -closed [15] if A ⊆ A. The interior of a subset A in (X, τ (I )) is denoted by Int (A). semi-I -open) set is said to be semi -I -closed [13] (resp. semi-I -closed [14]). For a subset A of an ideal topological space (X, τ, I ), the intersection of all semi-I -closed (resp. semi -I -closed) sets containing A is called the semi-I -closure [14] (resp. semi -I -closure [12]) of s Int I (A)).

Lemma 2.1. [6]
For a subset A of an ideal topological space (X, τ, I ), the following properties hold: Recall that a subset A of an ideal topological space (X, τ, I ) is said to be α--closed [2] if For a subset A of an ideal topological space (X, τ, I ), the intersection of all α--closed sets containing A is called the α--closure [6] of A and is denoted by αCl(A). The α--interior [6] of A is defined by the union of all α--open sets contained in A and is denoted by αInt(A).

Lemma 2.2. [6]
For a subset A of an ideal topological space (X, τ, I ), the following properties hold: For a subset A of an ideal topological space (X, τ, I ), the following properties are equivalent: For a subset A of an ideal topological space (X, τ, I ), the following properties hold: (2) sInt I (Cl (A)) = Cl (Int(Cl (A))).
By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y , and we always assume that F (x) = ∅ for all x ∈ X. For a multifunction F : X → Y , following [5] we shall denote the upper and lower inverse of a set B of Y by F + (B) and F − (B), respectively,

Upper and lower weakly α--continuous multifunctions
We begin this section by introducing the concepts of upper and lower weakly α--continuous multifunctions.
Proof. We prove the only implication (3) ⇒ (1). Suppose that F is lower weakly α--continuous. Let x ∈ X and V be any -open set of Y such that F (x) ∩ V = ∅. Since F is lower weakly α--continuous, there exists an α--open set U of X containing x such that F (z) ∩ Cl (V ) = ∅ for each z ∈ U. Since F (z) is -open, we have F (z) ∩ V = ∅ for each z ∈ U and hence F is lower α--continuous.