(τ1, τ2)α-Continuity for Multifunctions

Viriyapong, Chokchai; Boonpok, Chawalit

doi:https://doi.org/10.1155/2020/6285763

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Research Article | Open Access

Volume 2020 | Article ID 6285763 | https://doi.org/10.1155/2020/6285763

(τ₁, τ₂)α-Continuity for Multifunctions

Chokchai Viriyapong¹and Chawalit Boonpok¹

Academic Editor: Frédéric Mynard

Received05 Jan 2020

Accepted18 Feb 2020

Published27 Apr 2020

Abstract

The main goal of this article is to introduce the concepts of upper and lower (τ₁, τ₂)α-continuous multifunctions. Characterizations of upper and lower (τ₁, τ₂)α-continuous multifunctions are investigated. The relationships between upper and lower (τ₁, τ₂)α-continuous multifunctions and the other types of continuity are discussed.

1. Introduction

General topology is important in many fields of applied sciences as well as branches of mathematics. Continuity is a basic concept for the study in topological spaces. Generalization of this concept by using weaker forms of open sets such as α-sets [1], semi-open sets [2], preopen sets [3], and semi-preopen sets [4] is one of the main research topics of general topology. In 1965, Njåstad [1] introduced a weak form of open sets called α-sets. Mashhour et al. [5] defined a function to be α-continuous if the inverse image of each open set is an α-set. Noiri [6] called α-continuous functions strong semi-continuous and investigated α-continuous functions in [7] and investigated α-continuous functions. In [8, 9], the authors investigated a class of functions called almost α-continuous or almost feebly continuous. In 1987, Noiri [10] introduced a class of functions called weakly α-continuous functions. Some characterizations of weakly α-continuous functions are investigated in [11–13]. Neubrunn [14] extended these functions to multifunction and introduced the notions of upper and lower α-continuous multifunctions. Popa and Noiri [15] investigated several characterizations of upper and lower α-continuous multifunctions and some basic properties of such multifunctions. In [16], the present authors introduced upper and lower almost α-continuous multifunctions and obtained some characterizations of upper and lower almost α-continuous multifunctions and several properties of such multifunctions. Recently, Popa and Noiri [17] investigated some characterizations and several properties concerning upper and lower weakly α-continuous multifunctions. In this paper, we introduce the notions of upper and lower (τ₁, τ₂)α-continuous multifunctions and investigate some characterizations of upper and lower (τ₁, τ₂)α-continuous multifunctions. Section 4 is devoted to introducing and studying upper and lower almost (τ₁, τ₂)α-continuous multifunctions. In Section 5, several interesting characterizations of upper and lower weakly (τ₁, τ₂)α-continuous multifunctions are discussed.

2. Preliminaries

Throughout the present paper, spaces (X, τ₁, τ₂) and (Y, σ₁, σ₂) (or simply X and Y) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space (X, τ₁, τ₂). The closure of A and the interior of A with respect to τ_i are denoted by τ_i‐Cl(A) and τ_i‐Int(A), respectively, for i = 1, 2. A subset A of a bitopological space (X, τ₁, τ₂) is called τ₁τ₂-semi-open (resp. τ₁τ₂-regular open [18] and τ₁τ₂-regular closed [19]) if A ⊆ τ₁‐Cl(τ₂‐Int(A)) (resp. A = τ₁‐Int(τ₂‐Cl(A)) and A = τ₁‐Cl(τ₂‐Int(A))). The complement of a τ₁τ₂-semi-open set is said to be τ₁τ₂-semi-closed. The τ₁τ₂-semi-closure of A is defined by the intersection of τ₁τ₂-semi-closed sets containing A and is denoted by τ₁τ₂‐sCl(A). The τ₁τ₂-semi-interior of A is defined by the union of τ₁τ₂-semi-open sets contained in A and is denoted by τ₁τ₂‐sInt(A). A subset A of a bitopological space (X, τ₁, τ₂) is said to be τ₁τ₂-closed [20] if A = τ₁‐Cl(τ₂‐Cl(A)). The complement of a τ₁τ₂-closed set is said to be τ₁τ₂-open. The intersection of all τ₁τ₂-closed sets containing A is called the τ₁τ₂-closure of A and is denoted by τ₁τ₂‐Cl(A). The union of all τ₁τ₂-open sets contained in A is called the τ₁τ₂-interior of A and is denoted by τ₁τ₂‐Int(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 [21], we shall denote the upper and lower inverse of a set B of Y by F ⁺(B) and F⁻(B), respectively, that is, F⁺(B) = {x ∈ X∣F(x) ⊆ B} and F⁻(B) = {x ∈ X ∣ F(x) ∩ B ≠ ∅}. In particular, F⁻(y) = {x ∈ X∣y ∈ F(x)} for each point y ∈ Y. For each A ⊆ X, F(A) = ∪_x∈AF(x). Then, F is said to be surjection if F(X) = Y, or equivalent, if for each y ∈ Y there exists x ∈ X such that y ∈ F(x) and F is called injection if x ≠ y implies F(x) ∩ F(y) = ∅.

Definition 1. (see [22]). A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is said to be:(1)upper (τ₁, τ₂)-continuous at a point x ∈ X if for each σ₁σ₂-open subset V of Y such that F(x) ⊆ V, there exists a τ₁τ₂-open subset U of X containing x such that F(U) ⊆ V;(2)lower (τ₁, τ₂)-continuous at a point x ∈ X if for each σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a τ₁τ₂-open subset U of X containing x such that F(z) ∩ V ≠ ∅ for every z ∈ U;(3)upper (resp. lower) (τ₁, τ₂)-continuous if F has this property at each point of X.

Definition 2. A subset A of a bitopological space (X, τ₁, τ₂) is said to be:(i)(τ₁, τ₂)α-open if A ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(A)));(ii)(τ₁, τ₂)α-closed if its complement is (τ₁, τ₂)α-open.

Definition 3. Let A be a subset of a bitopological (X, τ₁, τ₂). Then,(i)The intersection of all (τ₁, τ₂)α-closed sets containing A is called the (τ₁, τ₂)α‐closure of A and is denoted by (τ₁, τ₂)α- Cl(A).(ii)The union of all (τ₁, τ₂)α‐open sets contained in A is called the (τ₁, τ₂)α – interior of A and is denoted by (τ₁, τ₂)α- Int(A).

Lemma 1. (see [20]). For a subset A of a bitopological space (X, τ₁, τ₂), the following properties hold:(1)τ₁τ₂‐sCl(A) = τ₁‐Int(τ₂‐Cl(A)) ∪ A(2)If A is τ₁τ₂-open, then τ₁τ₂‐sCl(A) = τ₁‐Int(τ₂‐Cl(A))

Lemma 2. For a subset A of a bitopological space (X, τ₁, τ₂), the following properties are equivalent:(1)A is (τ₁, τ₂)α-open(2)UAτ₁‐Int(τ₂‐Cl(U)) for some τ₁τ₂-open set U(3)UAτ₁τ₂‐sCl(U) for some τ₁τ₂-open set U(4)Aτ₁τ₂‐sCl(τ₁τ₂‐Int(A))

Proof. (1) ⟹ (2): Suppose that A is a (τ₁, τ₂)α-open set. Then, we have Put U = τ₁τ₂‐Int(A); then, UAτ₁‐Int(τ₂‐Cl(U)). (2) ⟹ (3): This follows from Lemma 1 (2). (3) ⟹ (4): Suppose that UAτ₁τ₂‐sCl(U) for some τ₁τ₂-open set U. Then, we have Uτ₁τ₂‐Int(A), and hence Aτ₁τ₂‐sCl(τ₁τ₂‐Int(A)). (4) ⟹ (1): Suppose that Aτ₁τ₂‐sCl(τ₁τ₂‐Int(A)). Since τ₁τ₂‐Int(A) is τ₁τ₂-open and by Lemma 1 (2), Aτ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(A))). Thus, A is (τ₁, τ₂)α-open.

Lemma 3. For a subset A of a bitopological space (X, τ₁, τ₂), the following properties hold:(1)A is (τ₂, τ₂)α-closed if and only if τ₁τ₂‐sInt(τ₁τ₂‐Cl(A)) A(2)τ₁τ₂‐sInt(τ₁τ₂‐Cl(A)) = τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A)))(3)α(τ₁, τ₂)‐Cl(A) = A ∪ τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A)))

Proof. (1)Let A be a (τ₁, τ₂)α-closed set. Then, X – A is (τ₁, τ₂)α–open and by Lemma 2, X – Aτ₁τ₂‐sCl(τ₁τ₂‐Int(X – A)) = X − τ₁τ₂‐sInt(τ₁τ₂‐Cl(A)). Thus, τ₁τ₂‐sInt(τ₁τ₂‐Cl(A)) ⊆ A. Conversely, suppose that τ₁τ₂‐sInt(τ₁τ₂‐Cl(A)) A. Then, By Lemma 2, we have X − A is (τ₁, τ₂)α-open, and hence A is (τ₁, τ₂)α-closed.(2)Since X – τ₁τ₂‐sInt(τ₁τ₂‐Cl(A)) = τ₁τ₂‐sCl(τ₁τ₂‐Int(X – A)), by Lemma 1 (2), and hence τ₁τ₂‐sInt(τ₁τ₂‐Cl(A)) = τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A))).(3)We observe thatThus, A ∪ τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A))) is (τ₁, τ₂)α-closed, and hence (τ₁, τ₂)α‐Cl(A) A ∪ τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A))). On the other hand, since (τ₁, τ₂)α‐Cl(A) is (τ₁, τ₂)α-closed, we haveTherefore, A ∪ τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A))) (τ₁, τ₂)α‐Cl(A). Consequently, we obtain (τ₁, τ₂)α‐Cl(A) = A ∪ τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A))).

3. Characterizations of Upper and Lower α(τ₁, τ₂)-Continuous Multifunctions

In this section, we introduce the notions of upper and lower (τ₁, τ₂)α-continuous multifunctions. Moreover, several characterizations of upper and lower (τ₁, τ₂)α-continuous multifunctions are discussed.

Definition 4. A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is said to be:(1)upper (τ₁, τ₂)α-continuous at a point x ∈ X if for each σ₁σ₂-open subset V of Y such that F(x) V, there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) V;(2)lower (τ₁, τ₂)α-continuous at a point x ∈ X if for each σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(z) ∩ V ≠ ∅ for every z ∈ U;(3)upper (resp. lower) (τ₁, τ₂)α-continuous if F has this property at each point of X.

Remark 1. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following implication holds:
upper (τ₁, τ₂)-continuity ⟹ upper (τ₁, τ₂)α-continuity.
The converse of the implication is not true in general. We give an example for the implication as follows.

Example 1. Let X = {a, b, c, d} with topologies τ₁ = {∅, {a}, {b}, {a, b}, X} and τ₂ = {∅, {b}, X}. Let Y = {−2, −1, 0, 1, 2} with topologiesand σ₂ = {∅, {−1, −2, 2}, Y}. A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is defined as follows:Then, F is upper (τ₁, τ₂)α-continuous but F is not upper (τ₁, τ₂)-continuous.

Theorem 1. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is upper (τ₁, τ₂)α-continuous at a point x of X(2)x ∈ τ₁τ₂‐sCl(τ₁τ₂‐Int(F⁺(V))) for each σ₁σ₂-open subset V of Y containing F(x)(3)For each τ₁τ₂-semiopen subset U of X containing x and each σ₁σ₂-open subset V of Y containing F(x), there exists a nonempty τ₁τ₂-open subset G of X such that G ⊆ U and F(G) ⊆ V

Proof. (1) ⟹ (2): Let V be any σ₁σ₂-open subset of Y such that F(x) V. Then, there exists a (τ₁, τ₂)α-open set U containing x such that F(U) V; hence, x ∈ UF⁺(V). Since U is (τ₁, τ₂)α-open, by Lemma 2, we have x ∈ U ⊆ τ₁τ₂‐sCl(τ₁τ₂‐Int(U)) ⊆ τ₁τ₂‐sCl(τ₁τ₂‐IntF⁺(V))). (2) ⟹ (3): Let V be any σ₁σ₂-open subset of Y such that F(x) ⊆ V. Then, we have x ∈ τ₁τ₂‐sCl (τ₁τ₂‐ Int(F⁺(V))). Let U be any τ₁τ₂-semi-open set containing x. Thus, U ∩ [τ₁τ₂‐Int(F⁺(V))] ≠ ∅ and U ∩ [τ₁τ₂‐Int(F⁺(V))] is τ₁τ₂-semi-open. Put G = τ₁τ₂‐ Int(U ∩ [τ₁τ₂‐ Int(F⁺(V))]); then, G is a nonempty τ₁τ₂-open subset of X such that G ⊆ U and F(G) ⊆ V. (3) ⟹ (1): Let τ₁τ₂‐SO(X, x) be the family of all τ₁τ₂-semi-open subsets of X containing x. Let V be any σ₁σ₂-open subset of Y containing F(x). For each U ∈ τ₁τ₂‐SO(X, x), there exists a nonempty τ₁τ₂-open set H_U such that H_U ⊆ U and F(H_U) ⊆ V. Let W = ∪{H_U∣U ∈ τ₁τ₂‐SO(X, x)}. Then, we have W which is τ₁τ₂-open in X, x ∈ τ₁τ₂‐sCl(W), and F(W) ⊆ V. Put G = W ∪ {x}. By Lemma 1, we haveand by Lemma 2, G is a (τ₁, τ₂)α-open subset of X containing x such that F(G) ⊆ V. This shows that F is upper (τ₁, τ₂)α-continuous at x.

Theorem 2. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is lower (τ₁, τ₂)α-continuous at a point x of X(2)x ∈ τ₁τ₂‐sCl(τ₁τ₂‐Int(F^‐(V))) for each σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅(3)For each τ₁τ₂-semiopen subset U of X containing x and each σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a nonempty τ₁τ₂-open subset G of X such that G ⊆ U and F(z) ∩ V ≠ ∅ for every z ∈ G

Proof. The proof is similar to that of Theorem 1.

Definition 5. A subset A of a bitopological space (X, τ₁, τ₂) is said to be a τ₁τ₂-neighbourhood (resp. (τ₁, τ₂)α-neighbourhood) of x ∈ X if there exists a τ₁τ₂-open (resp. (τ₁, τ₂)α-open) set U such that x ∈ U ⊆ A.

Theorem 3. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is upper (τ₁, τ₂)α-continuous(2)F⁺(V) is (τ₁, τ₂)α-open in X for each σ₁σ₂-open subset V of Y(3)F^‐(H) is (τ₁, τ₂)α-closed in X for each σ₁σ₂-closed subset H of Y(4)τ₁τ₂‐sInt(τ₁τ₂‐Cl(F^‐(B))) ⊆ F^‐(σ₁σ₂‐Cl(B)) for each subset B of Y(5)F⁺(V) ⊆ τ₁τ₂‐sCl(τ₁τ₂‐Int(F⁺(V))) for each σ₁σ₂-open subset V of Y(6)(τ₁, τ₂)α‐Cl(F^‐(B)) ⊆ F^‐(σ₁σ₂‐Cl(B)) for each subset B of Y(7)For each x ∈ X and each σ₁σ₂-neighbourhood V of F(x), F⁺(V) is a (τ₁, τ₂)α-neighbourhood of x(8)For each x ∈ X and each σ₁σ₂-neighbourhood V of F(x), there exists a (τ₁, τ₂)α-neighbourhood U of x such that F(U) ⊆ V

Proof. (1) ⟹ (2): Let V be any σ₁σ₂-open subset of Y and x ∈ F⁺(V). By Theorem 1, x ∈ τ₁τ₂‐sCl(τ₁τ₂‐Int(F⁺(V))), and hence It follows from Lemma 2 that F⁺(V) is (τ₁, τ₂)α-open in X. (2) ⟹ (3): This follows from the fact that F⁺(Y – B) = X – F^‐(B) for any subset B of Y. (3) ⟹ (4): Let B be any subset of Y. Then, σ₁σ₂‐Cl(B) is σ₁σ₂-closed in Y and by (3), F^‐(σ₁σ₂‐Cl(B)) is α(τ₁, τ₂)-closed in X. By Lemma 3 (1), (4) ⟹ (5): Let V be any σ₁σ₂-open subset of Y. Then, we have Y – V is σ₁σ₂-closed in Y. By (4), and hence F⁺(V) ⊆ τ₁τ₂‐sCl(τ₁τ₂‐Int(F⁺(V))).(5)⟹ (6): Let B be any subset of Y. Then, Y – σ₁σ₂‐Cl(B) is σ₁σ₂-open in Y, and by (5), Thus, τ₁τ₂‐sInt(τ₁τ₂‐Cl(F^‐(σ₁σ₂‐Cl(B)))) ⊆ F^‐(σ₁σ₂‐Cl(B)). By Lemma 3, (6) ⟹ (3): Let H be any σ₁σ₂-closed subset of Y. By (6), we have and hence F^‐(H) is (τ₁, τ₂)α-closed in X. (2) ⟹ (7): Let x ∈ X and V be a σ₁σ₂-neighbourhood of F(x). Then, there exists a σ₁σ₂-open subset U of Y such that F(x) ⊆ U ⊆ V. Thus, x ∈ F⁺(U) ⊆ F⁺(V). By (2), we have F⁺(U) is (τ₁, τ₂)α-open in X and hence F⁺(V) is a (τ₁, τ₂)α-neighbourhood of x. (7) ⟹ (8): Let x ∈ X and V be a σ₁σ₂-neighbourhood of F(x). By (7), F⁺(V) is a (τ₁, τ₂)α-neighbourhood of x. Put U = F⁺(V); then, U is a (τ₁, τ₂)α-neighbourhood of x and F(U) ⊆ V. (8) ⟹ (1): Let x ∈ X and V be any σ₁σ₂-open subset of Y such that F(x) ⊆ V. Then, V is a σ₁σ₂-neighbourhood of F(x) and by (8), there exists a (τ₁, τ₂)α-neighbourhood U of x such that F(U) ⊆ V. Since U is (τ₁, τ₂)α-neighbourhood of x, there exists a (τ₁, τ₂)α-open set W such that x ∈ W ⊆ U; hence, F(W) ⊆ V. This shows that F is upper (τ₁, τ₂)α-continuous at x. Consequently, we obtain that F is upper (τ₁, τ₂)α-continuous.

Theorem 4. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is lower (τ₁, τ₂)α-continuous(2)F^‐(V) is (τ₁, τ₂)α-open in X for every σ₁σ₂-open subset V of Y(3)F⁺(H) is (τ₁, τ₂)α-closed in X for every σ₁σ₂-closed subset H of Y(4)τ₁τ₂‐sInt(τ₁τ₂‐Cl(F⁺(B))) ⊆ F⁺(σ₁σ₂‐Cl(B)) for every subset B of Y(5)(τ₁, τ₂)α‐Cl(F⁺(B)) ⊆ F⁺(σ₁σ₂‐Cl(B)) for every subset B of Y(6)F((τ₁, τ₂)α‐Cl(A)) ⊆ σ₁σ₂‐Cl(F(A)) for every subset A of X(7)F(τ₁τ₂‐sInt(τ₁τ₂‐Cl(A))) ⊆ σ₁σ₂‐Cl(F(A)) for every subset A of X(8)F(τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A)))) ⊆ σ₁σ₂‐Cl(F(A)) for every subset A of X

Proof. (1) ⟹ (2): Let V be any σ₁σ₂-open subset of Y and x ∈ F^‐(V). By Theorem 2, x ∈ τ₁τ₂‐sCl(τ₁τ₂‐Int(F^‐(V))), and hence It follows from Lemma 2 that F^‐(V) is (τ₁, τ₂)α-open in X. (2) ⟹ (3): This follows from the fact that F⁺(Y – B) = X – F^‐(B) for any subset B of Y. (3) ⟹ (4): Let B be any subset of Y. Then, we have σ₁σ₂‐Cl(B) is σ₁σ₂-closed in Y and by (3), F⁺(σ₁σ₂‐Cl(B)) is α(τ₁, τ₂)-closed in X. By Lemma 3 (1), (4) ⟹ (5): Let B be any subset of Y. By (4) and Lemma 3, we have (5) ⟹ (6): Let A be any subset of X. Since A ⊆ F⁺(F(A)), we have and hence F((τ₁, τ₂)α‐Cl(A)) ⊆ σ₁σ₂‐Cl(F(A)). (6) ⟹ (7): This follows immediately from Lemma 3. (7) ⟹ (8): Let A be any subset of X. By (7) and Lemma 3 (2), we have (8) ⟹ (1): Let x ∈ X and V be any σ₁σ₂-open subset of Y such that F(x) ∩ V ≠ ∅. Then, we have x ∈ F^‐(V). We shall show that F^‐(V) is (τ₁, τ₂)α-open in X. By the hypothesis, F(τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁺(Y – V))))) ⊆ σ₁σ₂‐Cl(F(F⁺(Y – V))) ⊆ Y – V. Thus, τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁺(Y – V)))) ⊆ F⁺(Y – V) = X − F^‐(V), and hence F^‐(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F^‐(V)))). This shows that F^‐(V) is α(τ₁, τ₂)-open in X. Put U = F^‐(V); then, U is a (τ₁, τ₂)α-open set containing x such that F(z) ∩ V ≠ ∅ for every z ∈ U. Consequently, we obtain that F is lower (τ₁, τ₂)α-continuous.

Definition 6. A function f: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is said to be (τ₁, τ₂)α-continuous if for every σ₁σ₂-open subset V of Y, f⁻¹(V) is (τ₁, τ₂)α-open in X.

Corollary 1. For a function f: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)f is (τ₁, τ₂)α-continuous(2)f ^‐1(F) is (τ₁, τ₂)α-closed in X for each σ₁σ₂-closed subset F of Y(3)τ₁τ₂‐sInt(τ₁τ₂‐Cl(f ^‐1(B))) ⊆ f ^‐1(σ₁σ₂‐Cl(B)) for each subset B of Y(4)(τ₁, τ₂)α‐Cl(f ^‐1(B)) ⊆ f ^‐1(σ₁σ₂‐Cl(B)) for each subset B of Y(5)For each x ∈ X and each σ₁σ₂-neighbourhood V of f(x), f ^‐1(V) is a (τ₁, τ₂)α-neighbourhood of x(6)For each x ∈ X and each σ₁σ₂-neighbourhood V of f(x), there exists a (τ₁, τ₂)α-neighbourhood U of x such that f(U) ⊆ V(7)f((τ₁, τ₂)α‐Cl(A)) ⊆ σ₁σ₂‐Cl(f(A)) for each subset A of X(8)f(τ₁τ₂‐sInt(τ₁τ₂‐Cl(A))) ⊆ σ₁σ₂‐Cl(f(A)) for each subset A of X(9)f(τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A)))) ⊆ σ₁σ₂‐Cl(f(A)) for each subset A of X

Lemma 4. (see [20]). For subsets A, B of a bitopological space (X, τ₁, τ₂), the following properties hold:(1)A ⊆ τ₁τ₂‐ker(A)(2)If A ⊆ B, then τ₁τ₂‐ker(A) ⊆ τ₁τ₂‐ker(B)(3)If A is τ₁τ₂-open, then τ₁τ₂‐ker(A) = A(4)x ∈ τ₁τ₂‐ker(A) if and only if A ∩ H≠∅ for any τ₁τ₂-closed set H containing x

Theorem 5. Let F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) be a multifunction. Iffor every subset A of X, then F is upper (τ₁, τ₂)α-continuous.

Proof. Let V be any σ₁σ₂-open subset of Y. By Lemma 4 (3), we have F⁺(V) = F⁺(σ₁σ₂‐ker(V)) ⊆ (τ₁, τ₂)α‐Int(F⁺(V)), and hence (τ₁, τ₂)α‐Int(F⁺(V)) = F⁺(V). Thus, F⁺(V) is (τ₁, τ₂)α-open, by Theorem 3, F is upper (τ₁, τ₂)α-continuous.

Theorem 6. Let F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) be a multifunction. Iffor every subset A of X, then F is lower (τ₁, τ₂)α-continuous.

Proof. The proof is similar to that of Theorem 5.

Definition 7. (see [20]). A collection of subsets of a bitopological space (X, τ₁, τ₂) is said to be τ₁τ₂-locally finite if for each x ∈ X has a τ₁τ₂-neighbourhood which intersects only finitely many elements of .

Definition 8. (see [20]). A subset A of a bitopological space (X, τ₁, τ₂) is said to be(1)τ₁τ₂-paracompact if every cover of A by τ₁τ₂-open sets of X is refined by a cover of A which consists of τ₁τ₂-open sets of X and is τ₁τ₂-locally finite in X(2)τ₁τ₂-regular if for each x ∈ A and each τ₁τ₂-open set U of X containing x, there exists a τ₁τ₂-open set V of X such that x ∈ V ⊆ τ₁τ₂‐Cl(V) ⊆ U

Lemma 5. (see [20]). If A is a τ₁τ₂-regular τ₁τ₂-paracompact set of a bitopological space (X, τ₁, τ₂) and U is a τ₁τ₂-open neighbourhood of A, then there exists a τ₁τ₂-open set V of X such that A ⊆ V ⊆ τ₁τ₂‐Cl(V) ⊆ U.

Definition 9. A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is called punctually (τ₁, τ₂)-paracompact (resp. punctually (τ₁, τ₂)-regular) if for each x ∈ X, F(x) is τ₁τ₂-paracompact (resp. τ₁τ₂-regular).
For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), bywe shall denote a multifunction defined as follows: αClF_⊛(x) = (σ₁, σ₂)α‐Cl(F(x)) for each x ∈ X.

Lemma 6. Let (X, τ₁, τ₂) be a bitopological space. Then, (τ₁, τ₂)α‐Cl(A) ⊆ τ₁τ₂‐Cl(A) for every subset A of X.

Lemma 7. If F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is punctually (τ₁, τ₂)-paracompact and punctually (τ₁, τ₂)-regular, then for every σ₁σ₂-open subset V of Y.

Proof. Let V be any σ₁σ₂-open subset V of Y and . Then, we have (σ₁, σ₂)α‐Cl(F(x)) ⊆ V and F(x) ⊆ V. Thus, x ∈ F⁺(V), and hence . On the other hand, let x ∈ F⁺(V). Then, we have F(x) ⊆ V and by Lemma 5, there exists a σ₁σ₂-open subset U of Y such that F(x) ⊆ σ₁σ₂‐Cl(U) ⊆ U ⊆ V. By Lemma 6, (σ₁, σ₂)α‐Cl(F(x)) ⊆ σ₁σ₂‐Cl(U) ⊆ V, and hence . Thus, . Consequently, we obtain .

Theorem 7. Let F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) be punctually (τ₁, τ₂)-paracompact and punctually (τ₁, τ₂)-regular. Then, F is upper (τ₁, τ₂)α-continuous if and only if αClF_⊛: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is upper (τ₁, τ₂)α-continuous.

Proof. Suppose that F is upper α(τ₁, τ₂)-continuous. Let x ∈ X and V be any σ₁σ₂-open subset of Y such that αClF_⊛(x) ⊆ V. By Lemma 7, we have . Since F is upper (τ₁, τ₂)α-continuous, there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ V. Since F(z) is σ₁σ₂-paracompact and σ₁σ₂-regular for each z ∈ U, by Lemma 5, there exists a σ₁σ₂-open set H such that F(z) ⊆ H ⊆ σ₁σ₂‐Cl(H) ⊆ V. By Lemma 6, we have (σ₁, σ₂)α‐Cl(F(z)) ⊆ σ₁σ₂‐Cl(H) ⊆ V for each z ∈ U, and hence αClF_⊛(U) ⊆ V. This shows that αClF_⊛ is upper (τ₁, τ₂)α-continuous.
Conversely, suppose that αClF_⊛: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is upper (τ₁, τ₂)α-continuous. Let x ∈ X and V be any σ₁σ₂-open subset of Y such that F(x) ⊆ V. By Lemma 7, we have , and hence αClF_⊛(x) ⊆ V. Since αClF_⊛ is upper (τ₁, τ₂)α-continuous, there exists a (τ₁, τ₂)α-open subset U of X containing x such that αClF_⊛(U) ⊆ V; hence, F(U) ⊆ V. This shows that F is upper (τ₁, τ₂)α-continuous.

Lemma 8. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), it follows that for each (σ₁, σ₂)α-open subset V of.

Proof. Suppose that V is any (σ₁, σ₂)α-open subset of Y. Let . Then, we have (σ₁, σ₂)α‐Cl(F(x)) ∩ V ≠ ∅, and hence F(x) ∩ V ≠ ∅. Thus, x ∈ F⁻(V). This shows that . On the other hand, let x ∈ F⁻(V). Then, we have ∅ ≠ F(x) ∩ V ⊆ (σ₁, σ₂)α‐Cl(F(x)) ∩ V, and hence . Therefore, . Consequently, we obtain .

Theorem 8. A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is lower (τ₁, τ₂)α-continuous if and only if αClF_⊛: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is lower (τ₁, τ₂)α-continuous.

Proof. By utilizing Lemma 8, this can be proved similarly to that of Theorem 7.

Definition 10. A bitopological space (X, τ₁, τ₂) is said to be τ₁τ₂-compact [20] (resp. (τ₁, τ₂)α-compact) if every cover of X by τ₁τ₂-open (resp. (τ₁, τ₂)α-open) sets of X has a finite subcover.

Theorem 9. Let F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) be an upper (τ₁, τ₂)α-continuous surjective multifunction such that F(x) is σ₁σ₂-compact for each x ∈ X. If (X, τ₁, τ₂) is (τ₁, τ₂)α-compact, then (Y, σ₁, σ₂) is σ₁σ₂-compact.

Proof. Suppose that (X, τ₁, τ₂) is (τ₁, τ₂)α-compact. Let {V_γ∣γ ∈ Γ} be a σ₁σ₂-open cover of Y. For each x ∈ X, F(x) is σ₁σ₂-compact, there exists a finite subset Γ(x) of Γ such that F(x) ⊆ ∪{V_γ∣γ ∈Γ(x)}. Put V(x) = ∪{V_γ∣γ ∈Γ(x)}. Since F is upper (τ₁, τ₂)α-continuous, there exists a (τ₁, τ₂)α-open set U(x) containing x such that F(U(x)) ⊆ V(x). The family {U(x) ∣ x ∈ X} is a (τ₁, τ₂)α-open cover of X and there exists a finite number of points, say, x₁, x₂, …, x_n in X such that . Thus, . This shows that (Y, σ₁, σ₂) is σ₁σ₂-compact.

4. Characterizations of Upper and Lower Almost (τ₁, τ₂)α-Continuous Multifunctions

In this section, we introduce the notions of upper and lower almost (τ₁, τ₂)α-continuous multifunctions and investigate some characterizations of these multifunctions.

Definition 11. A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is said to be:(1)upper almost (τ₁, τ₂)α-continuous at a point x ∈ X if for each τ₁τ₂-semi-open subset U of X containing x and each σ₁σ₂-open subset V of Y containing F(x), there exists a nonempty (τ₁, τ₂)α-open subset G of X such that G ⊆ U and F(G) ⊆ σ₁σ₂‐sCl(V);(2)lower almost (τ₁, τ₂)α-continuous at a point x ∈ X if for each τ₁τ₂-semi-open subset U of X containing x and each σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a nonempty (τ₁, τ₂)α-open subset G of X such that G ⊆ U and σ₁σ₂‐sCl(V) ∩ F(z) ≠ ∅ for every z ∈ G;(3)upper almost (resp. lower almost) (τ₁, τ₂)α-continuous if F has this property at each point of X.

Remark 2. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following implication holds:
upper (τ₁, τ₂)α-continuity ⟹ upper almost (τ₁, τ₂)α-continuity.
The converse of the implication is not true in general. We give an example for the implication as follows.

Example 2. Let X = {a, b, c} with topologies τ₁ = {∅, {c}, {a, b}, X} and τ₂ = {∅, {c}, X}. Let Y = {−2, −1, 0, 1, 2} with topologies σ₁ = {∅, {−1, 0, 1}, {−2, 2}, Y} and σ₂ = {∅, {−1, 0, 1}, Y}. Define a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) as follows:Then, F is upper almost (τ₁, τ₂)α-continuous but F is not upper (τ₁, τ₂)α-continuous.

Theorem 10. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is upper almost (τ₁, τ₂)α-continuous at x ∈ X(2)For every σ₁σ₂-open subset V of Y containing F(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐sCl(V)(3)x ∈ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₁‐sCl(V))) for every σ₁σ₂-open subset V of Y containing F(x)(4)x ∈ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₂‐sCl(V))))) for every σ₁σ₂-open subset V of Y containing F(x)

Proof. (1) ⟹ (2): Let τ₁τ₂‐SO(X, x) be the family of all τ₁τ₂-semi-open subsets of X containing x. Let V be any σ₁σ₂-open subset of Y containing F(x). For each U ∈ τ₁τ₂‐SO(X, x), there exists a nonempty τ₁τ₂-open subset G_U such that G_U ⊆ U and F(G_U) ⊆ σ₁σ₂‐sCl(V). Let W = ∪ {G_U ∣ U ∈ τ₁τ₂‐SO(X, x)}. Then, W is τ₁τ₂-open, x ∈ τ₁τ₂‐sCl(W) and F(W) ⊆ σ₁σ₂‐sCl(V). Put H = W ∪ {x}; by Lemma 1, and by Lemma 2, H is a (τ₁, τ₂)α-open subset of X containing x such that F(H) ⊆ σ₁σ₂‐sCl(V). (2) ⟹ (3): Let V be any σ₁σ₂-open subset of Y containing F(x). By (2), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐Cl(V). Thus, x ∈ U ⊆ F⁺(σ₁σ₂‐Cl(V)), and hence x ∈ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₁‐sCl(V))). (3) ⟹ (4):Let V be any σ₁σ₂-open subset of Y containing F(x). By (3), we have x ∈ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₁‐sCl(V))). Put U = (τ₁, τ₂)α‐Int(F⁺(σ₁σ₁‐sCl(V))); then, U is a (τ₁, τ₂)α-open subset of X such that x ∈ U ⊆ F⁺(σ₁σ₂‐sCl(V)). Consequently, we obtain x ∈ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₂‐sCl(V))))). (4) ⟹ (1): Let U be any τ₁τ₂-semi-open subset of X containing x and V be any σ₁σ₂-open subset of Y containing F(x). By (4) and Lemma 1 (2),Thus, τ₁τ₂‐Int(F⁺(σ₁σ₂‐sCl(V))) ∩ U ≠ ∅ and τ₁τ₂‐Int(F⁺(σ₁σ₂‐sCl(V))) ∩ U is τ₁τ₂-semi-open. Put G = τ₁τ₂‐Int(τ₁τ₂‐Int(F⁺(σ₁σ₂‐sCl(V))) ∩ U); then, G is a nonempty τ₁τ₂-open subset of X such that G ⊆ U and F(G) ⊆ σ₁σ₂‐sCl(V). This shows that F is upper almost (τ₁, τ₂)α-continuous at x.

Theorem 11. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is lower almost (τ₁, τ₂)α-continuous at x ∈ X(2)For every σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a (τ₁, τ₂)α-open subset U of X containing x such that σ₁σ₂‐sCl(V) ∩ F(z) ≠ ∅ for every z ∈ U(3)x ∈ (τ₁, τ₂)α‐Int(F⁻(σ₁σ₁‐sCl(V))) for every σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅(4)x ∈ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁻(σ₁σ₂‐sCl(V))))) for every σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅

Proof. The proof is similar to that of Theorem 10.

Lemma 9. Let (X, τ₁, τ₂) be a bitopological space. If V is τ₁τ₂-open in X, then τ₁τ₂‐sCl(V) is τ₁τ₂-regular open.

Proof. Let V be a τ₁τ₂-open set. By Lemma 1 (2), τ₁τ₂‐sCl(V) = τ₁‐Int(τ₂‐Cl(V)), and henceThus, τ₁τ₂‐sCl(V) is τ₁τ₂-regular open.

Theorem 12. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is upper almost (τ₁, τ₂)α-continuous(2)For each x ∈ X and each σ₁σ₂-open subset V of Y containing F(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐sCl(V)(3)For each x ∈ X and each σ₁σ₂-regular open subset V of Y containing F(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ V(4)F⁺(V) is (τ₁, τ₂)α-open in X for every σ₁σ₂-regular open subset V of Y(5)F⁻(K) is (τ₁, τ₂)α-closed in X for every σ₁σ₂-regular closed subset K of Y(6)F⁺(V) ⊆ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₂‐sCl(V))) for every σ₁σ₂-open subset V of Y(7)(τ₁, τ₂)α‐Cl(F⁻(σ₁σ₂‐sInt(K))) ⊆ F⁻(K) for every σ₁σ₂-closed subset K of Y(8)(τ₁, τ₂)α‐Cl(F⁻(σ₁‐Cl(σ₂‐Int(K)))) ⊆ F⁻(K) for every σ₁σ₂-closed subset K of Y(9)(τ₁, τ₂)α‐Cl(F⁻(σ₁‐Cl(σ₂‐Int(σ₁σ₂‐Cl(B))))) ⊆ F⁻(σ₁σ₂‐Cl(B)) for every subset B of Y(10)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁻(σ₁‐Cl(σ₂‐Int(K)))))) ⊆ F⁻(K) for every σ₁σ₂-closed subset K of Y(11)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁻(σ₁σ₂‐sInt(K))))) ⊆ F⁻(K) for every σ₁σ₂-closed subset K of Y(12)F⁺(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₂‐sCl(K))))) for every σ₁σ₂-open subset V of Y

Proof. (1) ⟹ (2): The proof follows from Theorem 10. (2) ⟹ (3): This is obvious. (3) ⟹ (4): Let V be any σ₁σ₂-regular open subset of Y and x ∈ F⁺(V). Then, we have F(x) ⊆ V. By (3), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ V. Thus, x ∈ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(U))) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(V)))), and hence F⁺(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(V)))). This shows that F⁺(V) is (τ₁, τ₂)α-open in X. (4) ⟹ (5): This follows from the fact that F⁺(Y – B) = X – F^‐(B) for every subset B of Y. (5) ⟹ (6): Let V be any σ₁σ₂-open subset of Y and x ∈ F⁺(V). Then, we have F(x) ⊆ V ⊆ σ₁σ₂‐sCl(V), and hence By Lemma 9, Y – σ₁σ₂‐sCl(V) is σ₁σ₂-regular closed in Y, and by (5), F^‐(Y – σ₁σ₂‐sCl(V)) is (τ₁, τ₂)α-closed in X. This implies that F⁺(σ₁σ₂‐sCl(V)) is (τ₁, τ₂)α-open in X, and hence x ∈ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₂‐sCl(V))). Consequently, we obtain F⁺(V) ⊆ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₂‐sCl(V))). (6) ⟹ (7): Let K be any σ₁σ₂-closed subset of Y. Then, we have Y‐K is σ₁σ₂-open in Y and by (6), Therefore, (τ₁, τ₂)α‐Cl(F^‐(σ₁σ₂‐sInt(K))) ⊆ F^‐(K). (7) ⟹ (8): The proof is obvious since σ₁σ₂‐sInt(K) = σ₁‐Cl(σ₂‐Int(K)) for every σ₁σ₂-closed subset K of Y. (8) ⟹ (9): The proof is obvious. (9) ⟹ (10): Let K be any σ₁σ₂-closed subset of Y. By (9) and Lemma 3 (3), we obtain (10) ⟹ (11): The proof is obvious since σ₁σ₂‐sInt(K) = σ₁‐Cl(σ₂‐Int(K)) for every σ₁σ₂-closed subset K of Y. (11) ⟹ (12): Let V be any σ₁σ₂-open subset of Y. Then, we have Y – V is σ₁σ₂-closed in Y and by (11), Moreover, we have and hence F⁺(V) ⊆ τ₁ ‐ Int(τ₂ ‐ Cl(τ₁τ₂ ‐ Int(F⁺(σ₁σ₂ ‐ sCl(V))))).(12)⟹ (1): Let x ∈ X and V be any σ₁σ₂-open subset of Y containing F(x). By (12), we have x ∈ F⁺(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₂‐sCl(V))))), and hence F is upper almost (τ₁, τ₂)α-continuous at x by Theorem 10. Consequently, we obtain F is upper almost (τ₁, τ₂)α-continuous.

Theorem 13. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is lower almost (τ₁, τ₂)α-continuous(2)For each x ∈ X and each σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a (τ₁, τ₂)α-open subset U of X containing x such that U ⊆ F^‐(σ₁σ₂‐sCl(V))(3)For each x ∈ X and each σ₁σ₂-regular open subset V of Y such that there exists a (τ₁, τ₂)α-open subset U of X containing x such that U ⊆ F^‐(V)(4)F^‐(V) is (τ₁, τ₂)α-open in X for every σ₁σ₂-regular open subset V of Y(5)F⁺(K) is (τ₁, τ₂)α-closed in X for every σ₁σ₂-regular closed subset K of Y(6)F^‐(V) ⊆ α(τ₁, τ₂)‐Int(F^‐(σ₁σ₂‐sCl(V))) for every σ₁σ₂-open subset V of Y(7)(τ₁, τ₂)α‐Cl(F⁺(σ₁σ₂‐sInt(K))) ⊆ F⁺(K) for every σ₁σ₂-closed subset K of Y(8)(τ₁, τ₂)α‐Cl(F⁺(σ₁‐Cl(σ₂‐Int(K)))) ⊆ F⁺(K) for every σ₁σ₂-closed subset K of Y(9)(τ₁,τ₂) α‐Cl(F⁺(σ₁‐Cl(σ₂‐Int(σ₁σ₂‐Cl(B))))) ⊆ F⁺(σ₁σ₂‐Cl(B)) for every subset B of Y(10)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁺(σ₁‐Cl(σ₂‐Int(K)))))) ⊆ F⁺(K) for every σ₁σ₂-closed subset K of Y(11)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁺(σ₁σ₂‐sInt(K))))) ⊆ F⁺(K) for every σ₁σ₂-closed subset K of Y(12)F^‐(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F^‐(σ₁σ₂‐sCl(K))))) for every σ₁σ₂-open subset V of Y

Proof. The proof is similar to that of Theorem 12.

Definition 12. A function f: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is said to be almost (τ₁, τ₂)α-continuous if f^‐1(V) is (τ₁, τ₂)α-open in X for every σ₁σ₂-regular open subset V of Y.

Corollary 2. For a function f: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)f is almost (τ₁, τ₂)α-continuous(2)For each x ∈ X and each σ₁σ₂-open subset V of Y containing f(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that f(U) ⊆ σ₁σ₂‐sCl(V)(3)For each x ∈ X and each σ₁σ₂-regular open subset V of Y containing f(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that f(U) ⊆ V(4)For each x ∈ X and each σ₁σ₂-open subset V of Y containing f(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that f(U) ⊆ σ₁‐Int(σ₂‐Cl(V))(5)f⁻¹(F) is (τ₁, τ₂)α-closed in X for every σ₁σ₂-regular closed subset F of Y(6)f⁻¹(V) ⊆ (τ₁, τ₂)α‐Int(f⁻¹(σ₁σ₂‐sCl(V))) for every σ₁σ₂-open subset V of Y(7)(τ₁, τ₂)α‐Cl(f⁻¹(σ₁σ₂‐sInt(F))) ⊆ f⁻¹(F) for every σ₁σ₂-closed subset F of Y(8)(τ₁, τ₂)α‐Cl(f⁻¹(σ₁‐Cl(σ₂‐Int(F)))) ⊆ f⁻¹(F) for every σ₁σ₂-closed subset F of Y(9)(τ₁,τ₂)α‐Cl(f⁻¹(σ₁‐Cl(σ₂‐Int(σ₁σ₂‐Cl(B))))) ⊆ f⁻¹(σ₁σ₂‐Cl(B)) for every subset B of Y(10)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(f⁻¹(σ₁‐Cl(σ₂‐Int(F)))))) ⊆ f⁻¹(F) for every σ₁σ₂-closed subset F of Y(11)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(f⁻¹(σ₁σ₂‐sInt(F))))) ⊆ f⁻¹(F) for every σ₁σ₂-closed subset F of Y(12)f⁻¹(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(f⁻¹(σ₁σ₂‐sCl(V))))) for every σ₁σ₂-open subset V of YLet {(X_γ, τ₁(γ), τ₂(γ))∣γ ∈Γ} be a family of bitopological spaces. Let be the product space, where and denotes the product topology of {τ_i(γ) ∣ γ ∈ Γ} for i = 1, 2.

Lemma 10. Let be a nonempty subset of for k = 1, 2, …, n. Then, is α-open in X^∗ if and only if is (τ₁(γ_k), τ₂(γ_k))α-open in for each k = 1, 2, …, n.

Proof. The proof is similar to that of [23] (Theorem 4.6)

Let {(X_γ, τ₁(γ), τ₂(γ))∣γ ∈Γ} and {(Y_γ, σ₁(γ), σ₂(γ))∣γ ∈ Γ} be two arbitrary families of bitopological spaces with the same set of indices. Let F_γ: (X_γ, τ₁(γ), τ₂(γ)) ⟶ (Y_γ, σ₁(γ), σ₂(γ)) be a multifunction for each γ ∈ Γ. Let be the product multifunction defined by for each , where and denote the product topologies for i = 1, 2.

Theorem 14. If is upper almost -continuous, thenis upper almost (τ₁(γ), τ₂(γ))α-continuous for each γ ∈Γ.

Proof. Let V_γ be a σ₁(γ)σ₂(γ)-regular open subset of Y_γ. Since F^∗ is upper almost -continuous,is -open in X^∗, and by Lemma 10, is (τ₁(γ), τ₂(γ))α-open in X_γ. Thus, F_γ is upper almost (τ₁(γ), τ₂(γ))α-continuous.

Theorem 15. If is lower almost -continuous, thenis lower almost (τ₁(γ), τ₂(γ))α-continuous for each γ ∈Γ.

Proof. The proof is similar to that of Theorem 14.

Theorem 16. Let (X, τ₁, τ₂) and (Y_γ, σ₁(γ), σ₂(γ)) be bitopological spaces for each γ ∈Γ. Let F_γ: (X, τ₁, τ₂) ⟶ (Y_γ, σ₁(γ), σ₂(γ)) be a multifunction for each γ ∈Γ and a multifunction defined by for each x ∈ X. If F is upper almost (τ₁, τ₂)α-continuous, then F_γ is upper almost (τ₁, τ₂)α-continuous for each γ ∈ Γ.

Proof. Let x ∈ X, γ ∈Γ, and V_γ be any σ₁(γ)σ₂(γ)-regular open subset of Y_γ containing F_γ(x). Then, is -regular open in containing F(x), where is the projection for each γ ∈Γ. Since F is upper almost (τ₁, τ₂)α-continuous, there exists a (τ₁, τ₂)α-open subset U of X containing x such that . Therefore, . This shows that F_γ is upper almost (τ₁, τ₂)α-continuous for each γ ∈ Γ.

Theorem 17. Let (X, τ₁, τ₂) and (Y_γ, σ₁(γ), σ₂(γ)) be bitopological spaces for each γ ∈Γ. Let F_γ: (X, τ₁, τ₂) ⟶ (Y_γ, σ₁(γ), σ₂(γ)) be a multifunction for each γ ∈Γ and a multifunction defined by for each x ∈ X. If F is lower almost (τ₁, τ₂)α-continuous, then F_γ is lower almost (τ₁, τ₂)α-continuous for each γ ∈ Γ.

Proof. The proof is similar to that of Theorem 16.

5. Characterizations of Upper and Lower Weakly (τ₁, τ₂)α-Continuous Multifunctions

In this section, we introduce the concepts of upper and lower weakly (τ₁, τ₂)α-continuous multifunctions. Furthermore, some characterizations of upper and lower weakly (τ₁, τ₂)α-continuous multifunctions are investigated.

Definition 13. A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is said to be:(1)upper weakly (τ₁, τ₂)α-continuous at a point x ∈ X if for each τ₁τ₂-semi-open subset U of X containing x and each σ₁σ₂-open subset V of Y containing F(x), there exists a nonempty τ₁τ₂-open subset G of X such that G ⊆ U and F(G) ⊆ σ₁σ₂‐Cl(V);(2)lower weakly (τ₁, τ₂)α-continuous at a point x ∈ X if for each τ₁τ₂-semi-open subset U of X containing x and each σ₁σ₂-open subset V of Y such that F(x) ∩ V≠∅, there exists a nonempty τ₁τ₂-open subset G of X such that G ⊆ U and F(z) ∩ σ₁σ₂‐Cl(V) ≠ ∅ for every z ∈ G;(3)upper weakly (resp. lower weakly) (τ₁, τ₂)α-continuous if F has this property at each point of X.

Remark 3. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following implication holds:
upper almost (τ₁, τ₂)α-continuity ⟹ upper weakly (τ₁, τ₂)α-continuity.
The converse of the implication is not true in general. We give an example for the implication as follows.

Example 3. Let X = {a, b, c, d} with topologies τ₁ = {∅, {a}, X} and τ₂ = {∅, X}. Let Y = {−1, 0, 1, 2} with topologies σ₁ = {∅, {−1, 2}, {0, 1}, Y}, and σ₂ = {∅, {−1, 2}, Y}. A multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is defined as follows:Then, F is upper weakly (τ₁, τ₂)α-continuous but F is not upper almost (τ₁, τ₂)α-continuous.

Theorem 18. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is upper weakly (τ₁, τ₂)α-continuous at x ∈ X(2)For any σ₁σ₂-open subset V of Y containing F(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐Cl(V)(3)x ∈ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₁‐Cl(V))) for every σ₁σ₂-open subset V of Y containing F(x)(4)x ∈ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₁‐Cl(V))))) for every σ₁σ₂-open subset V of Y containing F(x)

Proof. (1) ⟹ (2): Let τ₁τ₂‐SO(X, x) be the family of all τ₁τ₂-semi-open subsets of X containing x. Let V be any σ₁σ₂-open subset of Y containing F(x). For each U ∈ τ₁τ₂‐SO(X, x), there exists a nonempty τ₁τ₂-open set H_U such that H_U ⊆ U and F(H_U) ⊆ V. Let W = ∪{H_U∣U ∈ τ₁τ₂‐SO(X, x)}. Then, W is τ₁τ₂-open in X, x ∈ τ₁τ₂ ‐ sCl(W), and F(W) ⊆ V. Put G = W ∪ {x}. By Lemma 1, we have and by Lemma 2, G is a (τ₁, τ₂)α-open subset of X containing x such that F(G) ⊆ σ₁σ₂‐Cl(V). (2) ⟹ (3): Let V be any σ₁σ₂-open subset of Y containing F(x). By (2), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐Cl(V). This implies that x ∈ U ⊆ F⁺(σ₁σ₂‐Cl(V)), and hence (3) ⟹ (4): Let V be any σ₁σ₂-open subset of Y. By (3), we have Then, there exists a (τ₁, τ₂)α-open subset U of X such that x ∈ U ⊆ F⁺(σ₁σ₁‐Cl(V)). Thus, x ∈ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₁‐Cl(V))))). (4) ⟹ (1): Let U be any τ₁τ₂-semi-open set containing x and V be any σ₁σ₂-open set containing F(x). By (4) and Lemma 1 (2), we haveThus, τ₁τ₂‐Int(F⁺(σ₁σ₁‐Cl(V))) ∩ U ≠ ∅ and τ₁τ₂‐Int(F⁺(σ₁σ₁‐Cl(V))) ∩ U is τ₁τ₂-semi-open. Put G = τ₁τ₂‐Int(τ₁τ₂‐Int(F⁺(σ₁σ₁‐Cl(V))) ∩ U); then G is a nonempty σ₁σ₂-open subset of X such that G ⊆ U and F(G) ⊆ σ₁σ₂‐Cl(V). This shows that F is upper weakly (τ₁, τ₂)α-continuous at x.

Theorem 19. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is lower weakly (τ₁, τ₂)α-continuous at x ∈ X(2)For any σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a (τ₁, τ₂)α-open subset U of X containing x such that σ₁σ₂‐Cl(V) ∩ F(z)≠∅ for every z ∈ U(3)x ∈ (τ₁, τ₂)α‐Int(F⁻(σ₁σ₁‐Cl(V))) for every σ₁σ₂-open subset V of Y such that F(x) ∩ V≠∅(4)x ∈ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁻(σ₁σ₁‐Cl(V))))) for every σ₁σ₂-open subset V of Y such that F(x) ∩ V≠∅

Proof. The proof is similar to that of Theorem 18.

Definition 14. Let A be a subset of a bitopological space (X, τ₁, τ₂). A point x ∈ X is called a (τ₁, τ₂)θ-cluster point of A if τ₁τ₂‐Cl(U) ∩ A≠∅ for every τ₁τ₂-open set U containing x. The set of all (τ₁, τ₂)θ-cluster point of A is called the (τ₁, τ₂)θ-closure of A and is denoted by (τ₁, τ₂)θ‐Cl(A).
A subset A of a bitopological space (X, τ₁, τ₂) is said to be (τ₁, τ₂)θ-closed if A = (τ₁, τ₂)θ‐Cl(A). The complement of a (τ₁, τ₂)θ-closed set is said to be (τ₁, τ₂)θ-open. The union of all (τ₁, τ₂)θ-open sets contained in A is called the (τ₁, τ₂)θ-interior of A and is denoted by (τ₁, τ₂)θ‐Int(A).

Lemma 11. For a subset A of a bitopological space (X, τ₁, τ₂), the following properties hold:(1)If A is τ₁τ₂-open in X, then τ₁τ₂‐Cl(A) = (τ₁, τ₂)θ‐Cl(A)(2)(τ₁, τ₂)θ‐Cl(A) is τ₁τ₂-closed in X

Proof. (1)In general, τ₁τ₂‐Cl(A) ⊆ (τ₁, τ₂)θ‐Cl(A). Suppose that x ∉ τ₁τ₂‐Cl(A). Then, there exists a τ₁τ₂-open set U containing x such that U ∩ A =∅; hence, τ₁τ₂‐Cl(U) ∩ A = ∅. This shows that x ∉ (τ₁, τ₂)θ‐Cl(A). Thus, (τ₁, τ₂)θ‐Cl(A) ⊆ τ₁τ₂‐Cl(A). Consequently, we obtain τ₁τ₂‐Cl(A) = (τ₁, τ₂)θ‐Cl(A).(2)Let x ∈ X – (τ₁, τ₂)θ‐Cl(A). Then, we have x∉(τ₁, τ₂)θ‐Cl(A). There exists a τ₁τ₂-open set U_x containing x such that τ₁τ₂‐Cl(U_x) ∩ A =∅. This implies that (τ₁, τ₂)θ‐Cl(A) ∩ U_x =∅, and hence x ∈ U_x ⊆ X – (τ₁, τ₂)θ‐Cl(A). Thus, . This shows that (τ₁, τ₂)θ‐Cl(A) is τ₁τ₂-closed.

Definition 15. A subset A of a bitopological space (X, τ₁, τ₂) is said to be (τ₁, τ₂)r-closed (resp. (τ₁, τ₂)r-open) if A = τ₁τ₂‐Cl(τ₁τ₂‐Int(A)) (resp. A = τ₁τ₂‐Int(τ₁τ₂‐Cl(A))).

Theorem 20. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is upper weakly (τ₁, τ₂)α-continuous(2)For each x ∈ X and each σ₁σ₂-open subset V of Y containing F(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐Cl(V)(3)F⁺(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₂‐Cl(V))))) for every σ₁σ₂-open subset V of Y(4)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁻(σ₁σ₂‐Int(K))))) ⊆ F⁻(K) for every σ₁σ₂-closed subset K of Y(5)(τ₁, τ₂)α‐Cl(F⁻(σ₁σ₂‐Int(K))) ⊆ F⁻(K) for every σ₁σ₂-closed subset K of Y(6)(τ₁, τ₂)α‐Cl(F⁻(σ₁σ₂‐Int(σ₁σ₂‐Cl(B)))) ⊆ F⁻(σ₁σ₂‐Cl(B)) for every subset B of Y(7)F⁺(σ₁σ₂‐Int(B)) ⊆ (τ₁, τ₂)α‐Int(F⁺ (σ₁σ₂‐Cl(σ₁σ₂‐Int(B)))) for every subset B of Y(8)F⁺(V) ⊆ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₂‐Cl(V))) for every σ₁σ₂-open subset V of Y(9)(τ₁, τ₂)α‐Cl(F⁻(σ₁σ₂‐Int(K))) ⊆ F⁻(K) for every (σ₁, σ₂)r-closed subset K of Y(10)(τ₁, τ₂)α‐Cl(F⁻(V)) ⊆ F⁻(σ₁σ₂‐Cl(V)) for every σ₁σ₂-open subset V of Y(11)(τ₁, τ₂)α‐Cl(F⁻(σ₁σ₂‐Int((σ₁, σ₂)θ‐Cl(B)))) ⊆ F⁻((σ₁, σ₂)θ‐Cl(B)) for every subset B of Y(12)(τ₁, τ₂)α‐Cl(F⁻(σ₁σ₂‐Int(σ₁σ₂‐Cl(V)))) ⊆ F⁻(σ₁σ₂‐Cl(V)) for every σ₁σ₂-open subset V of Y

Proof. (1) ⟹ (2): The proof follows immediately from Theorem 18. (2) ⟹ (3): Let V be any σ₁σ₂-open subset of Y and x ∈ F⁺(V). Then, we have F(x) ⊆ V and by (2), there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐Cl(V). Therefore, we have x ∈ U ⊆ F⁺(σ₁σ₂‐Cl(V)), and hence x ∈ U ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₂‐Cl(V))))). Consequently, we obtain F⁺(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁺(σ₁σ₂‐Cl(V))))). (3) ⟹ (4): Let K be any σ₁σ₂-closed subset of Y. Then, we have Y – K is σ₁σ₂-open in Y and by (3), Thus, τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F^‐(σ₁σ₂‐Int(K))))) ⊆ F^‐(K). (4) ⟹ (5): Let K be any σ₁σ₂-closed subset of Y. By (4), we have and hence (τ₁, τ₂)α‐Cl(F^‐(σ₁σ₂‐Int(K))) ⊆ F^‐(K) by Lemma 3 (3). (5) ⟹ (6): Let B be any subset of Y. Then, we have σ₁σ₂‐Cl(B) is σ₁σ₂-closed in Y, and by (5), (τ₁, τ₂)α‐Cl(F^‐(σ₁σ₂‐Int(σ₁σ₂‐Cl(B)))) ⊆ F^‐(σ₁σ₂‐Cl(B)). (6) ⟹ (7): Let B be any subset of Y. By (6), we have (7) ⟹ (8): The proof is obvious. (8) ⟹ (1): Let x ∈ X and V be any σ₁σ₂-open subset of Y containing F(x). By (8), we have and hence F is upper weakly (τ₁, τ₂)α-continuous at x by Theorem 18. Consequently, we obtain that F is upper weakly (τ₁, τ₂)α-continuous. (5) ⟹ (9): The proof is obvious. (9) ⟹ (10): Let V be any σ₁σ₂-open subset of Y. Then, we have σ₁σ₂‐Cl(V) is (σ₁, σ₂)r-closed in Y, and by (9), (10) ⟹ (8): Let V be any σ₁σ₂-open subset of Y. By (10), we have and hence F⁺(V) ⊆ F⁺(σ₁σ₂‐Int(σ₁σ₂‐Cl(V))) ⊆ (τ₁, τ₂)α‐Int(F⁺(σ₁σ₂‐Cl(V))). (10) ⟹ (11): Let B be any subset of Y. Then, σ₁σ₂‐Int((σ₁, σ₂)θ‐Cl(B)) is σ₁σ₂-open in Y. By (10) and Lemma 11 (1), (11) ⟹ (12): Let V be any σ₁σ₂-open subset of Y. By (11) and Lemma 11 (1), we have (τ₁, τ₂)α‐Cl(F^‐(σ₁σ₂‐Int(σ₁σ₂‐Cl(V)))) ⊆ F^‐(σ₁σ₂‐Cl(V)). (12) ⟹ (10): Let V be any σ₁σ₂-open subset of Y. By (12), we haveand hence

Theorem 21. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is lower weakly (τ₁, τ₂)α-continuous(2)For each x ∈ X and each σ₁σ₂-open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a (τ₁, τ₂)α-open subset U of X containing x such that U ⊆ F⁻(σ₁σ₂‐Cl(V))(3)F⁻(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(F⁻(σ₁σ₂‐Cl(V))))) for every σ₁σ₂-open subset V of Y(4)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(F⁺(σ₁σ₂‐Int(K))))) ⊆ F⁺(K) for every σ₁σ₂-closed subset K of Y(5)(τ₁, τ₂)α‐Cl(F⁺(σ₁σ₂‐Int(K))) ⊆ F⁺(K) for every σ₁σ₂-closed subset K of Y(6)(τ₁, τ₂)α‐Cl(F⁺(σ₁σ₂‐Int(σ₁σ₂‐Cl(B)))) ⊆ F⁺(σ₁σ₂‐Cl(B)) for every subset B of Y(7)F⁻(σ₁σ₂‐Int(B)) ⊆ (τ₁, τ₂)α‐Int(F⁻(σ₁σ₂‐Cl(σ₁σ₂‐Int(B)))) for every subset B of Y(8)F⁻(V) ⊆ (τ₁, τ₂)α‐Int(F⁻(σ₁σ₂‐Cl(V))) for every σ₁σ₂-open subset V of Y(9)(τ₁, τ₂)α‐Cl(F⁺(σ₁σ₂‐Int(K))) ⊆ F⁺(K) for every (σ₁, σ₂)r-closed subset K of Y(10)(τ₁, τ₂)α‐Cl(F⁺(V)) ⊆ F⁺(σ₁σ₂‐Cl(V)) for every σ₁σ₂-open subset V of Y(11)(τ₁, τ₂)α‐Cl(F⁺(σ₁σ₂‐Int((σ₁, σ₂)θ‐Cl(B)))) ⊆ F⁺((σ₁, σ₂)θ‐Cl(B)) for every subset B of Y(12)(τ₁, τ₂)α‐Cl(F⁺(σ₁σ₂‐Int(σ₁σ₂‐Cl(V)))) ⊆ F⁺(σ₁σ₂‐Cl(V)) for every σ₁σ₂-open subset V of Y

Proof. The proof is similar to that of Theorem 20.

Lemma 12. If F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is lower weakly (τ₁, τ₂)α-continuous, then for each x ∈ X and each subset B of Y such that (τ₁, τ₂)θ‐Int(B) ∩ F(x) ≠ ∅, there exists a (τ₁, τ₂)α-open subset U of X containing x such that U ⊆ F⁻(B).

Proof. Suppose that (τ₁, τ₂)θ‐Int(B) ∩ F(x) ≠ ∅. Then, there exists a nonempty σ₁σ₂-open subset V of Y such that V ⊆ σ₁σ₂‐Cl(V) ⊆ B and F(x) ∩ V ≠ ∅. Since F is lower weakly (τ₁, τ₂)α-continuous, there exists a (τ₁, τ₂)α-open subset U of X containing x such that σ₁σ₂‐Cl(V) ∩ F(z) ≠ ∅ for every z ∈ U, and hence U ⊆ F⁻(B).

Theorem 22. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)F is lower weakly (τ₁, τ₂)α-continuous(2)(τ₁, τ₂)α‐Cl(F⁺(B)) ⊆ F⁺((σ₁, σ₂)θ‐Cl(B)) for every subset B of Y(3)F((τ₁, τ₂)α‐Cl(A)) ⊆ (σ₁, σ₂)θ‐Cl(F(A)) for every subset A of X

Proof. (1) ⟹ (2): Let B be any subset of Y. Suppose that x ∉ F⁺((σ₁, σ₂)θ‐Cl(B)). Then, we have Therefore, F(x) ∈ (σ₁, σ₂)θ‐Int(Y‐B). By Lemma 12, there exists a (τ₁, τ₂)α-open subset U of X containing x such that U ⊆ F^1‐(Y − B) = X − F⁺(B), and hence U ∩ F⁺(B) = ∅. Thus, x∉(τ₁, τ₂)α‐Cl(F⁺(B)). Consequently, we obtain (τ₁, τ₂)α‐Cl(F⁺(B)) ⊆ F⁺((σ₁, σ₂)θ‐Cl(B)). (2) ⟹ (1): Let V be any σ₁σ₂-open subset of Y. By (2) and Lemma 11 (1), we have (τ₁, τ₂)α‐Cl(F⁺(V)) ⊆ F⁺((σ₁, σ₂)θ‐Cl(V)) = F⁺(σ₁σ₂‐Cl(V)), and by Theorem 21, F is lower weakly (τ₁, τ₂)α-continuous. (2) ⟹ (3): Let A be any subset of X. By (2), we have (τ₁, τ₂)α‐Cl(A) ⊆ (τ₁, τ₂)α‐Cl(F⁺(F(A))) ⊆ F⁺((σ₁, σ₂)θ‐Cl(F(A))). Thus, F((τ₁, τ₂)α‐Cl(A)) ⊆ (σ₁, σ₂)θ‐Cl(F(A)). (3) ⟹ (2): Let B be any subset of Y. By (3), we have F((τ₁, τ₂)α‐Cl(F⁺(B))) ⊆ (σ₁, σ₂)θ‐Cl(F(F⁺(B))) ⊆ (σ₁, σ₂)θ‐Cl(B), and hence

Definition 16. A function f: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) is said to be weakly (τ₁, τ₂)α-continuous if for each x ∈ X and each σ₁σ₂-open subset V containing f(x), there exists a (τ₁, τ₂)α-open subset U of X containing x such that f(U) ⊆ σ₁σ₂‐Cl(V).

Corollary 3. For a function f: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂), the following properties are equivalent:(1)f is weakly (τ₁, τ₂)α-continuous(2)f^‐1(V) ⊆ (τ₁, τ₂)α‐Int(f^‐1(σ₁σ₂‐Cl(V))) for every σ₁σ₂-open subset V of Y(3)(τ₁, τ₂)α‐Cl(f^‐1(σ₁σ₂‐Int(K))) ⊆ f^‐1(K) for every (σ₁, σ₂)r-closed subset K of Y(4)(τ₁, τ₂)α‐Cl(f^‐1(V)) ⊆ f^‐1(σ₁σ₂‐Cl(V)) for every σ₁σ₂-open subset V of Y(5)(τ₁, τ₂)α‐Cl(f^‐1(σ₁σ₂‐Int((σ₁, σ₂)θ‐Cl(B)))) ⊆ f^‐1((σ₁, σ₂)θ‐Cl(B)) for every subset B of Y(6)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(f^‐1(V)))) ⊆ f^‐1(σ₁σ₂‐Cl(V)) for every σ₁σ₂-open subset V of Y(7)f^‐1(V) ⊆ τ₁‐Int(τ₂‐Cl(τ₁τ₂‐Int(f^‐1(σ₁σ₂‐Cl(V))))) for every σ₁σ₂-open subset V of Y(8)f(τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(A)))) ⊆ (σ₁, σ₂)θ‐Cl(f(A)) for every subset A of X(9)τ₁‐Cl(τ₂‐Int(τ₁τ₂‐Cl(f^‐1(B)))) ⊆ f^‐1((σ₁, σ₂)θ‐Cl(B)) for every subset B of Y

Definition 17. A subset A of a bitopological space (X, τ₁, τ₂) is said to be τ₁τ₂-S-closed relative to X if for every cover {V_γ∣γ ∈Γ} of A by τ₁τ₂-open sets of X, there exists a finite subset Γ₀ of Γ such that A ⊆ ∪{τ₁τ₂‐Cl(V_γ)∣γ ∈ Γ₀}. If X is τ₁τ₂-S-closed relative to X, then the space (X, τ₁, τ₂) is called τ₁τ₂-S-closed.

Theorem 23. Let F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) be an upper weakly (τ₁, τ₂)α-continuous surjective multifunction such that F(x) is τ₁τ₂-compact for each x ∈ X. If (X, τ₁, τ₂) is (τ₁, τ₂)α-compact, then (Y, σ₁, σ₂) is σ₁σ₂-S-closed.

Proof. Let {V_γ∣γ ∈Γ} be any σ₁σ₂-open cover of Y. For each x ∈ X, F(x) is σ₁σ₂-compact, and hence there exists a finite subset Γ(x) of Γ such thatSince F is upper weakly (τ₁, τ₂)α-continuous, there exists a (τ₁, τ₂)α-open subset of X such that F(U(x)) ⊆∪{σ₁σ₂‐Cl(V_γ)∣γ ∈Γ(x)}. Since (X, τ₁, τ₂) is (τ₁, τ₂)α-compact, there exists a finite number of points, say, x₁, x₂, …, x_n in X such that X = ∪{U(x_i)∣1 ≤ i ≤ n}. Thus, Y = F(X) = F(∪{U(x_i)∣1 ≤ i ≤ n}) ⊆∪{σ₁σ₂‐Cl(V_γ)∣γ ∈ Γ(x_i); 1 ≤ i ≤ n}. This shows that (Y, σ₁, σ₂) is σ₁σ₂-S-closed.

Theorem 24. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) such that F(x) is σ₁σ₂-regular σ₁σ₂-paracompact for each x ∈ X, the following properties are equivalent:(1)F is upper (τ₁, τ₂)α-continuous(2)F is upper almost (τ₁, τ₂)α-continuous(3)F is upper weakly (τ₁, τ₂)α-continuous

Proof. We prove only the implication (3) ⟹ (1) since the others are obvious. Suppose that F is upper weakly (τ₁, τ₂)α-continuous. Let x ∈ X and V be any σ₁σ₂-open subset of Y such that F(x) ⊆ V. Since F(x) is σ₁σ₂-regular σ₁σ₂-paracompact, there exists a σ₁σ₂-open subset G of Y such that F(x) ⊆ G ⊆ σ₁σ₂‐Cl(G) ⊆ V. Since F is upper weakly (τ₁, τ₂)α-continuous at x and F(x) ⊆ G, there exists a (τ₁, τ₂)α-open subset U of X containing x such that F(U) ⊆ σ₁σ₂‐Cl(G), and hence F(U) ⊆ V. This shows that F is upper (τ₁, τ₂)α-continuous.

Lemma 13. Let A be a subset of a bitopological space (X, τ₁, τ₂). If A is τ₁τ₂-regular, then for every τ₁τ₂-open set G which intersects A, there exists a τ₁τ₂-open set W such that A ∩ W ≠ ∅ and τ₁τ₂‐Cl(W) ⊆ G.

Proof. Suppose that A is a τ₁τ₂-regular subset of X and G a τ₁τ₂-open set such that A ∩ G ≠ ∅. Let x ∈ A ∩ G. Then, we have x∉X – G and X – G is τ₁τ₂-closed. There exist disjoint τ₁τ₂-open subsets U and V of X such that x ∈ U and X – G ⊆ V, which implies τ₁τ₂ ‐ Cl(V) ∩ U =∅, and hence x∉τ₁τ₂-Cl(V). Let W = X – τ₁τ₂‐Cl(V). Thus, x ∈ W and A ∩ W ≠ ∅. Consequently, we obtain τ₁τ₂‐Cl(W) = τ₁τ₂‐Cl(X – τ₁τ₂‐Cl(V)) ⊆ τ₁τ₂‐Cl(X – V) = X – V ⊆ G.

Theorem 25. For a multifunction F: (X, τ₁, τ₂) ⟶ (Y, σ₁, σ₂) such that F(x) is σ₁σ₂-regular for each x ∈ X, the following properties are equivalent:(1)F is lower (τ₁, τ₂)α-continuous(2)F is lower almost (τ₁, τ₂)α-continuous(3)F is lower weakly (τ₁, τ₂)α-continuous

Proof. We prove only the implication (3) ⟹ (1) since the others are obvious. Suppose that F is lower weakly (τ₁, τ₂)α-continuous. Let x ∈ X and V be any σ₁σ₂-open subset of Y such that F(x) ∩ V ≠ ∅. Since F(x) is σ₁σ₂-regular, by Lemma 13, there exists a σ₁σ₂-open subset G of Y such that F(x) ∩ G ≠ ∅ and σ₁σ₂‐Cl(G) ⊆ V. Since F is upper weakly (τ₁, τ₂)α-continuous at x, there exists a (τ₁, τ₂)α-open subset U of X containing x such that σ₁σ₂‐Cl(G) ∩ F(z) ≠ ∅ for every z ∈ U and hence V ∩ F(z) ≠ ∅ for every z ∈ U as well, and this shows that F is lower (τ₁, τ₂)α-continuous at x. This shows that F is lower (τ₁, τ₂)α-continuous.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was financially supported by Mahasarakham University.

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Copyright © 2020 Chokchai Viriyapong and Chawalit Boonpok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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