EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2142-2154 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower Slight α(τ1, τ2)-Continuity Chokchai Viriyapong1, Supannee Sompong2, Chawalit Boonpok1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand 2 Department of Mathematics and Statistics, Faculty of Science and Technology, Sakon Nakhon Rajbhat University, Sakon Nakhon, 47000, Thailand Abstract. This paper deals with the notions of upper and lower slightly α(τ1, τ2)-continuous mul- tifunctions. Furthermore, several characterizations of upper and lower slightly α(τ1, τ2)-continuous multifunctions are discussed. 2020 Mathematics Subject Classifications: 54C08, 54C60, 54E55 Key Words and Phrases: α(τ1, τ2)-open set, upper slightly α(τ1, τ2)-continuous multifunction, lower slightly α(τ1, τ2)-continuous multifunction 1. Introduction In 1980, Jain [25] introduced the notion of slightly continuous functions. Nour [32] defined slightly semi-continuous functions as a weak form of slight continuity and inves- tigated some characterizations of slightly semi-continuous functions. Noiri and Chae [30] have further investigated slightly semi-continuous functions. Pal and Bhattacharyya [33] introduced and studied the concept of faintly precontinuous functions. Slight continu- ity implies both slight semi-continuity and faint precontinuity. Noiri [29] introduced and studied the notion of slight β-continuity which is implied by both slight semi-continuity and faint precontinuity. Duangphui et al. [21] introduced and investigated the notion of almost (µ, µ′)(m,n)-continuous functions. Thongmoon and Boonpok [42] introduced and studied the notion of strongly θ(Λ, p)-continuous functions. Moreover, several charac- terizations of almost (Λ, p)-continuous functions, almost strongly θ(Λ, p)-continuous func- tions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions, θ(⋆)-precontinuous functions, ⋆-continuous functions, θ-I -continuous functions, almost (g,m)-continuous functions, (Λ, sp)-continuous functions, δp(Λ, s)-continuous functions, (Λ, p(⋆))-continuous functions, pairwise almost M -continuous functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)-continuous functions and weakly (τ1, τ2)-continuous functions were presented in ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5321 Email addresses: chokchai.v@msu.ac.th (C. Viriyapong), s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 2142 © 2024 EJPAM All rights reserved. C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2143 [40], [11], [37], [16], [10], [9], [5], [2], [44], [41], [8], [3], [17], [15] and [12], respectively. Sangviset et al. [39] introduced the notion of slightly (m,µ)-continuous functions as func- tions from an m-spaces into a generalized topological space and investigated several char- acterizations of slightly (m,µ)-continuous functions. In 2005, Ekici [23] introduced and investigated the notion of upper (lower) slightly α-continuous multifunctions as a generalization of upper (lower) α-continuous multifunc- tions due to Neubrunn [28]. Popa and Noiri [36] introduced and studied the notion of upper (lower) β-continuous multifunctions. Furthermore, Ekici [22] introduced and stud- ied upper (lower) slightly β-continuous multifunctions as a generalization of upper (lower) semicontinuous multifunctions, upper (lower) α-continuous multifunctions, upper (lower) precontinuous multifunctions [35], upper (lower) quasi-continuous multifunctions [34], up- per (lower) γ-continuous multifunctions [24], upper (lower) β-continuous multifunctions and slightly β-continuous functions. Noiri and Popa [31] introduced the notion of slightly m-continuous multifunctions and established the relationships amongm-continuity, almost m-continuity, weak m-continuity and slight m-continuity for multifunctions. Laprom et al. [27] introduced and investigated the notion of β(τ1, τ2)-continuous multifunctions. Further- more, several characterizations of (τ1, τ2)δ-semicontinuous multifunctions, almost weakly ⋆-continuous multifunctions, weakly ⋆-continuous multifunctions, weakly α-⋆-continuous multifunctions, ı⋆-continuous multifunctions, almost β(⋆)-continuous multifunctions, al- most weakly (τ1, τ2)-continuous multifunctions, almost (τ1, τ2)-continuous multifunctions and (τ1, τ2)α-continuous multifunctions were investigated in [6], [18], [4], [14], [13], [7], [19], [26] and [43], respectively. Pue-on et al. [38] introduce and studied the notions of upper and lower (τ1, τ2)-continuous multifunctions. In this paper, we introduce the concepts of upper and lower slightly α(τ1, τ2)-continuous multifunctions. We also investigate several characterizations of upper and lower slightly α(τ1, τ2)-continuous multifunctions. 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 τi are denoted by τi-Cl(A) and τi-Int(A), respectively, for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed [20] if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [20] of A and is denoted by τ1τ2-Cl(A). The union of all τ1τ2-open sets of X contained in A is called the τ1τ2-interior [20] of A and is denoted by τ1τ2-Int(A). Lemma 1. [20] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- closure, the following properties hold: (1) A ⊆ τ1τ2-Cl(A) and τ1τ2-Cl(τ1τ2-Cl(A)) = τ1τ2-Cl(A). C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2144 (2) If A ⊆ B, then τ1τ2-Cl(A) ⊆ τ1τ2-Cl(B). (3) τ1τ2-Cl(A) is τ1τ2-closed. (4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A). (5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-clopen [20] if A is both τ1τ2-open and τ1τ2-closed. A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [43] (resp. (τ1, τ2)s-open [6], (τ1, τ2)p-open [6], (τ1, τ2)β-open [6]) if A = τ1τ2-Int(τ1τ2-Cl(A)) (resp. A ⊆ τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A)))). The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s- open, (τ1, τ2)p-open, (τ1, τ2)β-open) set is called (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed). A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [45] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is said to be α(τ1, τ2)-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all α(τ1, τ2)-closed sets of X containing A is called the α(τ1, τ2)-closure of A and is denoted by α(τ1, τ2)-Cl(A). The union of all α(τ1, τ2)- open sets of X contained in A is called the α(τ1, τ2)-interior of A and is denoted by α(τ1, τ2)-Int(A). Lemma 2. For subsets A and B of a bitopological space (X, τ1, τ2), the following properties hold: (1) A ⊆ α(τ1, τ2)-Cl(A) and α(τ1, τ2)-Cl(α(τ1, τ2)-Cl(A)) = α(τ1, τ2)-Cl(A). (2) If A ⊆ B, then α(τ1, τ2)-Cl(A) ⊆ α(τ1, τ2)-Cl(B). (3) α(τ1, τ2)-Cl(A) is α(τ1, τ2)-closed. (4) A is α(τ1, τ2)-closed if and only if A = α(τ1, τ2)-Cl(A). (5) α(τ1, τ2)-Cl(X −A) = X − α(τ1, τ2)-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 [1] 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). C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2145 3. Upper and lower slightly α(τ1, τ2)-continuous multifunctions In this section, we introduce the notions of upper and lower slightly α(τ1, τ2)-continuous multifunctions. Moreover, some characterizations of upper and lower slightly α(τ1, τ2)- continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper slightly α(τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-clopen set V of Y containing F (x), there exists an α(τ1, τ2)-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper slightly α(τ1, τ2)-continuous if F has this property at every point of X. Recall that a net (xγ) in a topological space (X, τ) is said to be eventually in the set U ⊆ X if there exists an index γ0 ∈ ∇ such that xγ ∈ U for all γ ≥ γ0. Definition 2. A sequence (xn) is called α(τ1, τ2)-converge to a point x if for every α(τ1, τ2)-open set V containing x, there exists an index n0 such that for n ≥ n0, xn ∈ V . Theorem 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper slightly α(τ1, τ2)-continuous; (2) for each x ∈ X and for each σ1σ2-clopen set V of Y such that x ∈ F+(V ), there exists an α(τ1, τ2)-open set U of X containing x such that U ⊆ F+(V ); (3) for each x ∈ X and for each σ1σ2-clopen set V of Y such that x ∈ F+(Y −V ), there exists an α(τ1, τ2)-closed set H of X such that x ∈ X −H and F−(V ) ⊆ H; (4) F+(V ) is α(τ1, τ2)-open in X for every σ1σ2-clopen set V of Y ; (5) F−(V ) is α(τ1, τ2)-closed in X for every σ1σ2-clopen set V of Y ; (6) F−(Y − V ) is α(τ1, τ2)-closed in X for every σ1σ2-clopen set V of Y ; (7) F+(Y − V ) is α(τ1, τ2)-open in X for every σ1σ2-clopen set V of Y ; (8) for each x ∈ X and for each net (xγ) which α(τ1, τ2)-converges to x in X and for each σ1σ2-clopen set V of Y such that x ∈ F+(V ), the net (xγ) is eventually in F+(V ). Proof. (1) ⇔ (2): Obvious. (2) ⇔ (3): Let x ∈ X and V be any σ1σ2-clopen set of Y such that x ∈ F+(Y − V ). By (2), there exists an α(τ1, τ2)-open set U of X containing x such that U ⊆ F+(Y − V ). Then, F−(V ) ⊆ X−U . Put H = X−U . Then, H is α(τ1, τ2)-closed in X and x ∈ X−H. The converse is similar. (1) ⇔ (4): Let V be any σ1σ2-clopen set of Y and x ∈ F+(V ). By (1), there exists an α(τ1, τ2)-open set Ux of X containing x such that Ux ⊆ F+(V ). It follows that F+(V ) = ∪x∈F+(V )Ux and hence F+(V ) is α(τ1, τ2)-open in X. The converse can be shown easily. C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2146 (4) ⇒ (5): Let V be any σ1σ2-clopen set of Y . Then, Y − V is σ1σ2-clopen in Y and by (4), F+(Y − V ) = X − F−(V ) is α(τ1, τ2)-open in X. Thus, F−(V ) is α(τ1, τ2)-closed in X. (5) ⇒ (4): It is similar to that of (4) ⇒ (5). (4) ⇔ (6) and (5) ⇔ (7): It follows from the fact that F−(Y −B) = X − F+(B) and F+(Y −B) = X − F−(B) for every subset B of Y . (1) ⇒ (8): Let (xγ) be a net which α(τ1, τ2)-converges to x in X and let V be any σ1σ2- clopen set of Y such that x ∈ F+(V ). Since F is an upper slightly α(τ1, τ2)-continuous multifunction, there exists an α(τ1, τ2)-open set U ofX containing x such that U ⊆ F+(V ). Since (xγ) α(τ1, τ2)-converges to x, it follows that there exists an index γ0 ∈ ∇ such that xγ ∈ U for all γ ≥ γ0. Therefore, xγ ∈ U ⊆ F+(V ) for all γ ≥ γ0. Thus, the net (xγ) is eventually in F+(V ). (8) ⇒ (1): Suppose that F is not upper slightly α(τ1, τ2)-continuous. There exists a point x and a σ1σ2-clopen set V of Y with x ∈ F+(V ) such that U ̸⊆ F+(V ) for each α(τ1, τ2)-open set U of X containing x. Let xU ∈ U and xU ̸∈ F+(V ) for each α(τ1, τ2)- open set U of X containing x. Then, for the α(τ1, τ2)-neighbourhood net (xU ), (xU ) α(τ1, τ2)-converges to x, but (xU ) is not eventually in F+(V ). This is a contradiction. Thus, F is upper slightly α(τ1, τ2)-continuous. Definition 3. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower slightly α(τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-clopen set V of Y such that F (x) ∩ V ̸= ∅, there exists an α(τ1, τ2)-open set U of X containing x such that F (z) ∩ V ̸= ∅ for each z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower slightly α(τ1, τ2)- continuous if F has this property at every point of X. Theorem 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower slightly α(τ1, τ2)-continuous; (2) for each x ∈ X and for each σ1σ2-clopen set V of Y such that x ∈ F−(V ), there exists an α(τ1, τ2)-open set U of X containing x such that U ⊆ F−(V ); (3) for each x ∈ X and for each σ1σ2-clopen set V of Y such that x ∈ F−(Y −V ), there exists an α(τ1, τ2)-closed set H of X such that x ∈ X −H and F+(V ) ⊆ H; (4) F−(V ) is α(τ1, τ2)-open in X for every σ1σ2-clopen set V of Y ; (5) F+(V ) is α(τ1, τ2)-closed in X for every σ1σ2-clopen set V of Y ; (6) F+(Y − V ) is α(τ1, τ2)-closed in X for every σ1σ2-clopen set V of Y ; (7) F−(Y − V ) is α(τ1, τ2)-open in X for every σ1σ2-clopen set V of Y ; (8) for each x ∈ X and for each net (xγ) which α(τ1, τ2)-converges to x in X and for each σ1σ2-clopen set V of Y such that x ∈ F−(V ), the net (xγ) is eventually in F−(V ). C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2147 Proof. The proof is similar to that of Theorem 1. Definition 4. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be slightly α(τ1, τ2)- continuous if for each x ∈ X and each σ1σ2-clopen set V of Y containing f(x), there exists an α(τ1, τ2)-open set U of X containing x such that f(U) ⊆ V . Corollary 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is slightly α(τ1, τ2)-continuous; (2) f−1(V ) is α(τ1, τ2)-open in X for each σ1σ2-clopen set V of Y ; (3) f−1(V ) is α(τ1, τ2)-closed in X for each σ1σ2-clopen set V of Y ; (4) for each x ∈ X and for each σ1σ2-clopen set V of Y containing f(x), there exists an α(τ1, τ2)-open set U of X containing x such that f(U) ⊆ V . Definition 5. A bitopological space (X, τ1, τ2) is said to be mildly τ1τ2-compact if every cover of X by τ1τ2-clopen sets of X has a finite subcover. Definition 6. A bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-compact if if every cover of X by α(τ1, τ2)-open sets of X has a finite subcover. Theorem 3. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be an upper slightly α(τ1, τ2)-continuous sur- jective multifunction such that F (x) is mildly σ1σ2-compact for each x ∈ X. If (X, τ1, τ2) is α(τ1, τ2)-compact, then (Y, σ1, σ2) is mildly σ1σ2-compact. Proof. Let {Vγ | γ ∈ Γ} be any σ1σ2-clopen cover of Y . Since F (x) is mildly σ1σ2- compact for each x ∈ X, there exists a finite subset Γ(x) of Γ such that F (x) ⊆ ∪{Vγ | γ ∈ Γ(x)}. Put V (x) = ∪{Vγ | γ ∈ Γ(x)}. Since F is upper slightly α(τ1, τ2)-continuous, there exists an α(τ1, τ2)-open set U(x) of X containing x such that F (U(x)) ⊆ V (x). Then, the family {U(x) | x ∈ X} is an α(τ1, τ2)-open cover of X. Since (X, τ1, τ2) is α(τ1, τ2)-compact, there exists a finite number of points, say, x1, x2, ..., xn inX such thatX = ∪{U(xi) | 1 ≤ i ≤ n}. Thus, Y = F (X) = ∪ {F (U(xi)) | 1 ≤ i ≤ n} ⊆ ∪{V (xi) | 1 ≤ i ≤ n} ⊆ ∪{Vγ | γ ∈ Γ(xi), 1 ≤ i ≤ n}. This shows that (Y, σ1, σ2) is mildly σ1σ2-compact. Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [20] if X cannot be written as the union of two disjoint nonempty τ1τ2-open sets. C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2148 Definition 7. A bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-connected provided that X is not the union of two disjoint nonempty α(τ1, τ2)-open sets. Definition 8. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called punctually τ1τ2- connected if, for each x ∈ X, F (x) is σ1σ2-connected. Theorem 4. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be an upper slightly α(τ1, τ2)-continuous multifunction such that F is punctually τ1τ2-connected. If (X, τ1, τ2) is α(τ1, τ2)-connected, then (Y, σ1, σ2) is σ1σ2-connected. Proof. Suppose that (Y, σ1, σ2) is not σ1σ2-connected. Then, there exist nonempty σ1σ2-open sets U and V of Y such that U∩V = ∅ and U∪V = Y . Since F is upper slightly α(τ1, τ2)-continuous, F +(U) and F+(V ) are α(τ1, τ2)-open sets of X. In view of the fact that F+(U), F+(V ) are disjoint and F is punctually τ1τ2-connected, X = F+(U)∪F+(V ) is a partition of X. This is contrary to the α(τ1, τ2)-connectedness of (X, τ1, τ2). This shows that (Y, σ1, σ2) is σ1σ2-connected. Theorem 5. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a lower slightly α(τ1, τ2)-continuous multifunction such that F is punctually τ1τ2-connected. If (X, τ1, τ2) is α(τ1, τ2)-connected, then (Y, σ1, σ2) is σ1σ2-connected. Proof. The proof is similar to that of Theorem 4. Definition 9. A bitopological space (X, τ1, τ2) is called strongly (τ1, τ2)-normal if, for any disjoint τ1τ2-closed sets F and K of X, there exist τ1τ2-clopen sets U and V of X such that F ⊆ U , K ⊆ V and U ∩ V = ∅. Definition 10. A bitopological space (X, τ1, τ2) is called α(τ1, τ2)-Hausdorff if, for each pair of distinct points x and y in X, there exist disjoint α(τ1, τ2)-open sets U and V of X such that x ∈ U and y ∈ V . Definition 11. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called punctually (τ1, τ2)- closed if, for each x ∈ X, F (x) is σ1σ2-closed. Theorem 6. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be an upper slightly α(τ1, τ2)-continuous mul- tifunction and punctually (τ1, τ2)-closed from a bitopological space (X, τ1, τ2) to a strongly (σ1, σ2)-normal bitopological space (Y, σ1, σ2) and let F (x) ∩ F (y) = ∅ for each pair of distinct points x, y ∈ X. Then, (X, τ1, τ2) is an α(τ1, τ2)-Hausdorff space. Proof. Let x and y be any two distinct points in X. Then, we have F (x) ∩ F (y) = ∅. Since (Y, σ1, σ2) is strongly (σ1, σ2)-normal, it follows that there exist disjoint σ1σ2-clopen sets U and V of Y containing F (x) and F (y), respectively. Thus, F+(U) and F+(V ) are disjoint α(τ1, τ2)-open sets of X containing x and y, respectively. This shows that (X, τ1, τ2) is an α(τ1, τ2)-Hausdorff space. C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2149 4. Slight α(τ1, τ2)-continuity and other forms of α(τ1, τ2)-continuity We begin this section by introducing the concept of upper α(τ1, τ2)-continuous multi- functions. Definition 12. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper α(τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing F (x), there exists an α(τ1, τ2)-open set U of X containing x such that F (U) ⊆ V . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper α(τ1, τ2)-continuous if F has this property at each point of X. Theorem 7. If a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper α(τ1, τ2)-continuous, then F is upper slightly α(τ1, τ2)-continuous. Proof. Let x ∈ X and V be any σ1σ2-clopen set of Y containing F (x). Since F is upper α(τ1, τ2)-continuous, there exists an α(τ1, τ2)-open set of X containing x such that F (U) ⊆ V . This shows that F is upper slightly α(τ1, τ2)-continuous. Definition 13. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower α(τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists an α(τ1, τ2)-open set U of X containing x such that F (z) ∩ V ̸= ∅ for every z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower α(τ1, τ2)-continuous if F has this property at each point of X. Theorem 8. If a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower α(τ1, τ2)-continuous, then F is lower slightly α(τ1, τ2)-continuous. Proof. The proof is similar to that of Theorem 7. Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-extremally disconnected [45] if the τ1τ2-closure of every τ1τ2-open set U of X is τ1τ2-open. Lemma 3. [45] For a bitopological space (X, τ1, τ2), the following properties are equivalent: (1) (X, τ1, τ2) is (τ1, τ2)-extremally disconnected. (2) Every (τ1, τ2)r-open set of X is τ1τ2-closed. (3) Every (τ1, τ2)r-closed set of X is τ1τ2-open. Definition 14. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost α(τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing F (x), there exists an α(τ1, τ2)-open set U of X containing x such that F (U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost α(τ1, τ2)-continuous if F has this property at each point of X. C. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2142-2154 2150 Lemma 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper almost α(τ1, τ2)-continuous; (2) for each x ∈ X and each (σ1, σ2)r-open set V of Y containing F (x), there exists an α(τ1, τ2)-open set of X containing x such that F (U) ⊆ V . Theorem 9. If a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper slightly α(τ1, τ2)- continuous and (Y, σ1, σ2) is (σ1, σ2)-extremally disconnected, then F is upper almost α(τ1, τ2)-continuous. Proof. Let x ∈ X and V be any (σ1, σ2)r-open set of Y containing F (x). Then, by Lemma 3 we have V is σ1σ2-clopen in Y . Since F is upper slightly α(τ1, τ2)-continuous, there exists an α(τ1, τ2)-open set of X containing x such that F (U) ⊆ V . By Lemma 4, F is upper almost α(τ1, τ2)-continuous. Definition 15. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower almost α(τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists an α(τ1, τ2)-open set U of X containing x such that σ1σ2-Int(σ1σ2-Cl(V )) ∩ F (z) ̸= ∅ for every z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower almost α(τ1, τ2)-continuous if F has this property at each point of X. Lemma 5. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower almost α(τ1, τ2)-continuous; (2) for each x ∈ X and each (σ1, σ2)r-open set V of Y such that F (x) ∩ V ̸= ∅, there exists an α(τ1, τ2)-open set of X containing x such that U ⊆ F−(V ). Theorem 10. If a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower slightly α(τ1, τ2)- continuous and (Y, σ1, σ2) is (σ1, σ2)-extremally disconnected, then F is lower almost α(τ1, τ2)-continuous. Proof. By utilizing Lemma 5, this can be proved similarly to that of Theorem 9. Definition 16. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly α(τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing F (x), there exists an α(τ1, τ2)-open set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly α(τ1, τ2)-continuous if F has this property at each point of X. REFERENCES 2151 Theorem 11. If a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper weakly α(τ1, τ2)- continuous, then F is upper slightly α(τ1, τ2)-continuous. Proof. Let x ∈ X and V be any σ1σ2-clopen set of Y containing F (x). Since F is upper weakly α(τ1, τ2)-continuous, there exists an α(τ1, τ2)-open set of X containing x such that F (U) ⊆ σ1σ2-Cl(V ) = V . This shows that F is upper slightly α(τ1, τ2)-continuous. Definition 17. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower weakly α(τ1, τ2)- continuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists an α(τ1, τ2)-open set U of X containing x such that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for every z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called lower weakly α(τ1, τ2)-continuous if F has this property at each point of X. Theorem 12. If a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower weakly α(τ1, τ2)- continuous, then F is lower slightly α(τ1, τ2)-continuous. Proof. The proof is similar to that of Theorem 11. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Berge. Espaces topologiques fonctions multivoques. Dunod, Paris, 1959. [2] C. Boonpok. Almost (g,m)-continuous functions. International Journal of Mathe- matical Analysis, 4(40):1957–1964, 2010. [3] C. Boonpok. M -continuous functions in biminimal structure spaces. 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