EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2210-2220 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and Lower s-(τ1, τ2)p-Continuous Multifunctions Nongluk 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. Our main purpose is to introduce the concepts of upper and lower s-(τ1, τ2)p-continuous multifunctions. Furthermore, several characterizations of upper and lower s-(τ1, τ2)p-continuous multifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08; 54C60; 54E55 Key Words and Phrases: (τ1, τ2)p-open set; upper s-(τ1, τ2)p-continuous multifunction; lower s-(τ1, τ2)p-continuous multifunction 1. Introduction In 1965, Lee [27] studied the notion of semiconnected functions. Kohli [24] intro- duced the notion of s-continuous functions and investigated several characterizations of semilocally connected spaces in terms of s-continuous functions. The class of s-continuity is a generalization of continuity and semiconnectedness. Furthermore, Kohli [25] intro- duced the concepts of s-regular spaces and completely s-regular spaces and proved that s-regularity and complete s-regularity are preserved under certain s-continuous functions. Duangphui et al. [21] introduced and investigated the notion of almost (µ, µ′)(m,n)- continuous functions. Thongmoon and Boonpok [35] introduced and studied the no- tion of strongly θ(Λ, p)-continuous functions. Moreover, several characterizations of al- most (Λ, p)-continuous functions, almost strongly θ(Λ, p)-continuous functions, 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 [33], [11], [31], [16], [10], [9], [5], [2], [37], [34], [8], [3], [17], [15] and [12], respectively. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5322 Email addresses: nongluk.h@msu.ac.th (N. Viriyapong), s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 2210 © 2024 EJPAM All rights reserved. N. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2210-2220 2211 In 1989, Lipski [28] extended the concept of s-continuous functions to the setting of multifunctions. Popa [29] introduced the concept of precontinuous multifunctions and showed that H-almost continuity and precontinuity are equivalent for multifunctions. Ew- ert and Lipski [22] introduced and investigated the concept of s-quasi-continuous multi- functions. Popa and Noiri [30] introduced and studied the notion of s-precontinuous multi- functions as a generalization of s-continuous multifunctions and precontinuous multifunc- tions. Laprom et al. [26] introduced and investigated the concept of β(τ1, τ2)-continuous multifunctions. In particular, some characterizations of (τ1, τ2)δ-semicontinuous multi- functions, almost weakly ⋆-continuous multifunctions, weakly ⋆-continuous multifunc- tions, weakly α-⋆-continuous multifunctions, ı⋆-continuous multifunctions, almost β(⋆)- continuous multifunctions, almost weakly (τ1, τ2)-continuous multifunctions, almost (τ1, τ2)- continuous multifunctions and (τ1, τ2)α-continuous multifunctions were established in [6], [18], [4], [14], [13], [7], [19], [23] and [36], respectively. Pue-on et al. [32] introduce and studied the concepts of upper and lower (τ1, τ2)-continuous multifunctions. In this paper, we introduce the notions of upper and lower s-(τ1, τ2)p-continuous multifunctions. We also investigate several characterizations of upper and lower s-(τ1, τ2)p-continuous multi- functions. 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. 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). (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 bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [20] if X cannot be writ- ten as the union of two nonempty disjoint τ1τ2-open sets. A subset A of a bitopo- logical space (X, τ1, τ2) is called (τ1, τ2)r-open [36] (resp. (τ1, τ2)s-open [6], (τ1, τ2)p- open [6], (τ1, τ2)β-open [6], α(τ1, τ2)-open) [38]) if A = τ1τ2-Int(τ1τ2-Cl(A)) (resp. A ⊆ N. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2210-2220 2212 τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))), A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A)))). The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open, α(τ1, τ2)-open) set is called (τ1, τ2)r-closed (resp. (τ1, τ2)s- closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed, α(τ1, τ2)-closed). Let A be a subset of a bitopo- logical space (X, τ1, τ2). The intersection of all (τ1, τ2)p-closed sets of X containing A is called the (τ1, τ2)p-closure of A and is denoted by (τ1, τ2)-pCl(A). The union of all (τ1, τ2)p-open sets of X contained in A is called the (τ1, τ2)p-interior of A and is denoted by (τ1, τ2)-pInt(A). Lemma 2. For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) A is (τ1, τ2)p-closed if and only if (τ1, τ2)-pCl(A) = A; (2) (τ1, τ2)-pCl(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∪A; (3) (τ1, τ2)-pCl((τ1, τ2)-pCl(A)) = (τ1, τ2)-pCl(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). 3. Upper and lower s-(τ1, τ2)p-continuous multifunctions In this section, we introduce the notions of upper and lower s-(τ1, τ2)p-continuous multifunctions. Moreover, some characterizations of upper and lower s-(τ1, τ2)p-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper s-(τ1, τ2)p- continuous if for x ∈ X and each σ1σ2-open set V of Y containing F (x) and having σ1σ2-connected complement, there exists a (τ1, τ2)p-open set U of X containing x such that F (U) ⊆ V . Theorem 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper s-(τ1, τ2)p-continuous; (2) F+(V ) is (τ1, τ2)p-open in X for every σ1σ2-open set V of Y having σ1σ2-connected complement; N. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2210-2220 2213 (3) F−(K) is (τ1, τ2)p-closed in X for every σ1σ2-connected σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(τ1τ2-Int(F −(B))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-connected σ1σ2-closure; (5) (τ1, τ2)-pCl(F −(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2- connected σ1σ2-closure; (6) F+(σ1σ2-Int(B)) ⊆ (τ1, τ2)-pInt(F +(B)) for every subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-connected. Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement and x ∈ F+(V ). Then, there exists a (τ1, τ2)p-open set U of X containing x such that F (U) ⊆ V . Therefore, we have x ∈ U ⊆ τ1τ2-Int(τ1τ2-Cl(F +(V ))). Thus, F+(V ) ⊆ τ1τ2-Int(τ1τ2-Cl(F +(V ))) and hence F+(V ) is (τ1, τ2)p-open in X. (2) ⇒ (3): The proof follows immediately from the fact that F+(Y −B) = X−F−(B) for every subset B of Y . (3) ⇒ (4): Let B be any subset of Y having the σ1σ2-connected σ1σ2-closure. Then, F−(σ1σ2-Cl(B)) is a (τ1, τ2)p-closed set of X. By Lemma 2, we have τ1τ2-Cl(τ1τ2-Int(F −(B))) ⊆ τ1τ2-Cl(τ1τ2-Int(F −(σ1σ2-Cl(B)))) ⊆ (τ1, τ2)-pCl(F −(σ1σ2-Cl(B))) = F−(σ1σ2-Cl(B)). (4) ⇒ (5): Let B be any subset of Y having the σ1σ2-connected σ1σ2-closure. It follows from Lemma 2 that (τ1, τ2)-pCl(F −(B)) = F−(B) ∪ τ1τ2-Cl(τ1τ2-Int(F −(B))) ⊆ F−(σ1σ2-Cl(B)). (5) ⇒ (6): Let B be any subset of Y such that Y −σ1σ2-Int(B) is σ1σ2-connected. By (5), X − (τ1, τ2)-Int(F +(B)) = (τ1, τ2)-pCl(X − F+(B)) = (τ1, τ2)-pCl(F −(Y −B)) ⊆ F−(σ1σ2-Cl(Y −B)) = F−(Y − σ1σ2-Int(B)) = X − F+(σ1σ2-Int(B)). N. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2210-2220 2214 Thus, F+(σ1σ2-Int(B)) ⊆ (τ1, τ2)-pInt(F +(B)). (6) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x) and having σ1σ2-connected complement. By (6), we have F+(V ) = F+(σ1σ2-Int(V )) ⊆ (τ1, τ2)-pInt(F +(V )). Put U = (τ1, τ2)-pInt(F +(V )). Then, U is a (τ1, τ2)p-open set of X containing x such that F (U) ⊆ V . This shows that F is upper s-(τ1, τ2)p-continuous. Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower s-(τ1, τ2)p- continuous if for each x ∈ X and each σ1σ2-open set V of Y having σ1σ2-connected complement such that F (x)∩ V ̸= ∅, there exists a (τ1, τ2)p-open set U of X containing x such that F (z) ∩ V ̸= ∅ for each z ∈ U . Theorem 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower s-(τ1, τ2)p-continuous; (2) F−(V ) is (τ1, τ2)p-open in X for every σ1σ2-open set V of Y having σ1σ2-connected complement; (3) F+(K) is (τ1, τ2)p-closed in X for every σ1σ2-connected σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(τ1τ2-Int(F +(B))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-connected σ1σ2-closure; (5) (τ1, τ2)-pCl(F +(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2- connected σ1σ2-closure; (6) F−(σ1σ2-Int(B)) ⊆ (τ1, τ2)-pInt(F −(B)) for every subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-connected. Proof. The proof is similar to that of Theorem 1. Definition 3. A function : (X, τ1, τ2) → (Y, σ1, σ2) is said to be s-(τ1, τ2)p-continuous if for each point x ∈ X and each σ1σ2-open set V of Y containing f(x) and having σ1σ2- connected complement, there exists a (τ1, τ2)p-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 s-(τ1, τ2)p-continuous; N. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2210-2220 2215 (2) f−1(V ) is (τ1, τ2)p-open in X for every σ1σ2-open set V of Y having σ1σ2-connected complement; (3) f−1(K) is (τ1, τ2)p-closed in X for every σ1σ2-connected σ1σ2-closed set K of Y ; (4) τ1τ2-Cl(τ1τ2-Int(f −1(B))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-connected σ1σ2-closure; (5) (τ1, τ2)-pCl(f −1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2- connected σ1σ2-closure; (6) f−1(σ1σ2-Int(B)) ⊆ (τ1, τ2)-pInt(f −1(B)) for every subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-connected. Corollary 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper s-(τ1, τ2)p-continuous if F−(V ) is (τ1, τ2)p-closed in X for every σ1σ2-connected set V of Y . Proof. Let V be any σ1σ2-open set of Y having σ1σ2-connected complement. Then, Y −V is σ1σ2-connected and F−(Y −V ) is (τ1, τ2)p-closed in X. Thus, F+(V ) is (τ1, τ2)p- open in X and by Theorem 1, F is upper s-(τ1, τ2)p-continuous. Corollary 3. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower s-(τ1, τ2)p-continuous if F+(V ) is (τ1, τ2)p-closed in X for every σ1σ2-connected set V of Y . Proof. The proof is similar to that of Corollary 2. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), by ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) [20] we denote a multifunction defined as follows: ClF⊛(x) = σ1σ2-Cl(F (x)) for each x ∈ X. Definition 4. [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 3. [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 . Lemma 4. [20] If F : (X, τ1, τ2) → (Y, σ1, σ2) is a multifunction such that F (x) is τ1τ2- regular and τ1τ2-paracompact for each x ∈ X, then ClF+ ⊛ (V ) = F+(V ) for each σ1σ2-open set V of Y . N. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2210-2220 2216 Theorem 3. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2- paracompact and σ1σ2-regular for each x ∈ X. Then, the following properties are equiva- lent: (1) F is upper s-(τ1, τ2)p-continuous; (2) ClF⊛ is upper s-(τ1, τ2)p-continuous. Proof. We put G = ClF⊛. Suppose that F is upper s-(τ1, τ2)p-continuous. Let x ∈ X and V be any σ1σ2-open set of Y containing G(x) and having σ1σ2-connected complement. By Lemma 4, we have x ∈ G+(V ) = F+(V ) and hence there exists a τ1τ2- open set 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 3 there exists a τ1τ2-open set W of X such that F (z) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V ; hence G(z) ⊆ σ1σ2-Cl(W ) ⊆ V for each z ∈ U . Thus, G(U) ⊆ V and hence G is upper s-(τ1, τ2)p-continuous. Conversely, suppose that G is upper s-(τ1, τ2)p-continuous. Let x ∈ X and V be any σ1σ2-open set of Y containing F (x) and having σ1σ2-connected complement. By Lemma 4, we have x ∈ F+(V ) = G+(V ) and hence G(x) ⊆ V . There exists a τ1τ2-open set U of X containing x such that G(U) ⊆ V . Thus, U ⊆ G+(V ) = F+(V ) and so F (U) ⊆ V . This shows that F is upper s-(τ1, τ2)p-continuous. Lemma 5. [20] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF − ⊛ (V ) = F−(V ) for each σ1σ2-open set V of Y . Theorem 4. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower s-(τ1, τ2)p-continuous; (2) ClF⊛ is lower s-(τ1, τ2)p-continuous. Proof. By using Lemma 5 this can be shown similarly to that of Theorem 3. The (τ1, τ2)p-frontier of a subset A of a bitopological space (X, τ1, τ2), denoted by (τ1, τ2)-pfr(A), is defined by (τ1, τ2)-pfr(A) = (τ1, τ2)-pCl(A) ∩ (τ1, τ2)-pCl(X −A) = (τ1, τ2)-pCl(A)− (τ1, τ2)-pInt(A). Theorem 5. The set of all points x of X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not upper s-(τ1, τ2)p-continuous is identical with the union of the (τ1, τ2)p-frontier of the upper inverse images of the σ1σ2-closures of σ1σ2-open sets containing F (x) and having σ1σ2-connected complement. N. Viriyapong, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2210-2220 2217 Proof. Suppose that F is not upper s-(τ1, τ2)p-continuous at x ∈ X. Then, there exists a σ1σ2-open set V of Y containing F (x) and having σ1σ2-connected complement such that U ∩ (X−F+(V )) ̸= ∅ for every (τ1, τ2)p-open set U of X containing x. Therefore, we have x ∈ (τ1, τ2)-pCl(X − F+(V )). On the other hand, we have x ∈ F+(V ) ⊆ (τ1, τ2)-pCl(F +(V )) and hence x ∈ (τ1, τ2)-pfr(F +(V )). Conversely, suppose that V is a σ1σ2-open set of Y containing F (x) and having σ1σ2-connected complement such that x ∈ (τ1, τ2)-pfr(F +(V )). If F is upper s-(τ1, τ2)p- continuous at x ∈ X, there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(V ); hence x ∈ (τ1, τ2)-pInt(F +(V )). This is a contradiction and so F is not upper s-(τ1, τ2)p-continuous at x. Theorem 6. The set of all points x of X at which a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is not lower s-(τ1, τ2)p-continuous is identical with the union of the (τ1, τ2)p-frontier of the lower inverse images of the σ1σ2-closures of σ1σ2-open sets meeting F (x) and having σ1σ2-connected complement. Proof. The proof is similar to that of Theorem 5. 4. Conclusion This paper deals with the notions of upper and lower s-(τ1, τ2)p-continuous multifunc- tions. Furthermore, some characterizations and several properties concerning upper and lower s-(τ1, τ2)p-continuous multifunctions are established. In the upcoming work, we plan to apply the concepts initiated in this paper to study a new generalization of upper (lower) s-(τ1, τ2)p-continuous multifunctions, namely upper (lower) almost s-(τ1, τ2)p-continuous multifunctions. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper (lower) al- most s-(τ1, τ2)p-continuous multifunctions if for each x ∈ X and each σ1σ2-open set V of Y having σ1σ2-connected complement such that x ∈ F+(V ) (x ∈ F−(V )), there ex- ists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))) (U ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V )))). The class of upper (lower) s-(τ1, τ2)p-continuous mul- tifunctions included in the class of upper (lower) almost s-(τ1, τ2)p-continuous multifunc- tions. Acknowledgements This research project was financially supported by Mahasarakham University. REFERENCES 2218 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. Far East Journal of Mathematical Sciences, 43(1):41–58, 2010. [4] C. Boonpok. On continuous multifunctions in ideal topological spaces. Lobachevskii Journal of Mathematics, 40(1):24–35, 2019. [5] C. Boonpok. On characterizations of ⋆-hyperconnected ideal topological spaces. Jour- nal of Mathematics, 2020:9387601, 2020. [6] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020. [7] C. Boonpok. Upper and lower β(⋆)-continuity. Heliyon, 7:e05986, 2021. [8] C. Boonpok. On some closed sets and low separation axioms via topological ideals. European Journal of Pure and Applied Mathematics, 15(3):300–309, 2022. [9] C. Boonpok. On some spaces via topological ideals. Open Mathematics, 21:20230118, 2023. [10] C. Boonpok. θ(⋆)-precontinuity. Mathematica, 65(1):31–42, 2023. [11] C. Boonpok and J. Khampakdee. Almost strong θ(Λ, p)-continuity for functions. European Journal of Pure and Applied Mathematics, 17(1):300–309, 2024. [12] C. Boonpok and C. Klanarong. On weakly (τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics, 17(1):416–425, 2024. [13] C. Boonpok and P. Pue-on. Continuity for multifunctions in ideal topological spaces. WSEAS Transactions on Mathematics, 19:624–631, 2020. [14] C. Boonpok and P. Pue-on. Upper and lower weakly α-⋆-continuous multifunctions. International Journal of Analysis and Applications, 21:90, 2023. [15] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous functions. International Journal of Analysis and Applications, 22:33, 2024. [16] C. Boonpok and N. Srisarakham. Weak forms of (Λ, b)-open sets and weak (Λ, b)- continuity. European Journal of Pure and Applied Mathematics, 16(1):29–43, 2023. [17] C. Boonpok and N. Srisarakham. (τ1, τ2)-continuity for functions. Asia Pacific Jour- nal of Mathematics, 11:21, 2024. REFERENCES 2219 [18] C. Boonpok and C. Viriyapong. Almost weak continuity for multifunctions in ideal topological spaces. WSEAS Transactions on Mathematics, 19:367–372, 2020. [19] C. Boonpok and C. Viriyapong. Upper and lower almost weak (τ1, τ2)-continuity. European Journal of Pure and Applied Mathematics, 14(1):1212–1225, 2021. [20] C. Boonpok, C. Viriyapong, and M. Thongmoon. On upper and lower (τ1, τ2)- precontinuous multifunctions. Journal of Mathematics and Computer Science, 18:282–293, 2018. [21] T. Duangphui, C. Boonpok, and C. Viriyapong. Continuous functions on bigeneral- ized topological spaces. International Journal of Mathematical Analysis, 5(24):1165– 1174, 2011. [22] J. Ewert and T. Lipski. On s-quasi-continuous multivalued maps. Review of Research, Faculty of Science, Mathematics Series, 20(1):167–183, 1990. [23] C. Klanarong, S. Sompong, and C. Boonpok. Upper and lower almost (τ1, τ2)- continuous multifunctions. European Journal of Pure and Applied Mathematics, 17(2):1244–1253, 2024. [24] J. K. Kohli. A class of mappings containing all continuous and all semi-connected mappings. Proceedings of the American Mathematical Society, 72:175–181, 1978. [25] J. K. Kohli. S-continuous functions and certain weak forms of regularity and complete regularity. Mathematics Nachrichten, 97:189–196, 1980. [26] K. Laprom, C. Boonpok, and C. Viriyapong. β(τ1, τ2)-continuous multifunctions on bitopological spaces. Journal of Mathematics, 2020:4020971, 2020. [27] Y. L. Lee. Some characterizations of semilocally connected spaces. Proceedings of the American Mathematical Society, 16:1318–1320, 1965. [28] T. Lipski. S-continuous multivalued maps. Mathematical Chronicle, 18:57–61, 1989. [29] V. Popa. Some properties of H-almost continuous multifunctions. Problemy Matem- atyczne, 10:9–26, 1988. [30] V. Popa and T. Noiri. On s-precontinuous multifunctions. Demonstratio Mathemat- ica, 33(3):679–687, 2000. [31] P. Pue-on and C. Boonpok. θ(Λ, p)-continuity for functions. International Journal of Mathematics and Computer Science, 19(2):491–495, 2024. [32] P. Pue-on, S. Sompong, and C. Boonpok. Upper and lower (τ1, τ2)-continuous multi- functions. International Journal of Mathematics and Computer Science, 19(4):1305– 1310, 2024. REFERENCES 2220 [33] N. Srisarakham and C. Boonpok. Almost (Λ, p)-continuous functions. International Journal of Mathematics and Computer Science, 18(2):255–259, 2023. [34] N. Srisarakham and C. Boonpok. On characterizations of δp(Λ, s)-D1 spaces. Inter- national Journal of Mathematics and Computer Science, 18(4):743–747, 2023. [35] M. Thongmoon and C. Boonpok. Strongly θ(Λ, p)-continuous functions. International Journal of Mathematics and Computer Science, 19(2):475–479, 2024. [36] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of Mathematics, 2020:6285763, 2020. [37] C. Viriyapong and C. Boonpok. (Λ, sp)-continuous functions. WSEAS Transactions on Mathematics, 21:380–385, 2022. [38] N. Viriyapong, S. Sompong, and C. Boonpok. (τ1, τ2)-extremal disconnectedness in bitopological spaces. International Journal of Mathematics and Computer Science, 19(3):855–860, 2024.