EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 156-168 ISSN 1307-5543 – ejpam.com Published by New York Business Global Upper and lower almost contra-(Λ, sp)-continuity Chawalit Boonpok1, Jeeranunt Khampakdee1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Abstract. Our main purpose is to introduce the concepts of upper and lower almost contra- (Λ, sp)-continuous multifunctions. Moreover, several characterizations of upper and lower almost contra-(Λ, sp)-continuous multifunctions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: (Λ, sp)-open set, upper almost contra-(Λ, sp)-continuous multifunc- tion, lower almost contra-(Λ, sp)-continuous multifunction 1. Introduction In 1996, Dontchev [8] introduced and studied the concept of contra-continuous func- tions. In 1999, Dontchev and Noiri [10] considered a slightly weaker form of contra- continuity called contra-semicontinuity and investigated the class of strongly S-closed spaces. In 2001, Caldas and Jafari [7] introduced and investigated the concept of contra- β-continuous functions. In 2002, Jafari and Noiri [16] introduced and studied a new form of functions called contra-precontinuous functions. In 2004, Ekici [11] introduced and inves- tigated almost contra-precontinuity as a new generalization of regular set-connectedness [9], contra-precontinuity [16], contra-continuity [8], almost s-continuity [19] and perfect continuity [18]. In 2005, Nasef [17] defined a new class of functions called contra-γ- continuous functions which lies between classes of contra-semicontinuous functions and contra-β-continuous functions. The first initiation of the concept of contra-continuous multifunctions has been done by Ekici et al. [12]. In 2009, Ekici et al. [13] introduced and studied a new generalization of contra-continuous multifunctions called almost contra- continuous multifunctions. In 2010, Ekici et al. [14] introduced and studied two new con- cepts namely contra-precontinuous multifunctions and almost contra-precontinuous multi- functions which are containing the class of contra-continuous multifunctions and contained in the class of weakly precontinuous multifunctions. In 2018, Boonpok et al. [6] introduced and studied the notions of upper and lower almost (τ1, τ2)-precontinuous multifunctions. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4581 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), jeeranunt.k@msu.ac.th (J. Khampakdee) https://www.ejpam.com 156 © 2023 EJPAM All rights reserved. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 157 Abd El-Monsef et al. [15] introduced a weak form of open sets called β-open sets. The notion of β-open sets is equivalent to that of semi-preopen sets [1]. Noiri and Hatir [20] introduced the concept of Λsp-sets in terms of the concept of β-open sets and investigated the notion of Λsp-closed sets by using Λsp-sets. In [3], the author introduced the concepts of (Λ, sp)-open sets and (Λ, sp)-closed sets which are defined by utilizing the notions of Λsp-sets and β-closed sets. The concept of (Λ, sp)-continuous multifunctions was intro- duced and investigated in [3]. The purpose of the present paper is to introduce the notions of upper and lower almost contra-(Λ, sp)-continuous multifunctions. In particular, several characterizations of upper and lower almost contra-(Λ, sp)-continuous multifunctions are discussed. 2. Preliminaries Let A be a subset of a topological space (X, τ). The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A of a topological space (X, τ) is said to be β-open [15] if A ⊆ Cl(Int(Cl(A))). The complement of a β-open set is called β-closed. The family of all β-open sets of a topological space (X, τ) is denoted by β(X, τ). A subset Λsp(A) [20] is defined as follows: Λsp(A) = ∩{U | A ⊆ U,U ∈ β(X, τ)}. A subset A of a topological space (X, τ) is called a Λsp-set [20] if A = Λsp(A). A subset A of a topological space (X, τ) is called (Λ, sp)-closed [3] if A = T ∩C, where T is a Λsp-set and C is a β-closed set. The complement of a (Λ, sp)-closed set is called (Λ, sp)-open. Let A be a subset of a topological space (X, τ). A point x ∈ X is called a (Λ, sp)-cluster point [3] of A if A ∩ U ̸= ∅ for every (Λ, sp)-open set U of X containing x. The set of all (Λ, sp)-cluster points of A is called the (Λ, sp)-closure [3] of A and is denoted by A(Λ,sp). The union of all (Λ, sp)-open sets contained in A is called the (Λ, sp)-interior [3] of A and is denoted by A(Λ,sp). Lemma 1. [3] Let A and B be subsets of a topological space (X, τ). For the (Λ, sp)-closure, the following properties hold: (1) A ⊆ A(Λ,sp) and [A(Λ,sp)](Λ,sp) = A(Λ,sp). (2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp). (3) A(Λ,sp) is (Λ, sp)-closed. (4) A is (Λ, sp)-closed if and only if A = A(Λ,sp). Lemma 2. [3] Let A and B be subsets of a topological space (X, τ). For the (Λ, sp)- interior, the following properties hold: (1) A(Λ,sp) ⊆ A and [A(Λ,sp)](Λ,sp) = A(Λ,sp). (2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp). (3) A(Λ,sp) is (Λ, sp)-open. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 158 (4) A is (Λ, sp)-open if and only if A(Λ,sp) = A. (5) [X −A](Λ,sp) = X −A(Λ,sp). (6) [X −A](Λ,sp) = X −A(Λ,sp). A subset A of a topological space (X, τ) is said to be s(Λ, sp)-open (resp. p(Λ, sp)-open, β(Λ, sp)-open, α(Λ, sp)-open, r(Λ, sp)-open) if A ⊆ [A(Λ,sp)] (Λ,sp) (resp. A ⊆ [A(Λ,sp)](Λ,sp), A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp), A ⊆ [[A(Λ,sp)] (Λ,sp)](Λ,sp), A = [A(Λ,sp)](Λ,sp)) [3]. The family of all s(Λ, sp)-open (resp. p(Λ, sp)-open, β(Λ, sp)-open, α(Λ, sp)-open, r(Λ, sp)-open) sets in a topological space (X, τ) is denoted by sΛspO(X, τ) (resp. pΛspO(X, τ), βΛspO(X, τ), αΛspO(X, τ), rΛspO(X, τ)). The complement of a s(Λ, sp)-open (resp. p(Λ, sp)-open, β(Λ, sp)-open, α(Λ, sp)-open, r(Λ, sp)-open) set is said to be s(Λ, sp)-closed (resp. p(Λ, sp)- closed, β(Λ, sp)-closed, α(Λ, sp)-closed, r(Λ, sp)-closed). The family of all s(Λ, sp)-closed (resp. p(Λ, sp)-closed, β(Λ, sp)-closed, α(Λ, sp)-closed, r(Λ, sp)-closed) sets in a topologi- cal space (X, τ) is denoted by sΛspC(X, τ) (resp. pΛspC(X, τ), βΛspC(X, τ), αΛspC(X, τ), rΛspC(X, τ)). Let A be a subset of a topological space (X, τ). The intersection of all s(Λ, sp)-closed (resp. p(Λ, sp)-closed, α(Λ, sp)-closed) sets containing A is called the s(Λ, sp)-closure [23] (resp. p(Λ, sp)-closure, α(Λ, sp)-closure [5, 22]) of A and is denoted by As(Λ,sp) (resp. Ap(Λ,sp), Aα(Λ,sp)). Throughout this paper, the spaces (X, τ) and (Y, σ) (or simply X and Y ) always mean topological spaces and F : X → Y (resp. f : X → Y ) presents a multivalued (resp. single valued) function. For a multifunction F : X → Y , following [2] 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). Moreover, F : X → Y is called upper semi-continuous (resp. lower semi-continuous) if F+(V ) (resp. F−(V )) is open in X for every open set V of Y [21]. 3. On upper and lower almost contra-(Λ, sp)-continuous multifunctions We begin this section by introducing the concepts of upper and lower almost contra- (Λ, sp)-continuous multifunctions. Definition 1. A multifunction F : (X, τ) → (Y, σ) is said to be: (i) lower almost contra-(Λ, sp)-continuous at x ∈ X if, for each r(Λ, sp)-closed set K of Y with x ∈ F−(K), there exists a (Λ, sp)-open set U of X containing x such that U ⊆ F−(K); (ii) upper almost contra-(Λ, sp)-continuous at x ∈ X if, for each r(Λ, sp)-closed set K of Y with x ∈ F+(K), there exists a (Λ, sp)-open set U of X containing x such that U ⊆ F+(K); (iii) lower (upper) almost contra-(Λ, sp)-continuous if F has this property at each point of X. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 159 Theorem 1. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is upper almost contra-(Λ, sp)-continuous; (2) F+(K) is (Λ, sp)-open in X for every r(Λ, sp)-closed set K of Y ; (3) F−(V ) is (Λ, sp)-closed in X for every r(Λ, sp)-open set V of Y ; (4) F−([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every (Λ, sp)-open set V of Y ; (5) F+([K(Λ,sp)] (Λ,sp)) is (Λ, sp)-open in X for every (Λ, sp)-closed set K of Y ; (6) for each x ∈ X and for each s(Λ, sp)-open set V of Y with F (x) ⊆ V , there exists a (Λ, sp)-open set U of X containing x such that F (U) ⊆ V (Λ,sp); (7) F+(V ) ⊆ [F+(V (Λ,sp))](Λ,sp) for every s(Λ, sp)-open set V of Y . Proof. (1) ⇒ (2): Let K be any r(Λ, sp)-closed set of Y and x ∈ F+(K). Since F is upper almost contra (Λ, sp)-continuous, there exists a (Λ, sp)-open set U of X containing x such that U ⊆ F+(K). Thus, F+(K) is (Λ, sp)-open in X. (2) ⇒ (1): The proof is obvious. (2) ⇔ (3): It follows from the fact that F+(Y −K) = X − F−(K) for every subset K of Y . (3) ⇔ (4): Let V be any (Λ, sp)-open set of Y . Then [V (Λ,sp)](Λ,sp) is r(Λ, sp)-open in Y and by (3), F−([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X. The converse is obvious. (4) ⇔ (5): It follows from the fact that F+(Y −K) = X − F−(K) for every subset K of Y . (5) ⇔ (2): It similar to that (3) ⇔ (4). (6) ⇒ (7): Let V be any s(Λ, sp)-open set of Y and x ∈ F+(V ). Then F (x) ⊆ V . By (6), there exists a (Λ, sp)-open set U of X containing x such that F (U) ⊆ V (Λ,sp). Thus, x ∈ U ⊆ F+(V (Λ,sp)) and hence x ∈ [F+(V (Λ,sp))](Λ,sp). This shows that F+(V ) ⊆ [F+(V (Λ,sp))](Λ,sp). (7) ⇒ (2): Let K be any r(Λ, sp)-closed set of Y . Then K is s(Λ, sp)-open in Y . By (7), we have F+(K) ⊆ [F+(K)](Λ,sp) and hence F+(K) is (Λ, sp)-open in X. (2) ⇒ (6): Let x ∈ X and V be any s(Λ, sp)-open set of Y with F (x) ⊆ V . Since V (Λ,sp) is r(Λ, sp)-closed and by (2), F+(V (Λ,sp)) is (Λ, sp)-open in X. Then, there exists a (Λ, sp)-open set U of X containing x such that U ⊆ F+(V (Λ,sp)). Thus, F (U) ⊆ V (Λ,sp). Theorem 2. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is lower almost contra-(Λ, sp)-continuous; (2) F−(K) is (Λ, sp)-open in X for every r(Λ, sp)-closed set K of Y ; (3) F+(V ) is (Λ, sp)-closed in X for every r(Λ, sp)-open set V of Y ; C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 160 (4) F+([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every (Λ, sp)-open set V of Y ; (5) F−([K(Λ,sp)] (Λ,sp)) is (Λ, sp)-open in X for every (Λ, sp)-closed set K of Y ; (6) for each x ∈ X and for each s(Λ, sp)-open set V of Y with F (x)∩V ̸= ∅, there exists a (Λ, sp)-open set U of X containing x such that F (z)∩ V (Λ,sp) ̸= ∅ for each z ∈ U ; (7) F−(V ) ⊆ [F−(V (Λ,sp))](Λ,sp) for every s(Λ, sp)-open set V of Y . Proof. The proof is similar to that of Theorem 1. Definition 2. A function f : (X, τ) → (Y, σ) is called almost contra-(Λ, sp)-continuous if, for each x ∈ X and each r(Λ, sp)-closed set K of Y containing f(x), there exists a (Λ, sp)-open set U of X containing x such that f(U) ⊆ K. Corollary 1. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is almost contra-(Λ, sp)-continuous; (2) f−1(K) is (Λ, sp)-open in X for every r(Λ, sp)-closed set K of Y ; (3) f−1(V ) is (Λ, sp)-closed in X for every r(Λ, sp)-open set V of Y ; (4) f−1([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every (Λ, sp)-open set V of Y ; (5) f−1([K(Λ,sp)] (Λ,sp)) is (Λ, sp)-open in X for every (Λ, sp)-closed set K of Y ; (6) for each x ∈ X and for each s(Λ, sp)-open set V of Y containing f(x), there exists a (Λ, sp)-open set U of X containing x such that f(U) ⊆ V (Λ,sp); (7) f−1(V ) ⊆ [f−1(V (Λ,sp))](Λ,sp) for every s(Λ, sp)-open set V of Y . Lemma 3. [4] Let V be a subset of a topological space (X, τ). If V ∈ βΛspO(X, τ), then V (Λ,sp) ∈ rΛspC(X, τ). Theorem 3. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is upper almost contra-(Λ, sp)-continuous; (2) F+(V (Λ,sp)) is (Λ, sp)-open in X for every β(Λ, sp)-open set V of Y ; (3) F+(V (Λ,sp)) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (4) F−([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every p(Λ, sp)-open set V of Y . C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 161 Proof. (1) ⇒ (2): Let V be any β(Λ, sp)-open set of Y . By Lemma 3, V (Λ,sp) is r(Λ, sp)-closed and by Theorem 1, F+(V (Λ,sp)) is (Λ, sp)-open in X. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let V be any p(Λ, sp)-open set of Y . Then Y − [V (Λ,sp)](Λ,sp) is r(Λ, sp)- closed and s(Λ, sp)-open. By (3), we have X − F−([V (Λ,sp)](Λ,sp)) = F+(Y − [V (Λ,sp)](Λ,sp)) = F+([Y − [V (Λ,sp)](Λ,sp)] (Λ,sp)) is (Λ, sp)-open and hence F−([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X. (4) ⇒ (1): Let V be any r(Λ, sp)-open set of Y . Then V is p(Λ, sp)-open in Y and by (4), F−(V ) = F−([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X. Thus, by Theorem 1, F is upper almost contra-(Λ, sp)-continuous. Theorem 4. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is lower almost contra-(Λ, sp)-continuous; (2) F−(V (Λ,sp)) is (Λ, sp)-open in X for every β(Λ, sp)-open set V of Y ; (3) F−(V (Λ,sp)) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (4) F+([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every p(Λ, sp)-open set V of Y . Proof. The proof is similar to that of Theorem 3. Corollary 2. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is almost contra-(Λ, sp)-continuous; (2) f−1(V (Λ,sp)) is (Λ, sp)-open in X for every β(Λ, sp)-open set V of Y ; (3) f−1(V (Λ,sp)) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (4) f−1([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every p(Λ, sp)-open set V of Y . Lemma 4. For a subset A of a topological space (X, τ), the following properties hold: (1) Aα(Λ,sp) = A ∪ [[A(Λ,sp)](Λ,sp)] (Λ,sp) [5, 22]. (2) As(Λ,sp) = A ∪ [A(Λ,sp)](Λ,sp) [23]. (3) Ap(Λ,sp) = A ∪ [A(Λ,sp)] (Λ,sp). Lemma 5. For a subset V of a topological space (X, τ), the following properties hold: (1) V α(Λ,sp) = V (Λ,sp) for every V ∈ βΛspO(X, τ). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 162 (2) V p(Λ,sp) = V (Λ,sp) for every V ∈ sΛspO(X, τ). (3) V s(Λ,sp) = V (Λ,sp) for every V ∈ pΛspO(X, τ). Theorem 5. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is upper almost contra-(Λ, sp)-continuous; (2) F+(V α(Λ,sp)) is (Λ, sp)-open in X for every β(Λ, sp)-open set V of Y ; (3) F+(V p(Λ,sp)) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (4) F−([V s(Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every p(Λ, sp)-open set V of Y . Proof. This is an immediate consequence of Theorem 3 and Lemma 5. Theorem 6. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is lower almost contra-(Λ, sp)-continuous; (2) F−(V α(Λ,sp)) is (Λ, sp)-open in X for every β(Λ, sp)-open set V of Y ; (3) F−(V p(Λ,sp)) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (4) F+([V s(Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every p(Λ, sp)-open set V of Y . Proof. This is an immediate consequence of Theorem 4 and Lemma 5. Corollary 3. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is almost contra-(Λ, sp)-continuous; (2) f−1(V α(Λ,sp)) is (Λ, sp)-open in X for every β(Λ, sp)-open set V of Y ; (3) f−1(V p(Λ,sp)) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (4) f−1([V s(Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X for every p(Λ, sp)-open set V of Y . Theorem 7. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is upper almost contra-(Λ, sp)-continuous; (2) [F−(V )](Λ,sp) ⊆ F−([V (Λ,sp)](Λ,sp)) for every (Λ, sp)-open set V of Y ; (3) [F−(V )](Λ,sp) ⊆ F−(V s(Λ,sp)) for every (Λ, sp)-open set V of Y . C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 163 Proof. (1) ⇒ (2): Let V be any (Λ, sp)-open set of Y . Then [V (Λ,sp)](Λ,sp) is r(Λ, sp)-open in Y . By Theorem 1, F−([V (Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X. Since V ⊆ [V (Λ,sp)](Λ,sp), F −(V ) ⊆ F−([V (Λ,sp)](Λ,sp)) and hence [F−(V )](Λ,sp) ⊆ F−([V (Λ,sp)](Λ,sp)). (2) ⇒ (1): Let V be any r(Λ, sp)-open set of Y . Then V is (Λ, sp)-open in Y . By (2), we have [F−(V )](Λ,sp) ⊆ F−([V (Λ,sp)](Λ,sp)) = F−(V ) and hence F−(V ) is (Λ, sp)-closed in X. Thus, by Theorem 1, F is upper almost contra-(Λ, sp)-continuous. (2) ⇔ (3): It follows from Lemma 4. Theorem 8. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is lower almost contra-(Λ, sp)-continuous; (2) [F+(V )](Λ,sp) ⊆ F+([V (Λ,sp)](Λ,sp)) for every (Λ, sp)-open set V of Y ; (3) [F+(V )](Λ,sp) ⊆ F+(V s(Λ,sp)) for every (Λ, sp)-open set V of Y . Proof. The proof is similar to that of Theorem 7. Corollary 4. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is almost contra-(Λ, sp)-continuous; (2) [f−1(V )](Λ,sp) ⊆ f−1([V (Λ,sp)](Λ,sp)) for every (Λ, sp)-open set V of Y ; (3) [f−1(V )](Λ,sp) ⊆ f−1(V s(Λ,sp)) for every (Λ, sp)-open set V of Y . Let A be a subset of a topological space (X, τ). A point x ∈ X is said to be in the θs(Λ, sp)-closure of A, denoted by Aθs(Λ,sp), if A ∩ U (Λ,sp) ̸= ∅ for each s(Λ, sp)-open set U of X containing x. A subset A of a topological space (X, τ) is called θs(Λ, sp)-closed if A = Aθs(Λ,sp). The complement of a θs(Λ, sp)-closed set is called θs(Λ, sp)-open. Theorem 9. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is lower almost contra-(Λ, sp)-continuous; (2) F−(V ) is (Λ, sp)-open in X for every θs(Λ, sp)-open set V of Y ; (3) F+(K) is (Λ, sp)-closed in X for every θs(Λ, sp)-closed set K of Y ; (4) [F+([B(Λ,sp)](Λ,sp))] (Λ,sp) ⊆ F+(Bs(Λ,sp)) for every subset B of Y ; (5) [F+(B)](Λ,sp) ⊆ F+(Bθs(Λ,sp)) for every subset B of Y ; (6) F (A(Λ,sp)) ⊆ [F (A)]θs(Λ,sp) for every subset A of X. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 164 Proof. (1) ⇒ (2): Let V be any θs(Λ, sp)-open set of Y . There exists a family of r(Λ, sp)-closed sets {Kγ | γ ∈ Γ} such that V = ∪{Kγ | γ ∈ Γ}. It follows from Theorem 2 that F−(V ) = ∪{F−(Kγ) | γ ∈ Γ} is (Λ, sp)-open in X. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let B be any subset of Y . Then [B(Λ,sp)](Λ,sp) is r(Λ, sp)-open and hence [B(Λ,sp)](Λ,sp) is θs(Λ, sp)-open in Y . By (3), F+([B(Λ,sp)](Λ,sp)) is (Λ, sp)-closed in X. Thus, [F+([B(Λ,sp)](Λ,sp))] (Λ,sp) = F+([B(Λ,sp)](Λ,sp)) ⊆ F+(Bs(Λ,sp)). (4) ⇒ (5): Let B be any subset of Y . For any r(Λ, sp)-open set V with B ⊆ V , we have [F+(B)](Λ,sp) ⊆ [F+(V )](Λ,sp) = [F+([V (Λ,sp)](Λ,sp))] (Λ,sp) ⊆ F+(V s(Λ,sp)) = F+(V ). Thus, [F+(B)](Λ,sp) ⊆ F+(∩{V ∈ rΛspO(X, τ) | B ⊆ V }) = F+(Bθs(Λ,sp)). (5) ⇒ (1): Let V be any s(Λ, sp)-open set of Y . By (5), X − [F−(V (Λ,sp))](Λ,sp) = [F+(Y − V (Λ,sp))](Λ,sp) ⊆ F+([Y − V (Λ,sp)]θs(Λ,sp)) = F+(Y − V (Λ,sp)) = X − F−(V (Λ,sp)) and hence F−(V ) ⊆ F−(V (Λ,sp)) ⊆ [F−(V (Λ,sp))](Λ,sp). By Theorem 2, F is lower almost contra-(Λ, sp)-continuous. (5) ⇒ (6): Let A be any subset of X and B = F (A). Then A ⊆ F+(B) and by (5), A(Λ,sp) ⊆ [F+(B)](Λ,sp) ⊆ F+(Bθs(Λ,sp)). Thus, F (A(Λ,sp)) ⊆ F (F+(Bθs(Λ,sp))) ⊆ Bθs(Λ,sp) = [F (A)]θs(Λ,sp). (6) ⇒ (5): Let B be any subset of Y . By (6), we have F ([F+(B)](Λ,sp)) ⊆ [F (F+(B))]θs(Λ,sp) ⊆ Bθs(Λ,sp) and hence [F+(B)](Λ,sp) ⊆ F+(Bθs(Λ,sp)). Corollary 5. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is almost contra-(Λ, sp)-continuous; (2) f−1(V ) is (Λ, sp)-open in X for every θs(Λ, sp)-open set V of Y ; (3) f−1(K) is (Λ, sp)-closed in X for every θs(Λ, sp)-closed set K of Y ; (4) [f−1([B(Λ,sp)](Λ,sp))] (Λ,sp) ⊆ f−1(Bs(Λ,sp)) for every subset B of Y ; (5) [f−1(B)](Λ,sp) ⊆ f−1(Bθs(Λ,sp)) for every subset B of Y ; (6) f(A(Λ,sp)) ⊆ [f(A)]θs(Λ,sp) for every subset A of X. Definition 3. A multifunction F : (X, τ) → (Y, σ) is said to be upper strongly s(Λ, sp)- continuous if, for each x ∈ X and each s(Λ, sp)-open set V of Y such that F (x) ⊆ V , there exists a (Λ, sp)-open set U of X containing x such that F (U) ⊆ V . C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 165 Theorem 10. For a multifunction F : (X, τ) → (Y, σ), the following properties are equivalent: (1) F is upper strongly s(Λ, sp)-continuous; (2) F+(V ) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (3) F−(K) is (Λ, sp)-closed in X for every s(Λ, sp)-closed set K of Y ; (4) [F−(B)](Λ,sp) ⊆ F−(Bs(Λ,sp)) for every subset B of Y ; (5) F+(Bs(Λ,sp)) ⊆ [F+(B)](Λ,sp) for every subset B of Y . Proof. (1) ⇒ (2): Let V be any s(Λ, sp)-open set of Y and x ∈ F+(V ). Then F (x) ⊆ V . Since F is upper strongly s(Λ, sp)-continuous, there exists a (Λ, sp)-open set U of X containing x such that U ⊆ F+(V ). Thus, F+(V ) ⊆ [F+(V )](Λ,sp) and hence F+(V ) is (Λ, sp)-open in X. (2) ⇒ (3): The proof is obvious. (3) ⇒ (4): Let B be any subset of Y . Then Bs(Λ,sp) is s(Λ, sp)-closed and by (3), F−(Bs(Λ,sp)) is (Λ, sp)-closed in X. Thus, [F−(B)](Λ,sp) ⊆ [F−(Bs(Λ,sp))](Λ,sp) = F−(Bs(Λ,sp)). (4) ⇒ (5): Let B be any subset of Y . By (4), X− [F+(B)](Λ,sp) = [X−F+(B)](Λ,sp) = [F−(Y −B)](Λ,sp) ⊆ F−([Y −B]s(Λ,sp)) = F−(Y −Bs(Λ,sp)) = X−F+(Bs(Λ,sp)). Therefore, F+(Bs(Λ,sp)) ⊆ [F+(B)](Λ,sp). (5) ⇒ (1): Let x ∈ X and V be any s(Λ, sp)-open set of Y such that F (x) ⊆ V . By (5), we have F+(V ) ⊆ [F+(V )](Λ,sp) and hence F+(V ) is (Λ, sp)-open in X. Put U = F+(V ), then U is a (Λ, sp)-open set of X containing x such that F (U) ⊆ V . This shows that F is upper strongly s(Λ, sp)-continuous. Definition 4. A multifunction F : (X, τ) → (Y, σ) is said to be lower strongly s(Λ, sp)- continuous if, for each x ∈ X and each s(Λ, sp)-open set V of Y such that F (x) ∩ V ̸= ∅, there exists a (Λ, sp)-open set U of X containing x such that F (z)∩V ̸= ∅ for each z ∈ U . Theorem 11. For a multifunction F : (X, τ) → (Y, σ), the following properties are equivalent: (1) F is lower strongly s(Λ, sp)-continuous; (2) F−(V ) is (Λ, sp)-open in X for every s(Λ, sp)-open set V of Y ; (3) F+(K) is (Λ, sp)-closed in X for every s(Λ, sp)-closed set K of Y ; (4) [F+(B)](Λ,sp) ⊆ F+(Bs(Λ,sp)) for every subset B of Y ; (5) F−(Bs(Λ,sp)) ⊆ [F−(B)](Λ,sp) for every subset B of Y . Proof. The proof is similar to that of Theorem 10. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 16 (1) (2023), 156-168 166 Definition 5. A topological space (X, τ) is called strongly s(Λ, sp)-regular if, for each s(Λ, sp)-closed set K and each x ∈ X −K, there exists a r(Λ, sp)-closed set F containing x such that F ∩K = ∅. Lemma 6. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is strongly s(Λ, sp)-regular; (2) for each s(Λ, sp)-open set W of X and each x ∈ W , there exists a s(Λ, sp)-open set V such that x ∈ V ⊆ V (Λ,sp) ⊆ W ; (3) for each s(Λ, sp)-open set W of X and each x ∈ W , there exists a r(Λ, sp)-closed set F such that x ∈ F ⊆ W ; (4) As(Λ,sp) = Aθs(Λ,sp) for every subset A of X; (5) every s(Λ, sp)-open set of X is θs(Λ, sp)-open. Theorem 12. Let (Y, σ) be a strongly s(Λ, sp)-regular space. For a multifunction F : (X, τ) → (Y, σ), the following properties are equivalent: (1) F is lower strongly s(Λ, sp)-continuous; (2) F+(Bθs(Λ,sp)) is (Λ, sp)-closed in X for every subset B of Y ; (3) F is lower almost contra-(Λ, sp)-continuous. Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 6, Bθs(Λ,sp) is s(Λ, sp)-closed and by Theorem 11, F+(Bθs(Λ,sp)) is (Λ, sp)-closed. (2) ⇒ (3): Let B be any subset of Y . By (2), we have [F+(B)](Λ,sp) ⊆ [F+(Bθs(Λ,sp))](Λ,sp) = F+(Bθs(Λ,sp)) and by Theorem 9, F is lower almost contra-(Λ, sp)-continuous. (3) ⇒ (1): Let V be any s(Λ, sp)-open set of Y . Since (Y, σ) is strongly s(Λ, sp)- regular, by Lemma 6, V is θs(Λ, sp)-open. By Theorem 9, F−(V ) is (Λ, sp)-open in X. Thus, by Theorem 11, F is lower strongly s(Λ, sp)-continuous. Theorem 13. If F : (X, τ) → (Y, σ) is an upper strongly s(Λ, sp)-continuous multifunc- tion and G : (Y, σ) → (Z, η) is an upper almost contra-(Λ, sp)-continuous multifunction, then G ◦ F : (X, τ) → (Z, η) is upper almost contra-(Λ, sp)-continuous. Proof. Let K be any r(Λ, sp)-closed set of Z. We have (G ◦ F )+(K) = F−(G+(K)). Since G is lower almost contra-(Λ, sp)-continuous, by Theorem 1, G+(K) is (Λ, sp)-open in X and hence G+(K) is s(Λ, sp)-open. Since F is lower strongly s(Λ, sp)-continuous, by Theorem 10, F+(G+(K)) is (Λ, sp)-open. Thus, G ◦ F is upper almost contra-(Λ, sp)- continuous. REFERENCES 167 Theorem 14. If F : (X, τ) → (Y, σ) is a lower strongly s(Λ, sp)-continuous multifunction and G : (Y, σ) → (Z, η) is a lower almost contra-(Λ, sp)-continuous multifunction, then G ◦ F : (X, τ) → (Z, η) is lower almost contra-(Λ, sp)-continuous. Proof. The proof is similar to that of Theorem 13. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] D. Andrijević. On b-open sets. Matematički Vesnik, 48:59–64, 1996. [2] C. Berge. Espaces topologiques fonctions multivoques. Dunod, Paris, 1959. [3] C. Boonpok. (Λ, sp)-closed sets and related topics in topological spaces. WSEAS Transactions on Mathematics, 19:321–322, 2020. [4] C. Boonpok and J. Khampakdee. (Λ, sp)-open sets in topological spaces. European Journal of Pure and Applied Mathematics, 15(2):572–588, 2022. [5] C. Boonpok and J. Khampakdee. On almost α(Λ, sp)-continuous multifunctions. European Journal of Pure and Applied Mathematics, 15(2):626–634, 2022. [6] 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. [7] M. Caldas and S. Jafari. Some properties of contra-β-continuous functions. Memoirs of the Faculty of Science, Kochi University. Series A, Mathematics, 22:19–28, 2001. [8] J. Dontchev. Contra-continuous functions and strongly S-closed spaces. International Journal of Mathematics and Mathematical Sciences, 19(2):303–310, 1996. [9] J. Dontchev, M. Ganster, and I. Reilly. More on almost s-continuity. Indian Journal of Mathematics, 41:139–146, 1999. [10] J. Dontchev and T. Noiri. Contra-semicontinuous functions. Mathematica Pannonica, 10(2):159–168, 1999. [11] E. Ekici. Almost contra-precontinuous functions. Bulletin of the Malaysian Mathe- matical Sciences Society, 27:53–65, 2004. [12] E. Ekici, S. Jafari, and T. Noiri. On upper and lower contra-continuous multifunc- tions. Analele Stiintifice ale Universitatii Al I Cuza din Iasi-Matematica, 54(1):75–85, 2008. REFERENCES 168 [13] E. Ekici, S. Jafari, and V. Popa. On almost contra-continuous multifunctions. Lobachevskii Journal of Mathematics, 30(2):124–131, 2009. [14] E. Ekici, S. Jafari, and V. Popa. On contra-precontinuous and almost contra- precontinuous multifunctions. Journal of Advanced Research in Pure Mathematics, 2(1):11–25, 2010. [15] M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud. β-open sets and β- continuous mappings. Bulletin of the Faculty of Science. Assiut University., 12:77–90, 1983. [16] S. Jafari and T. Noiri. On contra-precontinuous functions. Bulletin of the Malaysian Mathematical Sciences Society, 25:115–128, 2002. [17] A. A. Nasef. Some properties of contra-γ-continuous functions. Chaos, Solitons & Fractals, 24:471–477, 2005. [18] T. Noiri. Super-continuity and some strong forms of continuity. Indian Journal of Pure and Applied Mathematics, 15:241–250, 1984. [19] T. Noiri, B. Ahmad, and M. Khan. Almost s-continuous functions. Kyungpook Mathematical Journal, 35:311–322, 1995. [20] T. Noiri and E. Hatir. Λsp-sets and some weak separation axioms. Acta Mathematica Hungarica, 103(3):225–232, 2004. [21] V. I. Ponomarev. Properties of topological spaces preserved under multivalued continuous mappings on compacta. American Mathematical Society Translations, 38(2):119–140, 1964. [22] C. Viriyapong and C. Boonpok. On α(Λ, sp)-continuous multifunctions. International Journal of Mathematics and Computer Science, 17(3):1375–1381, 2022. [23] C. Viriyapong and C. Boonpok. Weak quasi (Λ, sp)-continuity for multifunctions. International Journal of Mathematics and Computer Science, 17(3):1201–1209, 2022.