EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1694-1704 ISSN 1307-5543 – ejpam.com Published by New York Business Global Contra-(Λ, sp)-continuity and δ(Λ, sp)-closed sets Chawalit Boonpok1, Chokchai Viriyapong1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Abstract. This paper deals with the concepts of upper and lower contra-(Λ, sp)-continuous mul- tifunctions. Moreover, some characterizations of upper and lower contra-(Λ, sp)-continuous multi- functions are investigated. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: δ(Λ, sp)-closed set, upper contra-(Λ, sp)-continuous multifunction, lower contra-(Λ, sp)-continuous multifunction 1. Introduction The field of the mathematical science which goes under the name of topology is con- cerned with all questions directly or indirectly related to continuity. The concept of contra- continuity was introduced and studied by Dontchev [7]. In 1999, Dontchev and Noiri [9] 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 [6] intro- duced and investigated the concept of contra-β-continuous functions. In 2002, Jafari and Noiri [14] introduced and studied a new form of functions called contra-precontinuous functions. In 2004, Ekici [10] presented and studied almost contra-precontinuity as a new generalization of regular set-connectedness [8], contra-precontinuity [14], contra-continuity [7], almost s-continuity [17] and perfect continuity [16]. In 2005, Nasef [15] 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. [10]. In 2009, Ekici et al. [11] introduced and studied a new generalization of contra-continuous multifunctions called almost contra-continuous multifunctions. Noiri and Popa [19] in- troduced and investigated the notion of weakly precontinuous multifunctions. In 2010, Ekici et al. [12] introduced and studied two new concepts namely contra-preconrinuous multifunctions and almost contra-precontinuous multifunctions which are containing the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4370 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), chokchai.v@msu.ac.th (C. Viriyapong) https://www.ejpam.com 1694 © 2022 EJPAM All rights reserved. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1695 class of contra-continuous multifunctions and contained in the class of weakly precontin- uous multifunctions. The concept of β-open sets due to Abd El-Monsef et al. [13] or semi-preopen sets in the sense of Andrijević [1] plays a significant role in general topology. Noiri and Hatir [18] 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. Moreover, some characterizations of Λsp-extremally disconnected spaces are investigated in [3]. The purpose of the present paper is to introduce the notions of upper and lower contra-(Λ, sp)-continuous multifunc- tions. In particular, several characterizations of upper and lower 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 [13] 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) [18] 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 [18] 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. The family of all (Λ, sp)-open sets in a topological space (X, τ) is denoted by ΛspO(X, τ). 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) = ∩{F |A ⊆ F and F is (Λ, sp)-closed}. (4) A(Λ,sp) is (Λ, sp)-closed. (5) 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). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1696 (2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp). (3) A(Λ,sp) is (Λ, sp)-open. (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 called r(Λ, sp)-open [3] if A = [A(Λ,sp)](Λ,sp). The complement of a r(Λ, sp)-open set is said to be r(Λ, sp)-closed. The family of all r(Λ, sp)-open sets in a topological space (X, τ) is denoted by rΛspO(X, τ). Throughout this paper, (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. By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y , and always assume that F (x) ̸= ∅ for all x ∈ X. 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). Then, F is said to be a surjection if F (X) = Y , or equivalently, if for each y ∈ Y , there exists an x ∈ X such that y ∈ F (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 [20]. 3. Upper and lower contra-(Λ, sp)-continuous multifunctions In this section, we introduce the notions of upper and lower contra-(Λ, sp)-continuous multifunctions. Moreover, several characterizations of upper and lower contra-(Λ, sp)- continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ) → (Y, σ) is said to be: (1) upper contra-(Λ, sp)-continuous at x ∈ X if, for each (Λ, sp)-closed set K of Y such that x ∈ F+(K), there exists a (Λ, sp)-open set U of X containing x such that U ⊆ F+(K); (2) lower contra-(Λ, sp)-continuous at x ∈ X if, for each (Λ, sp)-closed set K of Y such that x ∈ F−(K), there exists a (Λ, sp)-open set U of X containing x such that U ⊆ F−(K); (3) upper (resp. lower) contra-(Λ, sp)-continuous if F has this property at each point of X. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1697 Theorem 1. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is upper contra-(Λ, sp)-continuous; (2) F+(K) is (Λ, sp)-open in X for every (Λ, sp)-closed set K of Y ; (3) F−(V ) is (Λ, sp)-closed in X for every (Λ, sp)-open set V of Y ; (4) for each x ∈ X and each (Λ, sp)-closed set K of Y containing F (x), there exists U ∈ ΛspO(X, τ) containing x such that if y ∈ U , then F (y) ⊆ K. Proof. (1) ⇔ (2): Let K be any (Λ, sp)-closed set of Y and let x ∈ F+(K). Since F is upper contra-(Λ, sp)-continuous, there exists U ∈ ΛspO(X, τ) containing x such that F (U) ⊆ K. Thus, x ∈ U ⊆ F+(K) and hence F+(K) is (Λ, sp)-open in X. The converse is similar. (2) ⇔ (3): It follows from the fact that F+(Y −B) = X − F−(B) for every subset B of Y . (1) ⇔ (4): This is obvious. Theorem 2. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is lower contra-(Λ, sp)-continuous; (2) F−(K) is (Λ, sp)-open in X for every (Λ, sp)-closed set K of Y ; (3) F+(V ) is (Λ, sp)-closed in X for every (Λ, sp)-open set V of Y ; (4) for each x ∈ X and each (Λ, sp)-closed set K of Y such that F (x) ∩ K ̸= ∅, there exists U ∈ ΛspO(X, τ) containing x such that if y ∈ U , then F (y) ∩K ̸= ∅. Proof. The proof is similar to that of Theorem 1. Definition 2. A function f : (X, τ) → (Y, σ) is said to be contra-(Λ, sp)-continuous if, for each x ∈ X and each (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 contra-(Λ, sp)-continuous; (2) f−1(K) is (Λ, sp)-open in X for every (Λ, sp)-closed set K of Y ; (3) f−1(V ) is (Λ, sp)-closed in X for every (Λ, sp)-open set V of Y . C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1698 Let A be a subset of a topological space (X, τ). A point x ∈ X is called a δ(Λ, sp)- cluster point [21] of A if A∩ [U (Λ,sp)](Λ,sp) ̸= ∅ for every (Λ, sp)-open set U of X containing x. The set of all δ(Λ, sp)-cluster points of A is called the δ(Λ, sp)-closure [21] of A and is denoted by Aδ(Λ,sp). If A = Aδ(Λ,sp), then A is said to be δ(Λ, sp)-closed [21]. The complement of a δ(Λ, sp)-closed set is said to be δ(Λ, sp)-open [21]. The union of all δ(Λ, sp)-open sets contained in A is called the δ(Λ, sp)-interior [21] of A and is denoted by Aδ(Λ,sp). Definition 3. A topological space (X, τ) is said to be semi-(Λ, sp)-regular if, for each (Λ, sp)-open set U of X and each x ∈ U , there exists a r(Λ, sp)-open set V such that x ∈ V ⊆ U . Lemma 3. Let (X, τ) be a semi-(Λ, sp)-regular space. Then, the following properties hold: (1) A(Λ,sp) = Aδ(Λ,sp) for every subset A of X. (2) Every (Λ, sp)-open set is δ(Λ, sp)-open. Theorem 3. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a semi-(Λ, sp)- regular space, the following properties are equivalent: (1) F is upper contra-(Λ, sp)-continuous; (2) F+(Bδ(Λ,sp)) is (Λ, sp)-open in X for every subset B of Y ; (3) F+(K) is (Λ, sp)-open in X for every δ(Λ, sp)-closed set K of Y ; (4) F−(V ) is (Λ, sp)-closed in X for every δ(Λ, sp)-open set V of Y . Proof. (1) ⇒ (2): Let B be any subset of Y . Thus, by Lemma 3, Bδ(Λ,sp) is a (Λ, sp)-closed set of Y and by Theorem 1, F+(Bδ(Λ,sp)) is (Λ, sp)-open in X. (2) ⇒ (3): Let K be any δ(Λ, sp)-closed set of Y . Then, Kδ(Λ,sp) = K. By (2), F+(K) is (Λ, sp)-open in X. (3) ⇒ (4): Let V be any δ(Λ, sp)-open set of Y . Then, Y − V is a δ(Λ, sp)-closed set of Y . By (3), we have X − F−(V ) = F+(Y − V ) is (Λ, sp)-open in X and hence F−(V ) is (Λ, sp)-closed. (4) ⇒ (1): Let V be any (Λ, sp)-open set of Y . Since (Y, σ) is semi-(Λ, sp)-regular, by Lemma 3, V is a δ(Λ, sp)-open set of Y . By (4), F−(V ) is (Λ, sp)-closed in X and by Theorem 1, F is upper contra-(Λ, sp)-continuous. Theorem 4. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a semi-(Λ, sp)- regular space, the following properties are equivalent: (1) F is lower contra-(Λ, sp)-continuous; (2) F−(Bδ(Λ,sp)) is (Λ, sp)-open in X for every subset B of Y ; (3) F−(K) is (Λ, sp)-open in X for every δ(Λ, sp)-closed set K of Y ; C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1699 (4) F+(V ) is (Λ, sp)-closed in X for every δ(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), where (Y, σ) is a semi-(Λ, sp)-regular space, the following properties are equivalent: (1) f is contra-(Λ, sp)-continuous; (2) f−1(Bδ(Λ,sp)) is (Λ, sp)-open in X for every subset B of Y ; (3) f−1(K) is (Λ, sp)-open in X for every δ(Λ, sp)-closed set K of Y ; (4) f−1(V ) is (Λ, sp)-closed in X for every δ(Λ, sp)-open set V of Y . Definition 4. A subset K of a topological space (X, τ) is called strongly SΛsp-closed (resp. Λsp-compact) relative to X if every cover of K by (Λ, sp)-closed (resp. (Λ, sp)-open) sets of X has a finite subcover. A topological space (X, τ) is called strongly SΛsp-closed (resp. Λsp-compact [22]) if X is strongly SΛsp-closed (resp. Λsp-compact) relative to X. Theorem 5. Let F : (X, τ) → (Y, σ) be an upper contra-(Λ, sp)-continuous surjective multifunction such that F (x) is strongly SΛsp-closed relative to Y for each x ∈ X. If A is Λsp-compact relative to X, then F (A) is strongly SΛsp-closed relative to Y . Proof. Let {Vα | α ∈ ∇} be any cover of F (A) by (Λ, sp)-closed sets of Y . For each x ∈ A, F (x) is strongly SΛsp-closed relative to Y and there exists a finite subset ∇(x) of ∇ such that F (x) ⊆ ∪{Vα | α ∈ ∇(x)}. Put V (x) = ∪{Vα | α ∈ ∇(x)}. Then, F (x) ⊆ V (x). Since F is upper contra-(Λ, sp)-continuous, there exists a (Λ, sp)-open U(x) of X containing x such that F (U(x)) ⊆ V (x). Since {U(x) | x ∈ A} is a cover of A by (Λ, sp)-open sets of X, there exists a finite number of points of A, say, x1, x2, ..., xn such that A ⊆ ∪{U(xi) | 1 ≤ i ≤ n}. Thus, F (A) ⊆ F ( n ∪ i=1 U(xi)) ⊆ n ∪ i=1 F (U(xi)) ⊆ n ∪ i=1 V (xi) ⊆ n ∪ i=1 [∪α∈∇(xi)Vα] and hence F (A) is strongly SΛsp-closed relative to Y . Corollary 3. Let F : (X, τ) → (Y, σ) be an upper contra-(Λ, sp)-continuous surjective multifunction such that F (x) is strongly SΛsp-closed relative to Y for each x ∈ X. If X is Λsp-compact, then Y is strongly SΛsp-closed. Corollary 4. If f : (X, τ) → (Y, σ) is a contra-(Λ, sp)-continuous surjective function and A is Λsp-compact relative to X, then f(A) is strongly SΛsp-closed relative to Y . Definition 5. [4] Let A be a subset of a topological space (X, τ). The (Λ, sp)-frontier of A, denoted by (Λ, sp)-Fr(A), is defined by (Λ, sp)-Fr(A) = A(Λ,sp) ∩ [X −A](Λ,sp) = A(Λ,sp) −A(Λ,sp). C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1700 Theorem 6. The set of all points x of X at which a multifunction F : (X, τ) → (Y, σ) is not upper contra-(Λ, sp)-continuous is identical with the union of the (Λ, sp)-frontiers of the upper inverse images of (Λ, sp)-closed sets of Y containing F (x). Proof. Let x ∈ X at which F is not upper contra-(Λ, sp)-continuous. Then, there exists a (Λ, sp)-closed set V of Y containing F (x) such that U ∩ (X − F+(V )) ̸= ∅ for every U ∈ ΛspO(X, τ) containing x. Thus, x ∈ [X − F+(V )](Λ,sp). On the other hand, we have x ∈ F+(V ) ⊆ [F+(V )](Λ,sp) and hence x ∈ (Λ, sp)-Fr(F+(V )). Conversely, let V be any (Λ, sp)-closed set of Y containing F (x) such that x ∈ (Λ, sp)-Fr(F+(V )). If F is upper contra-(Λ, sp)-continuous at x, then there exists U ∈ ΛspO(X, τ) containing x such that U ⊆ F+(V ); hence x ∈ [F+(V )](Λ,sp). This is a contradiction and hence F is not upper contra-(Λ, sp)-continuous at x. Theorem 7. The set of all points x of X at which a multifunction F : (X, τ) → (Y, σ) is not lower contra-(Λ, sp)-continuous is identical with the union of the (Λ, sp)-frontiers of the lower inverse images of (Λ, sp)-closed sets of Y meeting F (x). Proof. The proof is similar to that of Theorem 6. Corollary 5. The set of all points x of X at which a function f : (X, τ) → (Y, σ) is not contra-(Λ, sp)-continuous is identical with the union of the (Λ, sp)-frontiers of the inverse images of (Λ, sp)-closed sets of Y containing f(x). Definition 6. [5] Let A be a subset of a topological space (X, τ). A subset Λ(Λ,sp)(A) is defined as follows: Λ(Λ,sp)(A) = ∩{U | A ⊆ U,U ∈ ΛspO(X, τ)}. Lemma 4. [5] For subsets A,B of a topological space (X, τ), the following properties hold: (1) A ⊆ Λ(Λ,sp)(A). (2) If A ⊆ B, then Λ(Λ,sp)(A) ⊆ Λ(Λ,sp)(B). (3) Λ(Λ,sp)[Λ(Λ,sp)(A)] = Λ(Λ,sp)(A). (4) If A is (Λ, sp)-open, Λ(Λ,sp)(A) = A. Theorem 8. Let F : (X, τ) → (Y, σ) be a multifunction. If [F−(B)](Λ,sp) ⊆ F−(Λ(Λ,sp)(B)) for every subset B of Y , then F is upper contra-(Λ, sp)-continuous. Proof. Let V be any (Λ, sp)-open set of Y . By Lemma 4, [F−(V )](Λ,sp) ⊆ F−(Λ(Λ,sp)(V )) = F−(V ) and hence [F−(V )](Λ,sp) = F−(V ). Thus, F−(V ) is (Λ, sp)-closed in X, by Theorem 1, F is upper contra-(Λ, sp)-continuous. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1701 Corollary 6. Let f : (X, τ) → (Y, σ) be a function. If [f−1(B)](Λ,sp) ⊆ f−1(Λ(Λ,sp)(B)) for every subset B of Y , then f is contra-(Λ, sp)-continuous. Theorem 9. Let F : (X, τ) → (Y, σ) be a multifunction. If F (B(Λ,sp)) ⊆ Λ(Λ,sp)(F (B)) for every subset B of Y , then F is lower contra-(Λ, sp)-continuous. Proof. Let V be any (Λ, sp)-open set of Y . Then, F ([F+(V )](Λ,sp)) ⊆ Λ(Λ,sp)(V ) and [F+(V )](Λ,sp) ⊆ F+(Λ(Λ,sp)(V )). By Lemma 4, [F+(V )](Λ,sp) ⊆ F+(Λ(Λ,sp)(V )) = F+(V ). Thus, [F+(V )](Λ,sp) = F+(V ) and hence F+(V ) is (Λ, sp)-closed in X, by Theorem 2, F is lower contra-(Λ, sp)-continuous. Corollary 7. Let f : (X, τ) → (Y, σ) be a function. If f(B(Λ,sp)) ⊆ Λ(Λ,sp)(f(B)) for every subset B of Y , then f is contra-(Λ, sp)-continuous. Definition 7. [3] A multifunction F : (X, τ) → (Y, σ) is said to be: (i) upper (Λ, sp)-continuous if, for each x ∈ X and each (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 ; (ii) lower (Λ, sp)-continuous if, for each x ∈ X and each (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 . Lemma 5. [3] For a multifunction F : (X, τ) → (Y, σ), the following properties are equivalent: (1) F is upper (Λ, sp)-continuous; (2) F+(V ) is (Λ, sp)-open in X for every (Λ, sp)-open set V of Y ; (3) F−(K) is (Λ, sp)-closed in X for every (Λ, sp)-closed set K of Y ; (4) [F−(B)](Λ,sp) ⊆ F−(B(Λ,sp)) for every subset B of Y ; (5) F+(B(Λ,sp)) ⊆ [F+(B)](Λ,sp) for every subset B of Y . Theorem 10. If F : (X, τ) → (Y, σ) is an upper (Λ, sp)-continuous multifunction and G : (Y, σ) → (Z, ρ) is an upper contra-(Λ, sp)-continuous multifunction, then G ◦ F : (X, τ) → (Z, ρ) is upper contra-(Λ, sp)-continuous. Proof. Let V be any (Λ, sp)-closed set of Z. From the definition of G ◦ F , we have (G ◦ F )+(V ) = F+(G+(V )). Since G is upper contra-(Λ, sp)-continuous, by Theorem 1, G+(V ) is (Λ, sp)-open in Y . Since F is upper (Λ, sp)-continuous, by Lemma 5, F+(G+(V )) is (Λ, sp)-open in X. Thus, (G ◦ F )+(V ) is (Λ, sp)-open in X and hence G ◦ F is upper contra-(Λ, sp)-continuous. C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 1694-1704 1702 Lemma 6. [3] For a multifunction F : (X, τ) → (Y, σ), the following properties are equivalent: (1) F is lower (Λ, sp)-continuous; (2) F−(V ) is (Λ, sp)-open in X for every (Λ, sp)-open set V of Y ; (3) F+(K) is (Λ, sp)-closed in X for every (Λ, sp)-closed set K of Y ; (4) [F+(B)](Λ,sp) ⊆ F+(B(Λ,sp)) for every subset B of Y ; (5) F (A(Λ,sp)) ⊆ [F (A)](Λ,sp) for every subset A of X; (6) F−(B(Λ,sp)) ⊆ [F−(B)](Λ,sp) for every subset B of Y . Theorem 11. If F : (X, τ) → (Y, σ) is a lower (Λ, sp)-continuous multifunction and G : (Y, σ) → (Z, ρ) is a lower contra-(Λ, sp)-continuous multifunction, then G ◦ F : (X, τ) → (Z, ρ) is lower contra-(Λ, sp)-continuous. Proof. Let V be any (Λ, sp)-closed set of Z. From the definition of G ◦ F , we have (G ◦ F )−(V ) = F−(G−(V )). Since G is lower contra-(Λ, sp)-continuous, by Theorem 2, G−(V ) is (Λ, sp)-open in Y . Since F is lower (Λ, sp)-continuous, by Lemma 6, F−(G−(V )) is (Λ, sp)-open in X. Thus, (G ◦ F )−(V ) is (Λ, sp)-open in X and hence G ◦ F is lower contra-(Λ, sp)-continuous. Definition 8. [22] A function f : (X, τ) → (Y, σ) is said to be (Λ, sp)-continuous at a point x ∈ X if, for each (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 . A function f : (X, τ) → (Y, σ) is said to be (Λ, sp)-continuous if f has this property at each point of X. Lemma 7. [22] For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is (Λ, sp)-continuous; (2) f−1(V ) is (Λ, sp)-open in X for every (Λ, sp)-open set V of Y ; (3) f(A(Λ,sp)) ⊆ [f(A)](Λ,sp) for every subset A of X; (4) [f−1(B)](Λ,sp) ⊆ f−1(B(Λ,sp)) for every subset B of Y ; (5) f−1(B(Λ,sp)) ⊆ [f−1(B)](Λ,sp) for every subset B of Y ; (6) f−1(F ) is (Λ, sp)-closed in X for every (Λ, sp)-closed set F of Y . REFERENCES 1703 Corollary 8. 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