EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1180-1188 ISSN 1307-5543 – ejpam.com Published by New York Business Global Slight (Λ, sp)-continuity and Λsp-extremally disconnectedness Chawalit Boonpok1, Jeeranunt Khampakdee1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Abstract. This paper is concerned with the concepts of upper and lower slightly (Λ, sp)-continuous multifunctions. Moreover, some characterizations of upper and lower slightly (Λ, sp)-continuous multifunctions are established. 2020 Mathematics Subject Classifications: 54C08, 54C60, 54G05 Key Words and Phrases: Upper slight (Λ, sp)-continuity, lower slight (Λ, sp)-continuity, Λsp- extremally disconnectedness 1. Introduction Stronger and weaker forms of open sets play an important role in the researches of generalizations of continuity for functions and multifunctions in topological spaces. The concept of slightly continuous functions was first introduced by Jain [6]. In 1995, Nour [11] defined slightly semi-continuous functions as a weak form of slight continuity and investigated several characterizations of slightly semi-continuous functions. Noiri and Chae [8] further investigated slight semi-continuity. Pal and Bhattacharya [12] defined a function to be faintly precontinuous if the preimage of each clopen set of the codomain is preopen and obtained some properties of such functions. Slight continuity implies both slight semi-continuity and faint precontinuity. In 2001, Noiri [7] introduced the concept of slight β-continuity which is implies by both slight semi-continuity and faint precontinuity. A unified theory of slight continuity is presented in [14], the present authors introduced and investigated the concept of slightly m-continuous functions. Noiri and Popa [10] introduced the notion of slightlym-continuous multifunctions and studied the relationships among m-continuity, almost m-continuity, weak m-continuity and slight m-continuity for multifunctions. The concept of β-open sets due to Abd El-Monsef et al. [5] or semi-preopen sets in the sense of Andrijević [1] plays a significant role in general topology. In 2004, Noiri and Hatir ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4369 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), jeeranunt.k@msu.ac.th (J. Khampakdee) https://www.ejpam.com 1180 © 2022 EJPAM All rights reserved. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (3) (2022), 1180-1188 1181 [9] introduced the notion 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. In particular, some characterizations of Λsp-extremally discon- nected spaces are investigated in [3]. The purpose of the present paper is to introduce the notions of upper and lower slightly (Λ, sp)-continuous multifunctions. Furthermore, some characterizations of upper and lower slightly (Λ, sp)-continuous multifunctions are discussed. 2. Preliminaries Throughout this paper, spaces (X, τ) and (Y, σ) (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. 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 [5] 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) [9] 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 [9] 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). (2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (3) (2022), 1180-1188 1182 (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 said to be s(Λ, sp)-open (resp. p(Λ, sp)- open, r(Λ, sp)-open, β(Λ, sp)-open) if A ⊆ [A(Λ,sp)] (Λ,sp) (resp. A ⊆ [A(Λ,sp)](Λ,sp), A = [A(Λ,sp)](Λ,sp), A ⊆ [[A(Λ,sp)](Λ,sp)] (Λ,sp)) [3]. The complement of a s(Λ, sp)-open (resp. p(Λ, sp)-open, r(Λ, sp)-open, β(Λ, sp)-open) set is called s(Λ, sp)-closed (resp. p(Λ, sp)- closed, r(Λ, sp)-closed, β(Λ, sp)-closed). The family of all s(Λ, sp)-open (resp. p(Λ, sp)-open, r(Λ, sp)-open, β(Λ, sp)-open) sets in a topological space (X, τ) is denoted by sΛspO(X, τ) (resp. pΛspO(X, τ), rΛspO(X, τ), βΛspO(X, τ)). A subset A of a topological space (X, τ) is called (Λ, sp)-clopen [4] if A is both (Λ, sp)-open and (Λ, sp)-closed. 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 [13]. 3. Upper and lower slightly (Λ, sp)-continuous multifunctions In this section, we introduce of the concepts of upper and lower slightly (Λ, sp)- continuous multifunctions. Moreover, some characterizations of upper and lower slightly (Λ, sp)-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ) → (Y, σ) is said to be: (i) upper slightly (Λ, sp)-continuous if, for each x ∈ X and each (Λ, sp)-clopen 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 slightly (Λ, sp)-continuous if, for each x ∈ X and each (Λ, sp)-clopen 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 1. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (3) (2022), 1180-1188 1183 (1) F is upper slightly (Λ, sp)-continuous; (2) F+(V ) is (Λ, sp)-open in X for every (Λ, sp)-clopen set V of Y ; (3) F−(V ) is (Λ, sp)-closed in X for every (Λ, sp)-clopen set V of Y . Proof. (1) ⇒ (2): Let V be any (Λ, sp)-clopen set of Y and let x ∈ F+(V ). Then, F (x) ⊆ V . Since F is upper slightly (Λ, sp)-continuous, there exists U ∈ ΛspO(X, τ) containing x such that F (U) ⊆ V . Thus, x ∈ U ⊆ F+(V ) and hence x ∈ [F+(V )](Λ,sp). Therefore, F+(V ) ⊆ [F+(V )](Λ,sp). This shows that F +(V ) is (Λ, sp)-open in X. (2) ⇒ (3) and (3) ⇒ (2): The proofs are obvious. (2) ⇒ (1): Let x ∈ X and let V be any (Λ, sp)-clopen set of Y containing F (x). Then, x ∈ F+(V ), by (2), 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 . Thus, F is upper slightly (Λ, sp)-continuous. Theorem 2. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv- alent: (1) F is lower slightly (Λ, sp)-continuous; (2) F−(V ) is (Λ, sp)-open in X for every (Λ, sp)-clopen set V of Y ; (3) F+(V ) is (Λ, sp)-closed in X for every (Λ, sp)-clopen 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 slightly (Λ, sp)-continuous if, for each x ∈ X and each (Λ, sp)-clopen set V of Y containing f(x), there exists a (Λ, sp)-open set U of X containing x such that f(U) ⊆ V . Corollary 1. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is slightly (Λ, sp)-continuous; (2) f−1(V ) is (Λ, sp)-open in X for every (Λ, sp)-clopen set V of Y ; (3) f−1(V ) is (Λ, sp)-closed in X for every (Λ, sp)-clopen set V of Y . Definition 3. Let A be a subset of a topological space (X, τ). The (Λ, sp)-frontier of A, denoted by (Λ, sp)-Fr(A), (Λ, sp)-Fr(A) = A(Λ,sp) ∩ [X −A](Λ,sp) = A(Λ,sp) −A(Λ,sp). Theorem 3. The set of all points x ∈ X at which a multifunction F : (X, τ) → (Y, σ) is not upper slightly (Λ, sp)-continuous is identical with the union of (Λ, sp)-frontiers of the upper inverse images of (Λ, sp)-clopen sets containing F (x). C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (3) (2022), 1180-1188 1184 Proof. Suppose that F is not upper slightly (Λ, sp)-continuous at x ∈ X. Then, there exists a (Λ, sp)-clopen 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, suppose that F is upper slightly (Λ, sp)-continuous at x ∈ X. Let V be any (Λ, sp)-clopen set of Y containing F (x). Then, there exists U ∈ ΛspO(X, τ) containing x such that U ⊆ F+(V ); hence x ∈ [F+(V )](Λ,sp). Thus, x ̸∈ (Λ, sp)-Fr(F+(V )) for every (Λ, sp)-clopen set V of Y containing F (x). Theorem 4. The set of all points x ∈ X at which a multifunction F : (X, τ) → (Y, σ) is not lower slightly (Λ, sp)-continuous is identical with the union of (Λ, sp)-frontiers of the lower inverse images of (Λ, sp)-clopen sets meeting F (x). Proof. The proof is similar to that of Theorem 3. Definition 4. [3] A topological space (X, τ) is called Λsp-extremally disconnected if V (Λ,sp) is (Λ, sp)-open in X for every (Λ, sp)-open set V of X. Theorem 5. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is upper slightly (Λ, sp)-continuous; (2) [F−(V )](Λ,sp) ⊆ F−(V (Λ,sp)) for every (Λ, sp)-open set V of Y ; (3) F+(K(Λ,sp)) ⊆ [F+(K)](Λ,sp) for every (Λ, sp)-closed set K of Y . Proof. (1) ⇒ (2): Let V be any (Λ, sp)-open set of Y . Since (Y, σ) is Λsp-extremally dis- connected, V (Λ,sp) is (Λ, sp)-open in Y . Thus, V (Λ,sp) is (Λ, sp)-clopen in Y . By Theorem 1, F−(V (Λ,sp)) is (Λ, sp)-closed and hence [F−(V )](Λ,sp) ⊆ [F−(V (Λ,sp))](Λ,sp) = F−(V (Λ,sp)). (2) ⇒ (3): Let K be any (Λ, sp)-closed set of Y . Then, Y −K is (Λ, sp)-open in Y , by (2), we have X − [F+(K)](Λ,sp) = [X − F+(K)](Λ,sp) = [F−(Y −K)](Λ,sp) ⊆ F−([Y −K](Λ,sp)) = F−(Y −K(Λ,sp)) = X − F+(K(Λ,sp)) and hence F+(K(Λ,sp)) ⊆ [F+(K)](Λ,sp). (3) ⇒ (1): Let x ∈ X and let V be any (Λ, sp)-clopen set of Y containing F (x). Thus, by (3), x ∈ F+(V ) = F+(V(Λ,sp)) ⊆ [F+(V )](Λ,sp). Then, there exists U ∈ ΛspO(X, τ) such that x ∈ U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper slightly (Λ, sp)-continuous. C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (3) (2022), 1180-1188 1185 Theorem 6. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is lower slightly (Λ, sp)-continuous; (2) [F+(V )](Λ,sp) ⊆ F+(V (Λ,sp)) for every (Λ, sp)-open set V of Y ; (3) F−(K(Λ,sp)) ⊆ [F−(K)](Λ,sp) for every (Λ, sp)-closed set K of Y . Proof. The proof is similar to that of Theorem 5. Lemma 3. [3] For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is Λsp-extremally disconnected. (2) The (Λ, sp)-closure of every s(Λ, sp)-open set of X is (Λ, sp)-open. (3) The (Λ, sp)-closure of every p(Λ, sp)-open set of X is (Λ, sp)-open. (4) The (Λ, sp)-closure of every r(Λ, sp)-open set of X is (Λ, sp)-open. Theorem 7. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is upper slightly (Λ, sp)-continuous; (2) [F−(V )](Λ,sp) ⊆ F−(V (Λ,sp)) for every s(Λ, sp)-open set V of Y ; (3) F+(K(Λ,sp)) ⊆ [F+(K)](Λ,sp) for every s(Λ, sp)-closed set K of Y . Proof. The proof is similar to that of Theorem 5 and it follows from Theorem 1 and Lemma 3. Theorem 8. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is lower slightly (Λ, sp)-continuous; (2) [F+(V )](Λ,sp) ⊆ F+(V (Λ,sp)) for every s(Λ, sp)-open set V of Y ; (3) F−(K(Λ,sp)) ⊆ [F−(K)](Λ,sp) for every s(Λ, sp)-closed set K of Y . Proof. The proof is similar to that of Theorem 6 and it follows from Theorem 2 and Lemma 3. Corollary 2. For a function f : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally discon- nected space, the following properties are equivalent: (1) f is slightly (Λ, sp)-continuous; C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (3) (2022), 1180-1188 1186 (2) [f−1(V )](Λ,sp) ⊆ f−1(V (Λ,sp)) for every s(Λ, sp)-open set V of Y ; (3) f−1(K(Λ,sp)) ⊆ [f−1(K)](Λ,sp) for every s(Λ, sp)-closed set K of Y . Theorem 9. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is upper slightly (Λ, sp)-continuous; (2) [F−(V )](Λ,sp) ⊆ F−(V (Λ,sp)) for every p(Λ, sp)-open set V of Y ; (3) F+(K(Λ,sp)) ⊆ [F+(K)](Λ,sp) for every p(Λ, sp)-closed set K of Y . Proof. The proof is similar to that of Theorem 5 and it follows from Theorem 1 and Lemma 3. Theorem 10. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is lower slightly (Λ, sp)-continuous; (2) [F+(V )](Λ,sp) ⊆ F+(V (Λ,sp)) for every p(Λ, sp)-open set V of Y ; (3) F−(K(Λ,sp)) ⊆ [F−(K)](Λ,sp) for every p(Λ, sp)-closed set K of Y . Proof. The proof is similar to that of Theorem 6 and it follows from Theorem 2 and Lemma 3. Corollary 3. For a function f : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally discon- nected space, the following properties are equivalent: (1) f is slightly (Λ, sp)-continuous; (2) [f−1(V )](Λ,sp) ⊆ f−1(V (Λ,sp)) for every p(Λ, sp)-open set V of Y ; (3) f−1(K(Λ,sp)) ⊆ [f−1(K)](Λ,sp) for every p(Λ, sp)-closed set K of Y . Theorem 11. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is upper slightly (Λ, sp)-continuous; (2) [F−(V )](Λ,sp) ⊆ F−(V (Λ,sp)) for every β(Λ, sp)-open set V of Y ; (3) F+(K(Λ,sp)) ⊆ [F+(K)](Λ,sp) for every β(Λ, sp)-closed set K of Y . Proof. The proof is similar to that of Theorem 5 and it follows from Theorem 1 and Lemma 3. REFERENCES 1187 Theorem 12. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally disconnected space, the following properties are equivalent: (1) F is lower slightly (Λ, sp)-continuous; (2) [F+(V )](Λ,sp) ⊆ F+(V (Λ,sp)) for every β(Λ, sp)-open set V of Y ; (3) F−(K(Λ,sp)) ⊆ [F−(K)](Λ,sp) for every β(Λ, sp)-closed set K of Y . Proof. The proof is similar to that of Theorem 6 and it follows from Theorem 2 and Lemma 3. Corollary 4. For a function f : (X, τ) → (Y, σ), where (Y, σ) is a Λsp-extremally discon- nected space, the following properties are equivalent: (1) f is slightly (Λ, sp)-continuous; (2) [f−1(V )](Λ,sp) ⊆ f−1(V (Λ,sp)) for every β(Λ, sp)-open set V of Y ; (3) f−1(K(Λ,sp)) ⊆ [f−1(K)](Λ,sp) for every β(Λ, sp)-closed set K of Y . 4. Conclusion 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. This paper deals with the concepts of upper and lower slight (Λ, sp)-continuity. In particular, some character- izations of upper and lower slightly (Λ, sp)-continuous multifunctions are obtained. The ideas and results of this paper may motivate further research. 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. 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