EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 29-43 ISSN 1307-5543 – ejpam.com Published by New York Business Global Weak forms of (Λ, b)-open sets and weak (Λ, b)-continuity Chawalit Boonpok1, Napassanan Srisarakham1,∗ 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 s(Λ, b)-open sets, p(Λ, b)-open sets, α(Λ, b)-open sets, β(Λ, b)-open sets and b(Λ, b)-open sets. Some properties of s(Λ, b)-open sets, p(Λ, b)-open sets, α(Λ, b)-open sets, β(Λ, b)-open sets and b(Λ, b)-open sets are investigated. More- over, several characterizations of weakly (Λ, b)-continuous functions are established. 2020 Mathematics Subject Classifications: 54A05, 54C08, 54G05 Key Words and Phrases: (Λ, b)-open set, (Λ, b)-submaximal space, (Λ, b)-extremally discon- nected space, weakly (Λ, b)-continuous function 1. Introduction The concepts of openness and continuity are fundamental with respect to the investi- gation of topological spaces. The notion of continuity is one of the most important tools in Mathematics and many different forms of generalizations of continuity have been in- troduced and investigated. The concept of weakly continuous functions was introduced by Levine [10]. Rose [13] introduced the notion of subweakly continuous functions and investigated the relationships between subweak continuity and weak continuity. In [4] has obtained further properties of subweakly continuous functions. In 1970, Willard [16] intro- duced the concept of extremally disconnected spaces. Extremally disconnected spaces play a prominent role in set-theoretical topology. Due to their peculiar properties extremally disconnected spaces provide crucial applications in the theory of Boolean algebra, in ax- iomatic set theory and in some branches of functional analysis. Sivaraj [14] investigated some characterizations of extremally disconnected spaces by utilizing semi-open sets due to Levine [11]. The concept of submaximality of generalized topological spaces was intro- duced by Hewitt [9]. He discovered a general way of constructing maximal topologies. The first systematic study of submaximal spaces was undertaken in the paper of Arhangel’skĭı and Collins [3]. They gave various necessary and suffient conditions for a space to be submaximal and showed that every submaximal space is left-separated. In 1996, An- drijević [2] introduced a new class of generalized open sets called b-open sets into the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4571 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), napassanan.sri@msu.ac.th (N. Srisarakham) https://www.ejpam.com 29 © 2023 EJPAM All rights reserved. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 30 field of topology. This class is a subset of the class of semi-preopen sets [1], i.e. a sub- set of a topological space which is contained in the closure of the interior of its closure. Also the class of b-open sets in a superset of the class of semi-open sets [11], i.e. a set which is contained in the closure of its interior and the class of locally dense set [8] or preopen sets [12], i.e. a set which is contained in the interior of its closure. Moreover, Andrijević studied several fundamental and interesting properties of b-open sets. Caldas et al. [7] introduced the concept of Λb-sets which is the intersection of b-open sets and investigated the fundamental properties of Λb-sets and Vb-sets. In [5], the present au- thors introduced and studied the notions of (Λ, b)-closed sets and (Λ, b)-open sets. The notions of (Λ, b)-extremally disconnected spaces, (Λ, b)-hyperconnected spaces and (Λ, b)- submaximal spaces were introduced by Viriyapong and Boonpok [15]. Recently, Boonpok and Viriyapong [6] introduced and studied the notion of weakly (Λ, p)-continuous func- tions. In this paper, we introduce new classes of sets called s(Λ, b)-open sets, p(Λ, b)-open sets, α(Λ, b)-open sets, β(Λ, b)-open sets and b(Λ, b)-open sets. We also investigate some of their fundamental properties. Furthermore, we investigate several characterizations of (Λ, b)-submaximal spaces and (Λ, b)-extremally disconnected spaces by utilizing the no- tion of (Λ, b)-open sets. In particular, some characterizations of weakly (Λ, b)-continuous functions are discussed. 2. Preliminaries Throughout this paper, the spaces (X, τ) and (X,σ) (or simply X and Y ) always means topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of X, the closure of A and the interior of A in (X, τ) are denoted by Cl(A) and Int(A), respectively. A subset A of a topological space (X, τ) is called b-open [2] if A ⊆ Cl(Int(A)) ∪ Int(Cl(A)). The complement of a b-open set is called b-closed. By BO(X, τ), we denote the collection of all b-open sets of a topological space (X, τ). Let A be a subset of a topological space (X, τ). A subset AΛb [7] is defined to be the set ∩{U ∈ BO(X, τ)|A ⊆ U}. A subset A of a topological space (X, τ) is called a Λb-set [7] if A = AΛb . A subset A of a topological space (X, τ) is called (Λ, b)-closed [5] if A = T ∩C, where T is a Λb-set and C is a b-closed set. The complement of a (Λ, b)-closed set is called (Λ, b)-open. The collection of all (Λ, b)-closed (resp. (Λ, b)-open) sets in a topological space (X, τ) is denoted by ΛbC(X, τ) (resp. ΛbO(X, τ)). Let A be a subset of a topological space (X, τ). A point x ∈ X is called a (Λ, b)-cluster point [5] of A if for every (Λ, b)-open set U of X containing x we have A∩U ̸= ∅. The set of all (Λ, b)-cluster points of A is called the (Λ, b)-closure [5] of A and is denoted by A(Λ,b). The union of all (Λ, b)-open sets contained in A is called the (Λ, b)-interior [5] of A and is denoted by A(Λ,b). Let A be a subset of a topological space (X, τ). A subset Λ(Λ,b)(A) [5] is defined as follows: Λ(Λ,b)(A) = ∩{U ∈ ΛbO(X, τ) | A ⊆ U}. Lemma 1. [5] Let A and B be subsets of a topological space (X, τ). For the (Λ, b)-closure, the following properties hold: (1) A ⊆ A(Λ,b) and [A(Λ,b)](Λ,p) = A(Λ,b). C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 31 (2) If A ⊆ B, then A(Λ,b) ⊆ B(Λ,b). (3) A(Λ,b) = ∩{F |A ⊆ F and F is (Λ, b)-closed}. (4) A(Λ,b) is (Λ, b)-closed. (5) A is (Λ, b)-closed if and only if A = A(Λ,b). Lemma 2. [5] Let A and B be subsets of a topological space (X, τ). For the (Λ, b)-interior, the following properties hold: (1) A(Λ,b) ⊆ A and [A(Λ,b)](Λ,b) = A(Λ,b). (2) If A ⊆ B, then A(Λ,b) ⊆ B(Λ,b). (3) A(Λ,b) is (Λ, b)-open. (4) A is (Λ, b)-open if and only if A(Λ,b) = A. Lemma 3. [5] For subsets A,B of a topological space (X, τ), the following properties hold: (1) A ⊆ Λ(Λ,b)(A). (2) If A ⊆ B, then Λ(Λ,b)(A) ⊆ Λ(Λ,b)(B). (3) Λ(Λ,b)[Λ(Λ,b)(A)] = Λ(Λ,b)(A). (4) If A is (Λ, b)-open, then Λ(Λ,b)(A) = A. 3. Some properties of generalized (Λ, b)-open sets In this section, we introduce new classes of sets called s(Λ, b)-open sets, p(Λ, b)-open sets, α(Λ, b)-open sets, β(Λ, b)-open sets and b(Λ, b)-open sets. We also investigate some of their fundamental properties. Definition 1. A subset A of a topological space (X, τ) is said to be: (i) s(Λ, b)-open if A ⊆ [A(Λ,b)] (Λ,b); (ii) p(Λ, b)-open if A ⊆ [A(Λ,b)](Λ,b); (iii) α(Λ, b)-open if A ⊆ [[A(Λ,b)] (Λ,b)](Λ,b); (iv) β(Λ, b)-open if A ⊆ [[A(Λ,b)](Λ,b)] (Λ,b). Example 1. Let X = {−1, 0, 1} with the topology τ = {∅, {−1}, {0, 1}, X}. Let A = {−1}. Then, A is s(Λ, b)-open and p(Λ, b)-open. On the other hand, let B = {0, 1}. Then, B is α(Λ, b)-open and β(Λ, b)-open. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 32 The family of all s(Λ, b)-open (resp. p(Λ, b)-open, α(Λ, b)-open, β(Λ, b)-open) sets in a topological space (X, τ) is denoted by sΛbO(X, τ) (resp. pΛbO(X, τ), αΛbO(X, τ), βΛbO(X, τ)). The complement of a s(Λ, b)-open (resp. p(Λ, b)-open, α(Λ, b)-open, β(Λ, b)- open) set is said to be s(Λ, b)-closed (resp. p(Λ, b)-closed, α(Λ, b)-closed, β(Λ, b)-closed). The family of all s(Λ, b)-closed (resp. p(Λ, b)-closed, α(Λ, b)-closed, β(Λ, b)-closed) sets in a topological space (X, τ) is denoted by sΛbC(X, τ) (resp. pΛbC(X, τ), αΛbC(X, τ), βΛbO(X, τ)). Proposition 1. For a topological space (X, τ), the following properties hold: (1) ΛbO(X, τ) ⊆ αΛbO(X, τ) ⊆ sΛbO(X, τ) ⊆ βΛbO(X, τ). (2) αΛbO(X, τ) ⊆ pΛbO(X, τ) ⊆ βΛbO(X, τ). (3) αΛbO(X, τ) = sΛbO(X, τ) ∩ pΛbO(X, τ). A subset A of a topological space (X, τ) is called r(Λ, b)-open [15] if A = [A(Λ,b)](Λ,b). The complement of a r(Λ, b)-open set is called r(Λ, b)-closed. The family of all r(Λ, b)-open (resp. r(Λ, b)-closed) sets in a topological space (X, τ) is denoted by rΛbO(X, τ) (resp. rΛbC(X, τ)). Proposition 2. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is r(Λ, b)-open. (2) A is (Λ, b)-open and s(Λ, b)-closed. (3) A is α(Λ, b)-open and s(Λ, b)-closed. (4) A is p(Λ, b)-open and s(Λ, b)-closed. (5) A is (Λ, b)-open and β(Λ, b)-closed. (6) A is α(Λ, b)-open and β(Λ, b)-closed. Proof. (1) ⇒ (2) ⇒ (3) ⇒ (4): Obvious. (4) ⇒ (5): Let A be (Λ, b)-open and s(Λ, b)-closed. Then, A ⊆ [A(Λ,b)](Λ,b) and [A(Λ,b)](Λ,b) ⊆ A. This implies that A = [A(Λ,b)](Λ,b). Thus, A is r(Λ, b)-open and hence A is (Λ, b)-open. Since every s(Λ, b)-closed set is β(Λ, b)-closed. This shows that A is (Λ, b)-open and β(Λ, b)-closed. (5) ⇒ (6): Obvious. (6) ⇒ (1): Let A be α(Λ, b)-open and β(Λ, b)-closed. Then, A ⊆ [[A(Λ,b)] (Λ,b)](Λ,b) and [[A(Λ,b)] (Λ,b)](Λ,b) ⊆ A. This shows that A = [[A(Λ,b)] (Λ,b)](Λ,b). Thus, A(Λ,b) = [[A(Λ,b)] (Λ,b)](Λ,b) = A and hence [A(Λ,b)](Λ,b) = [[A(Λ,b)] (Λ,b)](Λ,b) = A. Therefore, A is r(Λ, b)-open. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 33 Corollary 1. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A is r(Λ, b)-closed. (2) A is (Λ, b)-closed and s(Λ, b)-open. (3) A is α(Λ, b)-closed and s(Λ, b)-open. (4) A is p(Λ, b)-closed and s(Λ, b)-open. (5) A is (Λ, b)-closed and β(Λ, b)-open. (6) A is α(Λ, b)-closed and β(Λ, b)-open. Definition 2. A subset A of a topological space (X, τ) is said to be b(Λ, b)-open if A ⊆ [A(Λ,b)] (Λ,b)∪ [A(Λ,b)](Λ,b). The complement of a b(Λ, b)-open set is said to be b(Λ, b)-closed. Example 2. Let X = {−2,−1, 0, 1, 2} with the topology τ = {∅, {−2,−1}, {0, 1, 2}, X}. Let C = {−2,−1}. Then, C is b(Λ, b)-open. The family of all b(Λ, b)-open (resp. b(Λ, b)-closed) sets in a topological space (X, τ) is denoted by bΛbO(X, τ) (resp. bΛbC(X, τ)). Proposition 3. Let A be a subset of a topological space (X, τ). If A is b(Λ, b)-open, then A(Λ,b) is r(Λ, b)-closed. Proof. Suppose that A is b(Λ, b)-open. Then, A ⊆ [A(Λ,b)] (Λ,b) ∪ [A(Λ,b)](Λ,b). Thus, A(Λ,b) ⊆ [[A(Λ,b)] (Λ,b) ∪ [A(Λ,b)](Λ,b)] (Λ,b) ⊆ [[A(Λ,b)] (Λ,b)](Λ,b) ∪ [[A(Λ,b)](Λ,b)] (Λ,b) = [[A(Λ,b)](Λ,b)] (Λ,b) ⊆ A(Λ,b) and hence A(Λ,b) = [[A(Λ,b)](Λ,b)] (Λ,b). Therefore, A(Λ,b) is r(Λ, b)-closed. Corollary 2. For a subset A of a topological space (X, τ), the following properties hold: (1) If A is s(Λ, b)-open, then A(Λ,b) is r(Λ, b)-closed. (2) If A is p(Λ, b)-open, then A(Λ,b) is r(Λ, b)-closed. (3) If A is α(Λ, b)-open, then A(Λ,b) is r(Λ, b)-closed. Proposition 4. For a subset A of a topological space (X, τ), the following properties are equivalent: (1) A ∈ βΛbO(X, τ). (2) A(Λ,b) ∈ rΛbC(X, τ). (3) A(Λ,b) ∈ βΛbO(X, τ). C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 34 (4) A(Λ,b) ∈ sΛbO(X, τ). (5) A(Λ,b) ∈ bΛbO(X, τ). Proof. (1) ⇒ (2): Let A ∈ βΛbO(X, τ). Then, A ⊆ [[A(Λ,b)](Λ,b)] (Λ,b). Thus, A(Λ,b) ⊆ [[A(Λ,b)](Λ,b)] (Λ,b) ⊆ A(Λ,b) and hence A(Λ,b) = [[A(Λ,b)](Λ,b)] (Λ,b). Therefore, A(Λ,b) ∈ rΛbC(X, τ). (2) ⇒ (3) ⇒ (4) ⇒ (5): Obvious. (5) ⇒ (1): LetA(Λ,b) ∈ bΛbO(X, τ). Then, A(Λ,b) ⊆ [[A(Λ,b)](Λ,b)](Λ,b)∪[[A(Λ,b)](Λ,b)] (Λ,b) = [A(Λ,b)](Λ,b) ∪ [[A(Λ,b)](Λ,b)] (Λ,b) = [[A(Λ,b)](Λ,b)] (Λ,b). Thus, A ⊆ [[A(Λ,b)](Λ,b)] (Λ,b) and hence A ∈ βΛbO(X, τ). Definition 3. Let (X, τ) be a topological space and x ∈ X. A subset ⟨x⟩b is defined as follows: ⟨x⟩b = Λ(Λ,b)({x}) ∩ {x}(Λ,b). Example 3. Let X = {−1, 1} with the topology τ = {∅, {−1}, X}. Then, ⟨1⟩b = {1}. Recall that a subset N of a topological space (X, τ) is said to be a (Λ, b)-neighbourhood [5] of a point x ∈ X if there exists a (Λ, b)-open set U of X such that x ∈ U ⊆ N . Theorem 1. Let (X, τ) be a topological space. Then, the following properties hold: (1) Λ(Λ,b)(A) = {x ∈ X | A ∩ {x}(Λ,b) ̸= ∅} for each subset A of X. (2) For each x ∈ X, Λ(Λ,b)(⟨x⟩b) = Λ(Λ,b)({x}). (3) For each x ∈ X, (⟨x⟩b)(Λ,b) = {x}(Λ,b). (4) If U is (Λ, b)-open in (X, τ) and x ∈ U , then ⟨x⟩b ⊆ U . (5) If F is (Λ, b)-closed in (X, τ) and x ∈ F , then ⟨x⟩b ⊆ F . Proof. (1) Suppose that A ∩ {x}(Λ,b) = ∅. Then, we have x ̸∈ X − {x}(Λ,b) which is a (Λ, b)-open set containing A. Therefore, x ̸∈ Λ(Λ,b)(A). Consequently, we obtain Λ(Λ,b)(A) ⊆ {x ∈ X | A ∩ {x}(Λ,b) ̸= ∅}. Next, let x ∈ X such that A ∩ {x}(Λ,b) ̸= ∅ and suppose that x ̸∈ Λ(Λ,b)(A). Then, there exists a (Λ, b)-open set U containing A and x ̸∈ U . Let y ∈ A ∩ {x}(Λ,b). Thus, U is a (Λ, b)-neighbourhood of y which does not contain x. By this contradiction x ∈ Λ(Λ,b)(A). (2) Let x ∈ X, Then, we have {x} ⊆ {x}(Λ,b) ∩ Λ(Λ,b)({x}) = ⟨x⟩b. By Lemma 3, Λ(Λ,b)({x}) ⊆ Λ(Λ,b)(⟨x⟩b). Next, we show the opposite implication. Suppose that y ̸∈ Λ(Λ,b)({x}). There exists a (Λ, b)-open set V such that x ∈ V and y ̸∈ V . Since ⟨x⟩b ⊆ Λ(Λ,b)({x}) ⊆ Λ(Λ,b)(V ) = V , we have Λ(Λ,b)(⟨x⟩b) ⊆ V . Since y ̸∈ V, y ̸∈ Λ(Λ,b)(⟨x⟩b). Thus, Λ(Λ,b)(⟨x⟩b) ⊆ Λ(Λ,b)({x}) and hence Λ(Λ,b)({x}) = Λ(Λ,b)(⟨x⟩b). (3) By the definition of ⟨x⟩b, we have {x} ⊆ ⟨x⟩b and {x}(Λ,b) ⊆ (⟨x⟩b)(Λ,b) by Lemma 1. On the other hand, we have ⟨x⟩b ⊆ {x}(Λ,b) and (⟨x⟩b)(Λ,b) ⊆ ({x}(Λ,b))(Λ,b) = {x}(Λ,b). Thus, (⟨x⟩b)(Λ,b) ⊆ {x}(Λ,b). C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 35 (4) Since x ∈ U and U is (Λ, b)-open, we have Λ(Λ,b)({x}) ⊆ U . Thus, ⟨x⟩b ⊆ U . (5) Since x ∈ F and F is (Λ, b)-closed, ⟨x⟩b = {x}(Λ,b) ∩ Λ(Λ,b)({x}) ⊆ {x}(Λ,b) ⊆ F (Λ,b) = F . Definition 4. [15] A subset A of a topological space (X, τ) is said to be: (i) (Λ, b)-dense if A(Λ,b) = X; (ii) (Λ, b)-codense if its complement is (Λ, b)-dense. Recall that a topological space (X, τ) is called (Λ, b)-submaximal [15] if each (Λ, b)- dense subset of X is (Λ, b)-open. Proposition 5. Let (X, τ) be a topological space. If each p(Λ, b)-open set is s(Λ, b)-open and each α(Λ, b)-open set is (Λ, b)-open, then (X, τ) is (Λ, b)-submaximal. Proof. Let D be (Λ, b)-dense subset of X. Since D(Λ,b) = X, we have D is p(Λ, b)- open. Thus, D is s(Λ, b)-open and hence D is α(Λ, b)-open. Since each α(Λ, b)-open set is (Λ, b)-open, D is (Λ, b)-open. This shows that (X, τ) is (Λ, b)-submaximal. Proposition 6. Let (X, τ) be a topological space. If each p(Λ, b)-open set is (Λ, b)-open, then (X, τ) is (Λ, b)-submaximal. Proof. Suppose that each p(Λ, b)-open set is (Λ, b)-open. It follows that every p(Λ, b)- open set is s(Λ, b)-open. Since each α(Λ, b)-open set is p(Λ, b)-open, then each α(Λ, b)-open set is (Λ, b)-open. Thus, by Proposition 5, (X, τ) is (Λ, b)-submaximal. Proposition 7. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is (Λ, b)-submaximal. (2) Each (Λ, b)-codense subset of X is (Λ, b)-closed. Recall that a topological space (X, τ) is called (Λ, b)-extremally disconnected [15] if U (Λ,b) is (Λ, b)-open in X for every (Λ, b)-open set U of X. Lemma 4. Let A be a subset of a topological space (X, τ). If U ∈ ΛbO(X, τ) and U∩A = ∅, then U ∩A(Λ,b) = ∅. Theorem 2. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is (Λ, b)-extremally disconnected. (2) U (Λ,b) ∩ V (Λ,b) = ∅ for all (Λ, b)-open subsets U and V of X with U ∩ V = ∅. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 36 Proof. (1) ⇒ (2): Suppose that U and V are (Λ, b)-open sets of X such that U∩V = ∅. Thus, by Lemma 4, U (Λ,b) ∩ V = ∅ which implies that [U (Λ,b)](Λ,b) ∩ V (Λ,b) = ∅ and hence U (Λ,b) ∩ V (Λ,b) = ∅. (2) ⇒ (1): Let U be any (Λ, b)-open set of X. Then, X − U is (Λ, b)-closed and [X−U ](Λ,b) is (Λ, b)-open such that U∩[X−U ](Λ,b) = ∅. Thus, U (Λ,b)∩[[X−U ](Λ,b)] (Λ,b) = ∅ which implies that U (Λ,b) ∩ [X − [U (Λ,b)](Λ,b) = ∅. Therefore, U (Λ,b) ⊆ [U (Λ,b)](Λ,b) and hence U (Λ,b) = [U (Λ,b)](Λ,b). This shows that U (Λ,b) is (Λ, b)-open. Thus, (X, τ) is (Λ, b)- extremally disconnected. Lemma 5. Let A be a subset of a topological space (X, τ). If U ∈ ΛbO(X, τ), then U (Λ,b) ∩A ⊆ [U ∩A](Λ,b). Lemma 6. For a subset A of a topological space (X, τ), the following properties hold: (1) [X −A](Λ,b) = X −A(Λ,b). (2) [X −A](Λ,b) = X −A(Λ,b). Theorem 3. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is (Λ, b)-extremally disconnected. (2) U (Λ,b) ∩ V (Λ,b) = [U ∩ V ](Λ,b) for all (Λ, b)-open sets U and V of X. (3) E(Λ,b) ∪ F(Λ,b) = [E ∪ F ](Λ,b) for all (Λ, b)-closed sets E and F of X. Proof. (1) ⇒ (2): Let U and V be (Λ, b)-open sets of X. Thus, by Lemma 5, U (Λ,b) ∩ V (Λ,b) ⊆ [U ∩ V (Λ,b)](Λ,b) ⊆ [[U ∩ V ](Λ,b)](Λ,b) = [U ∩ V ](Λ,b) and hence U (Λ,b) ∩ V (Λ,b) = [U ∩ V ](Λ,b). (2) ⇒ (3): Let E and F be (Λ, b)-closed sets of X. Then, X − E and X − F are (Λ, b)-open. By (2) and Lemma 6, we have E(Λ,b) ∪ F(Λ,b) = X − [X − [E(Λ,b) ∪ F(Λ,b)]] = X − [[X − E(Λ,b)] ∩ [X − F(Λ,b)]] = X − [X − [E ∪ F ](Λ,b)] = [E ∪ F ](Λ,b). (3) ⇒ (2): The proof is obvious. (2) ⇒ (1): Let U be any (Λ, b)-open set of X. Then, X − U is (Λ, b)-closed and [X − U ](Λ,b) is (Λ, b)-open. By (2), U (Λ,b) ∩ [[X − U ](Λ,b)] (Λ,b) = [U ∩ [X − U ](Λ,b)] (Λ,b) which implies that [X − [U (Λ,b)](Λ,b)] ∩ U (Λ,b) = ∅(Λ,b) = ∅. Thus, U (Λ,b) ⊆ [U (Λ,b)](Λ,b) and hence U (Λ,b) = [U (Λ,b)](Λ,b). Therefore, U (Λ,b) is (Λ, b)-open. This shows that (X, τ) is (Λ, b)-extremally disconnected. Theorem 4. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is (Λ, b)-extremally disconnected. (2) U (Λ,b) ∩ V (Λ,b) = [U ∩ V ](Λ,b) for all (Λ, b)-open sets U and V of X. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 37 (3) U (Λ,b) ∩ V (Λ,b) = ∅ for all (Λ, b)-open sets U and V of X with U ∩ V = ∅. Proof. This follows from Theorem 2 and Theorem 3. Theorem 5. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is (Λ, b)-extremally disconnected. (2) The (Λ, b)-closure of every β(Λ, b)-open set of X is (Λ, b)-open. (3) The (Λ, b)-closure of every p(Λ, b)-open set of X is (Λ, b)-open. Proof. This follows immediately since A(Λ,b) = [[A(Λ,b)](Λ,b)] (Λ,b). Theorem 6. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is (Λ, b)-extremally disconnected. (2) If U ∈ βΛbO(X, τ), V ∈ sΛbO(X, τ) and U ∩ V = ∅, then U (Λ,b) ∩ V (Λ,b) = ∅. (3) If U ∈ bΛbO(X, τ), V ∈ sΛbO(X, τ) and U ∩ V = ∅, then U (Λ,b) ∩ V (Λ,b) = ∅. (4) If U ∈ pΛbO(X, τ), V ∈ sΛbO(X, τ) and U ∩ V = ∅, then U (Λ,b) ∩ V (Λ,b) = ∅. (5) If U ∈ rΛbO(X, τ), V ∈ sΛbO(X, τ) and U ∩ V = ∅, then U (Λ,b) ∩ V (Λ,α) = ∅. (6) If U, V ∈ rΛbO(X, τ) and U ∩ V = ∅, then U (Λ,b) ∩ V (Λ,b) = ∅. Proof. (1) ⇒ (2): Suppose that U ∈ βΛbO(X, τ), V ∈ sΛbO(X, τ) and U ∩ V = ∅. Thus, U (Λ,b) ∩ V(Λ,b) = ∅, by Theorem 5, U (Λ,b) is (Λ, b)-open and hence U (Λ,b) ∩ V (Λ,b) = U (Λ,p) ∩ [V(Λ,b)] (Λ,b) = ∅. (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6): Obvious. (6) ⇒ (1): Let U and V be (Λ, b)-open sets of X. Then, U (Λ,b), V (Λ,b) ∈ rΛbO(X, τ) and U (Λ,b) ∩ V (Λ,b) = ∅. By (6), [U (Λ,b)](Λ,b) ∩ [V (Λ,b)](Λ,b) = U (Λ,b) ∩ V (Λ,b) = ∅. Thus, (X, τ) is (Λ, b)-extremally disconnected. 4. Characterizations of weakly (Λ, b)-continuous functions In this section, we introduce the notion of weakly (Λ, b)-continuous functions. More- over, several characterizations of weakly (Λ, b)-continuous functions are discussed. Definition 5. A function f : (X, τ) → (Y, σ) is called weakly (Λ, b)-continuous at a point x ∈ X if, for each (Λ, b)-open set V of Y containing f(x), there exists a (Λ, b)-open set U of X containing x such that f(U) ⊆ V (Λ,b). A function f : (X, τ) → (Y, σ) is called (Λ, b)-continuous if f has this property at each point x ∈ X. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 38 Example 4. Let X = {a, b, c} with the topology τ = {∅, {a}, {b, c}, X}. Let Y = {−1, 0, 1} with the topology σ = {∅, {−1}, {0, 1}, Y }. Let f : (X, τ) → (Y, σ) be a function defined as follows: f(a) = −1, f(b) = 0 and f(c) = 1. Then, f is weakly (Λ, b)-continuous. Theorem 7. A function f : (X, τ) → (Y, σ) is weakly (Λ, b)-continuous at x ∈ X if and only if for each (Λ, b)-open set V of Y containing f(x), x ∈ [f−1(V (Λ,b))](Λ,b). Proof. Let V be any (Λ, b)-open set of Y containing f(x). Then, there exists a (Λ, b)- open set U of X containing x such that f(U) ⊆ V (Λ,b). Thus, x ∈ U ⊆ f−1(V (Λ,b)) and hence x ∈ [f−1(V (Λ,b))](Λ,b). Conversely, let V be any (Λ, b)-open set of Y containing f(x). Then, x ∈ [f−1(V (Λ,b))](Λ,b) and there exists a (Λ, b)-open set U of X such that x ∈ U ⊆ f−1(V (Λ,b)); hence f(U) ⊆ V (Λ,b). Thus, f is weakly (Λ, b)-continuous at x ∈ X. Theorem 8. A function f : (X, τ) → (Y, σ) is weakly (Λ, b)-continuous if and only if f−1(V ) ⊆ [f−1(V (Λ,b))](Λ,b) for every (Λ, b)-open set V of Y . Proof. Let V be any (Λ, b)-open set of Y and x ∈ f−1(V ). Then, f(x) ∈ V . Since f is weakly (Λ, b)-continuous at x and by Theorem 7, x ∈ [f−1(V (Λ,b))](Λ,b). Therefore, f−1(V ) ⊆ [f−1(V (Λ,b))](Λ,b). Conversely, let x ∈ X and V be any (Λ, b)-open set of Y containing f(x). Then, x ∈ f−1(V ) ⊆ [f−1(V (Λ,b))](Λ,b) and hence x ∈ [f−1(V (Λ,b))](Λ,b). By Theorem 7, f is weakly (Λ, b)-continuous. Theorem 9. A function f : (X, τ) → (Y, σ) is weakly (Λ, b)-continuous if and only if [f−1(V )](Λ,b) ⊆ f−1(V (Λ,b)) for every (Λ, b)-open set V of Y . Proof. This is obvious from Theorem 8. Conversely, let V be any (Λ, b)-open set of Y containing f(x). Since V ∩(Y−V (Λ,b)) = ∅, f(x) ̸∈ [Y − V (Λ,b)](Λ,b). Since Y − V (Λ,b) is (Λ, b)-open and x ∈ f−1([Y − V (Λ,b)](Λ,b)). By the hypothesis, x ̸∈ [f−1([Y − V (Λ,b)](Λ,b))](Λ,b) = [X − f−1(V (Λ,b))](Λ,b) and there exists a (Λ, b)-open set U of X containing x such that U ∩ (X− f−1(V (Λ,b))) = ∅. This shows that f(U) ⊆ V (Λ,b). Thus, f is is weakly (Λ, b)-continuous. Theorem 10. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, b)-continuous; (2) f−1(U) ⊆ [f−1(U (Λ,b))](Λ,b) for every (Λ, b)-open set U of Y ; (3) [f−1(F(Λ,b))] (Λ,b) ⊆ f−1(F ) for every (Λ, b)-closed set F of Y ; (4) [f−1([A(Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(A(Λ,b)) for every subset A of Y ; (5) f−1(A(Λ,b)) ⊆ [f−1([A(Λ,b)] (Λ,b))](Λ,b) for every subset A of Y ; C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 39 (6) [f−1(U)](Λ,b) ⊆ f−1(U (Λ,b)) for every (Λ, b)-open set U of Y . Proof. (1) ⇒ (2): Let U be any (Λ, b)-open set of Y and x ∈ f−1(U). Then, there exists a (Λ, b)-open set V of X containing x such that f(V ) ⊆ U (Λ,b). Since x ∈ V ⊆ f−1(U (Λ,b)), x ∈ [f−1(U (Λ,b))](Λ,b). Thus, f −1(U) ⊆ [f−1(U (Λ,b))](Λ,b). (2) ⇒ (3): Let F be any (Λ, b)-closed set of Y . Then, Y −F is (Λ, b)-open and by (2), f−1(X − F ) ⊆ [f−1([X − F ](Λ,b))](Λ,b) = [f−1(Y − F(Λ,b))](Λ,b) = X − [f−1(F(Λ,b))] (Λ,b). Therefore, [f−1(F(Λ,b))] (Λ,b) ⊆ f−1(F ). (3) ⇒ (4): Let A be any subset of Y . Since A(Λ,b) is (Λ, b)-closed in Y and by (3), [f−1([A(Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(A(Λ,b)). (4) ⇒ (5): Let A be any subset of Y . By (4), f−1(A(Λ,b)) = X − f−1([Y − A](Λ,b)) ⊆ X − [f−1([[Y −A](Λ,b)](Λ,b))] (Λ,b) = [f−1([A(Λ,b)] (Λ,b))](Λ,b). (5) ⇒ (6): Let U be any (Λ, b)-open set of Y . Suppose that x ̸∈ f−1(U (Λ,b)). Then, f(x) ̸∈ U (Λ,b) and so there exists a (Λ, b)-open set V of Y containing f(x) such that U ∩ V = ∅ and hence U ∩ V (Λ,b) = ∅. By (5), we have x ∈ f−1(V ) ⊆ [f−1(V (Λ,b))](Λ,b). There exists a (Λ, b)-open set W containing x such that x ∈ W ⊆ f−1(V (Λ,b)). Since U ∩ V (Λ,b) = ∅ and f(W ) ⊆ V (Λ,b), we have W ∩ f−1(U) = ∅. Thus, x ̸∈ [f−1(U)](Λ,b) and hence [f−1(U)](Λ,b) ⊆ f−1(U (Λ,b)). (6) ⇒ (1): Let x ∈ X and U be any (Λ, b)-open set of Y containing f(x). Then, we have U = U(Λ,b) ⊆ [U (Λ,b)](Λ,b). Thus, by (6), x ∈ f−1(U) ⊆ f−1([U (Λ,b)](Λ,b)) = X − f−1([Y − U (Λ,b)](Λ,b)) ⊆ X − [f−1(Y − U (Λ,b))](Λ,b) = [f−1(U (Λ,b))](Λ,b). Then, there exists a (Λ, b)-open set V of X containing x such that V ⊆ f−1(U (Λ,b)). This shows that f is weakly (Λ, b)-continuous. Theorem 11. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, b)-continuous; (2) [f−1(F(Λ,b))] (Λ,b) ⊆ f−1(F ) for every r(Λ, b)-closed set F of Y ; (3) [f−1([U (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(U (Λ,b)) for every β(Λ, b)-open set U of Y ; (4) [f−1([U (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(U (Λ,b)) for every s(Λ, b)-open set U of Y . Proof. (1) ⇒ (2): Let F be any r(Λ, b)-closed set of Y . Then, F(Λ,b) is (Λ, b)-open and by Theorem 10, [f−1(F(Λ,b))] (Λ,b) ⊆ f−1([F(Λ,b)] (Λ,b)). Since F is r(Λ, b)-closed, we have [f−1(F(Λ,b))] (Λ,b) ⊆ f−1([F(Λ,b)] (Λ,b)) ⊆ f−1(F ). (2) ⇒ (3): Let U be any β(Λ, b)-open set of Y . Then, U (Λ,b) ⊆ [[U (Λ,b)](Λ,b)] (Λ,b) ⊆ U (Λ,b) and hence U (Λ,b) is r(Λ, b)-closed. By (2), [f−1([U (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(U (Λ,b)). (3) ⇒ (4): The proof is obvious. (4) ⇒ (1): Let U be any (Λ, b)-open set of Y . Thus, by (4), [f−1(U)](Λ,b) ⊆ [f−1([U (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(U (Λ,b)) and by Theorem 10, f is weakly (Λ, b)-continuous. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 40 Theorem 12. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, b)-continuous; (2) [f−1([U(Λ,b)] (Λ,b))](Λ,b) ⊆ f−1(U (Λ,b)) for every p(Λ, b)-open set U of Y ; (3) [f−1(U)](Λ,b) ⊆ f−1(U (Λ,b)) for every p(Λ, b)-open set U of Y ; (4) f−1(U) ⊆ [f−1(U (Λ,b))](Λ,b) for every p(Λ, b)-open set U of Y . Proof. (1) ⇒ (2): Let U be any p(Λ, b)-open set of Y . Then, U (Λ,b) = [[U (Λ,b)](Λ,b)] (Λ,b) and hence U (Λ,b) is r(Λ, b)-closed. By Theorem 11, [f−1([U (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(U (Λ,b)). (2) ⇒ (3): Let U be any p(Λ, b)-open set of Y . Then, U ⊆ [U (Λ,b)](Λ,b) and by (2), we have [f−1(U)](Λ,b) ⊆ [f−1([U (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(U (Λ,b)). (3) ⇒ (4): Let U be any p(Λ, b)-open set of Y . By (3), f−1(U) ⊆ f−1([U (Λ,b)](Λ,b)) = X − f−1([Y − U (Λ,b)](Λ,b)) = X − [f−1(Y − U (Λ,b))](Λ,b) = [f−1(U (Λ,b))](Λ,b). (4) ⇒ (1): Since every (Λ, b)-open set is p(Λ, b)-open, by (4) and Theorem 10, f is weakly (Λ, b)-continuous. Theorem 13. For a function f : (X, τ) → (Y, σ), the following properties are equivalent: (1) f is weakly (Λ, b)-continuous; (2) [f−1([A(Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(A(Λ,b)) for every subset A of Y ; (3) [f−1(F(Λ,b))] (Λ,b) ⊆ f−1(F ) for every r(Λ, b)-closed set F of Y ; (4) [f−1(U)](Λ,b) ⊆ f−1(U (Λ,b)) for every (Λ, b)-open set U of Y ; (5) f−1(U) ⊆ [f−1(U (Λ,b))](Λ,b) for every (Λ, b)-open set U of Y ; (6) [f−1(U)](Λ,b) ⊆ f−1(U (Λ,b)) for every p(Λ, b)-open set U of Y ; (7) f−1(U) ⊆ [f−1(U (Λ,b))](Λ,b) for every p(Λ, b)-open set U of Y . Proof. (1) ⇒ (2): Let A be any subset of Y and x ∈ X − f−1(A(Λ,b)). Then, f(x) ∈ Y − A(Λ,b) and there exists a (Λ, b)-open set U of Y containing f(x) such that U ∩A = ∅. Thus, U (Λ,b)∩ [A(Λ,b)](Λ,b) = ∅. Since f is weakly (Λ, b)-continuous, there exists a (Λ, b)-open setW containing x such that f(W ) ⊆ U (Λ,b). Then, W∩f−1([A(Λ,b)](Λ,b)) = ∅ and hence x ∈ X−[f−1([A(Λ,b)](Λ,b))] (Λ,b). Therefore, [f−1([A(Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(A(Λ,b)). (2) ⇒ (3): Let F be any r(Λ, b)-closed set of Y . By (2), we have [f−1(F(Λ,b))] (Λ,b) = [f−1([[F(Λ,b)] (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1([F(Λ,b)] (Λ,b)) = f−1(F ). (3) ⇒ (4): Let U be any (Λ, b)-open set of Y . Since U (Λ,b) is r(Λ, b)-closed, we have [f−1(U)](Λ,b) ⊆ [f−1([U (Λ,b)](Λ,b))] (Λ,b) ⊆ f−1(U (Λ,b)). C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 41 (4) ⇒ (5): Let U be any (Λ, b)-open set of Y . Since Y − U (Λ,b) is (Λ, b)-open and by (4), X − [f−1(U (Λ,b))](Λ,b) = [f−1(Y −U (Λ,b))](Λ,b) ⊆ f−1([Y −U (Λ,b)](Λ,b)) ⊆ X − f−1(U). Thus, f−1(U) ⊆ [f−1(U (Λ,b))](Λ,b). (5) ⇒ (1): Let x ∈ X and U be any (Λ, b)-open set of Y containing f(x). Then, x ∈ f−1(U) ⊆ [f−1(U (Λ,b))](Λ,b). Put W = [f−1(U (Λ,b))](Λ,b). Then, W is a (Λ, b)-open set of X containing x such that f(W ) ⊆ U (Λ,b). Thus, f is weakly (Λ, b)-continuous at x. This shows that f is weakly (Λ, b)-continuous. (1) ⇒ (6): Let U be any p(Λ, b)-open set of Y and x ∈ X−f−1(U (Λ,b)). There exists a (Λ, b)-open set V of Y containing f(x) such that V ∩U = ∅. Thus, [V ∩U ](Λ,b) = ∅. Since U is (Λ, b)-open, U ∩ V (Λ,b) ⊆ [U ∩ V ](Λ,b) = ∅. Since f is weakly (Λ, b)-continuous and V is a (Λ, b)-open set of Y containing f(x), there exists a (Λ, b)-open set W of X containing x such that f(W ) ⊆ V (Λ,b). Therefore, f(W ) ∩ U = ∅. Thus, W ∩ f−1(U) = ∅ and hence x ∈ X − [f−1(U)](Λ,b). This shows that [f−1(U)](Λ,b) ⊆ f−1(U (Λ,b)). (6) ⇒ (7): Let U be any p(Λ, b)-open set of Y . Since Y −U (Λ,b) is (Λ, b)-open, we have X − [f−1(U (Λ,b))](Λ,b) = [f−1(Y −U (Λ,b))](Λ,b) ⊆ f−1([Y −U (Λ,b)](Λ,b)) ⊆ X − f−1(U) and hence f−1(U) ⊆ [f−1(U (Λ,b))](Λ,b). (7) ⇒ (1): Let x ∈ X and V be any (Λ, b)-open set of Y containing f(x). Then, we have x ∈ f−1(V ) ⊆ [f−1(V (Λ,b))](Λ,b). Put U = [f−1(V (Λ,b))](Λ,b). Then, U is a (Λ, b)-open set of X containing x such that f(U) ⊆ V (Λ,b). Thus, f is weakly (Λ, b)-continuous at x and hence f is weakly (Λ, b)-continuous. Definition 6. A topological space (X, τ) is called (Λ, b)-connected if X cannot be written as a disjoint union of two nonempty (Λ, b)-open sets. Example 5. Let X = {a, b} with the topology τ = {∅, {a}, X}. Then, (X, τ) is (Λ, b)- connected. Lemma 7. For a topological space (X, τ), the following properties are equivalent: (1) (X, τ) is (Λ, b)-connected. (2) The only subsets of X, which are both (Λ, b)-open and (Λ, b)-closed are ∅ and X. Theorem 14. Let f : (X, τ) → (Y, σ) is a weakly (Λ, b)-continuous surjection. If (X, τ) is (Λ, b)-connected, then (Y, σ) is (Λ, b)-connected. Proof. Assume that (Y, σ) is not (Λ, b)-connected. Then, there exist nonempty (Λ, b)- open sets V1, V2 such that V1 ∩ V2 = ∅ and V1 ∪ V2 = Y . By Theorem 10, f−1(Vi) ⊆ [f−1[V (Λ,b)]](Λ,b) for i = 1, 2. Since Vi is (Λ, b)-closed in Y for each i = 1, 2. Therefore, f−1(Vi) ⊆ [f−1(Vi)](Λ,b) and by Lemma 2, f−1(Vi) is (Λ, b)-open for each i = 1, 2. More- over, X is union of nonempty disjoint sets f−1(V1) and f−1(V2). This shows that (X, τ) is not (Λ, b)-connected. This is contrary to the hypothesis that (X, τ) is (Λ, b)-connected. Thus, (Y, σ) is (Λ, b)-connected. C. Boonpok, N. Srisarakham / Eur. J. Pure Appl. Math, 16 (1) (2023), 29-43 42 Definition 7. A subset K of a topological spaces (X, τ) is said to be Λb-closed (resp. Λb- compact) relative to (X, τ) if for any cover {Vi : i ∈ I} of K by (Λ, b)-open sets of X, there exist a finite subset I0 of I such that K ⊆ ∪{V (Λ,b) i | i ∈ I0} (resp. K ⊆ ∪{Vi | i ∈ I0}). If X is Λb-closed (resp. Λb-compact) relative to (X, τ), then (X, τ) is said to be Λb- closed (resp. Λb-compact). Theorem 15. If f : (X, τ) → (Y, σ) is weakly (Λ, b)-continuous and K is Λb-compact, then f(K) is Λb-closed relative to (Y, σ). Proof. Let {Vα : α ∈ ∇} be any cover of f(K) by (Λ, b)-open sets of Y . For each x ∈ K, there exists i(x) ∈ I such that f(x) ∈ Vi(x). Since f is weakly (Λ, b)-continuous, there exists a (Λ, b)-open set U(x) containing x such that f(U(x)) ⊆ [Vi(x)] (Λ,b). The family {U(x) : x ∈ K} is a cover of K by (Λ, b)-open sets of X. Since K is Λb-compact, there exist a finite number of points, say, x1, x2, ..., xn in K such that K ⊆ ∪{U(xk) : xk ∈ K, 1 ≤ k ≤ n}. Thus, f(K) ⊆ ∪{f(U(xk)) : xk ∈ K, 1 ≤ k ≤ n} ⊆ ∪{[Vi(xk)] (Λ,b) : xk ∈ K, 1 ≤ k ≤ n}. This shows that f(K) is Λb-closed relative to (Y, σ). Corollary 3. If f : (X, τ) → (Y, σ) is a weakly (Λ, b)-continuous surjection and (X, τ) is Λb-compact, then (Y, σ) is Λb-closed. Definition 8. [5] Let A be a subset of a topological space (X, τ). The (Λ, b)-frontier of A, ΛbFr(A), is defined as follows: ΛbFr(A) = A(Λ,b) ∩ [X −A](Λ,b). Theorem 16. The set of all points x ∈ X at which a function f : (X, τ) → (Y, σ) is not weakly (Λ, b)-continuous is identical with the union of the (Λ, b)-frontiers of the inverse images of the (Λ, b)-closure of the (Λ, b)-open sets of Y containing f(x). Proof. Suppose that f is not weakly (Λ, b)-continuous at x ∈ X. There exists a (Λ, b)-open set V of Y containing f(x) such that f(U) is not contained in V (Λ,b) for every (Λ, b)-open set U containing x. Then, U ∩ (X − f−1(V (Λ,b))) ̸= ∅ for every (Λ, b)-open set U of X containing x and hence x ∈ [X − f−1(V (Λ,b))](Λ,b). On the other hand, we have x ∈ f−1(V ) ⊆ [f−1(V (Λ,b))](Λ,b). Thus, x ∈ ΛbFr(f−1(V (Λ,b))). Conversely, suppose that f is weakly (Λ, b)-continuous at x ∈ X and let V be any (Λ, b)-open set of Y containing f(x). Thus, by Theorem 7, we have x ∈ [f−1(V (Λ,b))](Λ,b). 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