EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5649 ISSN 1307-5543 – ejpam.com Published by New York Business Global Rarely s-(τ1, τ2)p-continuous Multifunctions Butsakorn Kong-ied1, Supunnee Sompong2, Chawalit Boonpok1,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand 2 Department of Mathematics and Statistics, Faculty of Science and Technology, Sakon Nakhon Rajbhat University, Sakon Nakhon, 47000, Thailand Abstract. This paper is concerned with the concepts of upper rarely s-(τ1, τ2)p-continuous mul- tifunctions and lower rarely s-(τ1, τ2)p-continuous multifunctions. Furthermore, some character- izations and several properties concerning upper rarely s-(τ1, τ2)p-continuous multifunctions and lower rarely s-(τ1, τ2)p-continuous multifunctions are established. 2020 Mathematics Subject Classifications: 54C08, 54C60 Key Words and Phrases: (τ1, τ2)p-open set, upper rarely s-(τ1, τ2)p-continuous multifunction, lower rarely s-(τ1, τ2)p-continuous multifunction 1. Introduction Weaker and stronger forms of open sets such as semi-open sets [48], preopen sets [50], α-open sets [51], β-open sets [38], δ-open sets [67] and θ-open sets [67] play an impor- tant role in the research of generalizations of continuity in topological spaces. By using these sets, many authors introduced and studied various types of continuity for functions and multifunctions. Viriyapong and Boonpok [69] investigated some characterizations of (Λ, sp)-continuous functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)- closed sets due to Boonpok and Khampakdee [12]. Dungthaisong et al. [35] introduced and studied the concept of g(m,n)-continuous functions. Duangphui et al. [34] intro- duced and investigated the notion of (µ, µ′)(m,n)-continuous functions. Moreover, some characterizations of almost (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous func- tions, almost strongly θ(Λ, p)-continuous functions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions, θ(⋆)-precontinuous functions, (Λ, p(⋆))-continuous functions, ⋆-continuous functions, θ-I -continuous functions, almost (g,m)-continuous functions, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5649 Email addresses: butsakorn.k@msu.ac.th (B. Kong-ied), s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 2 of 13 pairwise almost M -continuous functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)- continuous functions, weakly (τ1, τ2)-continuous functions, faintly (τ1, τ2)-continuous func- tions, almost quasi (τ1, τ2)-continuous functions and weakly quasi (τ1, τ2)-continuous func- tions were presented in [61], [64], [16], [56], [25], [11], [8], [10], [4], [1], [2], [26], [23], [18], [62], [46] and [33], respectively. Popa [54] introduced the concept of rare continuity as a generalization of weak continuity [47] which has been further investigated by Long and Herrington [49] and Jafari [39, 40]. Jafari [41] also generalized the concept of rare conti- nuity to rare β-continuity by involving the notion of β-open sets. Caldas [30] introduced a new class of functions called rarely βθ-continuous functions by utilizing the notion of β-θ-open sets and investigated some characterizations of rarely βθ-continuous functions. Jafari [42] introduced and studied the concept of rare α-continuity as a generalization of rare continuity and weak α-continuity [52]. Caldas and Jafari [31] introduced and investi- gated a new class of functions called rarely g-continuous functions which is a generalization of both the class of rarely continuous functions and the class of weakly g-continuous func- tions. Quite recently, Thongmoon et al. [66] introduced and studied the concept of rarely (τ1, τ2)-continuous functions. In 2005, Caldas et al. [32] introduced and studied the new notion of rarely g-continuous multifunctions is a generalization of weakly continuous multifunctions [53]. Viriyapong and Boonpok [70] introduced and studied the concept of weakly quasi (Λ, sp)-continuous mul- tifunctions. Furthermore, several characterizations of (τ1, τ2)δ-semicontinuous multifunc- tions, almost weakly (τ1, τ2)-continuous multifunctions, ⋆-continuous multifunctions, β(⋆)- continuous multifunctions, α-⋆-continuous multifunctions, almost α-⋆-continuous multi- functions, almost quasi ⋆-continuous multifunctions, weakly α-⋆-continuous multifunc- tions, sβ(⋆)-continuous multifunctions, weakly sβ(⋆)-continuous multifunctions, θ(⋆)-quasi continuous multifunctions, almost ı⋆-continuous multifunctions, weakly (Λ, sp)-continuous multifunctions, α(Λ, sp)-continuous multifunctions, almost α(Λ, sp)-continuous multifunc- tions, weakly α(Λ, sp)-continuous multifunctions, almost β(Λ, sp)-continuous multifunc- tions, slightly (Λ, sp)-continuous multifunctions, (τ1, τ2)-continuous multifunctions, al- most (τ1, τ2)-continuous multifunctions, weakly (τ1, τ2)-continuous multifunctions, weakly quasi (τ1, τ2)-continuous multifunctions, almost quasi (τ1, τ2)-continuous multifunctions, c-(τ1, τ2)-continuous multifunctions and c-quasi (τ1, τ2)-continuous multifunctions were es- tablished in [5], [28], [3], [7], [17], [24], [6], [21], [20], [15], [9], [19], [22], [43], [13], [27], [63], [14], [59], [45], [65], [60], [58], [44] and [57], respectively. Popa and Noiri [55] introduced and studied the notion of s-precontinuous multifunctions is a generalization of s-continuous multifunctions and precontinuous multifunctions. Ekici and Park [37] introduced and in- vestigated the concept of weakly s-precontinuous multifunctions. The notion of weakly s-precontinuous multifunctions is a generalization of s-precontinuous multifunctions due to Popa and Noiri [55]. Ekici and Jafari [36] introduced and investigated the notion of rarely s-precontinuous multifunctions which is a generalization of weakly s-precontinuous multifunctions due to Ekici and Park [37]. In this paper, we introduce the notions of upper rarely s-(τ1, τ2)p-continuous multifunctions and lower rarely s-(τ1, τ2)p-continuous multifunctions. We also investigate several characterizations of upper rarely s-(τ1, τ2)p- continuous multifunctions and lower rarely s-(τ1, τ2)p-continuous multifunctions. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 3 of 13 2. Preliminaries Throughout the present paper, spaces (X, τ1, τ2) and (Y, σ1, σ2) (or simply X and Y ) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space (X, τ1, τ2). The closure of A and the interior of A with respect to τi are denoted by τi-Cl(A) and τi-Int(A), respectively, for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed [29] if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. The intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [29] of A and is denoted by τ1τ2-Cl(A). The union of all τ1τ2-open sets of X contained in A is called the τ1τ2-interior [29] of A and is denoted by τ1τ2-Int(A). Lemma 1. [29] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2- closure, the following properties hold: (1) A ⊆ τ1τ2-Cl(A) and τ1τ2-Cl(τ1τ2-Cl(A)) = τ1τ2-Cl(A). (2) If A ⊆ B, then τ1τ2-Cl(A) ⊆ τ1τ2-Cl(B). (3) τ1τ2-Cl(A) is τ1τ2-closed. (4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A). (5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A). A bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [29] if X cannot be writ- ten as the union of two nonempty disjoint τ1τ2-open sets. A subset A of a bitopo- logical space (X, τ1, τ2) is called (τ1, τ2)r-open [68] (resp. (τ1, τ2)s-open [5], (τ1, τ2)p- open [5], (τ1, τ2)β-open [5], α(τ1, τ2)-open) [71]) if A = τ1τ2-Int(τ1τ2-Cl(A)) (resp. A ⊆ τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A))), A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A)))). The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open, α(τ1, τ2)-open) set is called (τ1, τ2)r-closed (resp. (τ1, τ2)s- closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed, α(τ1, τ2)-closed). A subset R of a bitopological space (X, τ1, τ2) is said to be τ1τ2-rare set [66] if τ1τ2-Int(R) = ∅. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all (τ1, τ2)p-closed sets of X containing A is called the (τ1, τ2)p-closure of A and is denoted by (τ1, τ2)-pCl(A). The union of all (τ1, τ2)p-open sets of X contained in A is called the (τ1, τ2)p-interior of A and is denoted by (τ1, τ2)-pInt(A). Lemma 2. [72] For a subset A of a bitopological space (X, τ1, τ2), the following properties hold: (1) A is (τ1, τ2)p-closed if and only if (τ1, τ2)-pCl(A) = A; (2) (τ1, τ2)-pCl(A) = τ1τ2Cl(τ1τ2Int(A)) ∪A; (3) (τ1, τ2)-pCl((τ1, τ2)-pCl(A)) = (τ1, τ2)-pCl(A). B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 4 of 13 By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y , and we always assume that F (x) ̸= ∅ for all x ∈ X. For a multifunction F : X → Y , we shall denote the upper and lower inverse of a set B of Y by F+(B) and F−(B), respectively, that is, F+(B) = {x ∈ X | F (x) ⊆ B} and F−(B) = {x ∈ X | F (x) ∩ B ̸= ∅}. In particular, F−(y) = {x ∈ X | y ∈ F (x)} for each point y ∈ Y . For each A ⊆ X, F (A) = ∪x∈AF (x). 3. Upper and lower rarely s-(τ1, τ2)p-continuous multifunctions In this section, we introduce the notions of upper rarely s-(τ1, τ2)p-continuous multi- functions and lower rarely s-(τ1, τ2)p-continuous multifunctions. Moreover, some char- acterizations of upper rarely s-(τ1, τ2)p-continuous multifunctions and lower rarely s- (τ1, τ2)p-continuous multifunctions are discussed. Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper rarely s-(τ1, τ2)p-continuous at x ∈ X if for each σ1σ2-open set V of Y having σ1σ2-connected complement such that F (x) ⊆ V , there exists a σ1σ2-rare set RV with τ1τ2-Cl(RV )∩V = ∅ and a (τ1, τ2)p-open set U of X containing x such that F (U) ⊆ V ∪RV . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper rarely s-(τ1, τ2)p-continuous if F is upper rarely s-(τ1, τ2)p-continuous at each point x of X. Theorem 1. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is upper rarely s-(τ1, τ2)p-continuous at x ∈ X; (2) for every σ1σ2-open set V of Y having σ1σ2-connected complement with F (x) ⊆ V , there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ (τ1, τ2)-pInt(F +(V ∪RV )); (3) for every σ1σ2-open set V of Y having σ1σ2-connected complement with F (x) ⊆ V , there exists a σ1σ2-rare set RV with σ1σ2-Cl(V ) ∩RV = ∅ such that x ∈ (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ) ∪RV )); (4) for every σ1σ2-open set V of Y having σ1σ2-connected complement with F (x) ⊆ V , there exists a (τ1, τ2)p-open set U of X containing x such that σ1σ2-Int(F (U) ∩ (Y − V )) = ∅; (5) for every σ1σ2-open set V of Y having σ1σ2-connected complement with F (x) ⊆ V , there exists a (τ1, τ2)p-open set U of X containing x such that σ1σ2-Int(F (U)) ⊆ σ1σ2-Cl(V ); B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 5 of 13 (6) for every σ1σ2-open set V of Y having σ1σ2-connected complement with F (x) ⊆ V , there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ F+(V ∪RV ) ∩ τ1τ2-Int(τ1τ2-Cl(F +(V ∪RV ))). Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement such that F (x) ⊆ V . Since F is upper rarely s-(τ1, τ2)p-continuous at x ∈ X, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ and a (τ1, τ2)p-open set U of X containing x such that F (U) ⊆ V ∪RV . Thus, x ∈ U ⊆ F+(V ∪RV ) and hence x ∈ (τ1, τ2)-pInt(F +(V ∪RV )). (2) ⇒ (3): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement such that F (x) ⊆ V . By (2), there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ (τ1, τ2)-pInt(F +(V ∪ RV )). Since σ1σ2-Cl(RV ) ∩ V = ∅, we have RV ⊆ Y − V and Y − V = [Y − σ1σ2-Cl(V )] ∪ [σ1σ2-Cl(V )− V ]. Thus, RV ⊆ [RV ∩ (Y − σ1σ2-Cl(V ))] ∪ [σ1σ2-Cl(V )− V ]. Put WV = RV ∩(Y −σ1σ2-Cl(V )). Then, WV is a τ1τ2-rare set with σ1σ2-Cl(V )∩WV = ∅. Therefore, x ∈ (τ1, τ2)-pInt(F +(V ∪RV )) ⊆ (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ) ∪WV )). (3) ⇒ (4): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement such that F (x) ⊆ V . By (3), there exists a σ1σ2-rare set RV with σ1σ2-Cl(V )∩RV = ∅ such that x ∈ (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ) ∪ RV )). Let U = (τ1, τ2)-pInt(F +(σ1σ2-Cl(V ) ∪ RV )). Then, U is a (τ1, τ2)p-open set of X containing x and F (U) ⊆ σ1σ2-Cl(V ) ∪RV . Thus, σ1σ2-Int(F (U) ∩ (Y − V )) = σ1σ2-Int(F (U)) ∩ σ1σ2-Int(Y − V ) ⊆ σ1σ2-Int(σ1σ2-Cl(V ) ∪RV ) ∩ (Y − σ1σ2-Cl(V )) = σ1σ2-Int([σ1σ2-Cl(V ) ∪RV ] ∩ [Y − σ1σ2-Cl(V )]) = σ1σ2-Int([σ1σ2-Cl(V ) ∩ (Y − σ1σ2-Cl(V ))] ∪ [RV ∩ (Y − σ1σ2-Cl(V ))]) = σ1σ2-Int(RV ∩ (Y − σ1σ2-Cl(V ))) = σ1σ2-Int(RV ) ∩ σ1σ2-Int(Y − σ1σ2-Cl(V )) = ∅. (4) ⇒ (5): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement such that F (x) ⊆ V . By (4), there exists a (τ1, τ2)p-open set U of X containing x such that σ1σ2-Int(F (U) ∩ (Y − V )) = ∅. Since σ1σ2-Int(F (U) ∩ (Y − V )) = ∅, we have σ1σ2-Int(F (U)) ⊆ V ⊆ σ1σ2-Cl(V ). (5) ⇒ (1): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement with F (x) ⊆ V . By (5), there exists a (τ1, τ2)p-open set U of X containing x such that σ1σ2-Int(F (U)) ⊆ σ1σ2-Cl(V ). Thus, F (U) = (F (U)− σ1σ2-Int(F (U))) ∪ σ1σ2-Int(F (U)) B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 6 of 13 ⊆ (F (U)− σ1σ2-Int(F (U))) ∪ σ1σ2-Cl(V ) = (F (U)− σ1σ2-Int(F (U))) ∪ V ∪ (σ1σ2-Cl(V )− V ) = [(F (U)− σ1σ2-Int(F (U))) ∩ (Y − V )] ∪ V ∪ (σ1σ2-Cl(V )− V ). Let WV = (F (U)−σ1σ2-Int(F (U)))∩ (Y −V ) and W ′ V = σ1σ2-Cl(V )−V . Then, WV and W ′ V are σ1σ2-rare sets and RV = WV ∪W ′ V is a σ1σ2-rare set such that σ1σ2-Cl(RV )∩V = ∅ and F (U) ⊆ V ∪RV . Thus, F is upper rarely s-(τ1, τ2)p-continuous at x. (2) ⇔ (6): It follows from the fact that (τ1, τ2)-pInt(F +(V ∪RV )) = τ1τ2-Int(τ1τ2-Cl(F +(V ∪RV ))) ∩ F+(V ∪RV ). Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly s-(τ1, τ2)p-continuous at x ∈ X if for each σ1σ2-open set V of Y having σ1σ2-connected complement such that x ∈ F+(V ), there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F+(σ1σ2-Cl(V )). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly s-(τ1, τ2)p-continuous if F is upper weakly s-(τ1, τ2)p-continuous at each point x of X. Definition 3. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower weakly s-(τ1, τ2)p-continuous at x ∈ X if for each σ1σ2-open set V of Y having σ1σ2-connected complement such that x ∈ F−(V ), there exists a (τ1, τ2)p-open set U of X containing x such that U ⊆ F−(σ1σ2-Cl(V )). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower weakly s-(τ1, τ2)p-continuous if F is lower weakly s-(τ1, τ2)p-continuous at each point x of X. Definition 4. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called strongly (τ1, τ2)p-open if F (U) is σ1σ2-open in Y for every (τ1, τ2)p-open set U of X. Theorem 2. If F : (X, τ1, τ2) → (Y, σ1, σ2) is upper rarely s-(τ1, τ2)p-continuous and strongly (τ1, τ2)p-open, then F is upper weakly s-(τ1, τ2)p-continuous. Proof. Let x ∈ X and V be any σ1σ2-open set of Y having σ1σ2-connected complement with F (x) ⊆ V . Since F is upper rarely s-(τ1, τ2)p-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that σ1σ2-Int(F (U)) ⊆ σ1σ2-Cl(V ). It follows from that F (U) ⊆ σ1σ2-Cl(σ1σ2-Int(F (U))) ⊆ σ1σ2-Cl(V ). Thus, F is upper weakly s-(τ1, τ2)p- continuous. Definition 5. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower rarely s-(τ1, τ2)p-continuous at x ∈ X if for each σ1σ2-open set V of Y having σ1σ2-connected complement such that F (x)∩V ̸= ∅, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV )∩V = ∅ and a (τ1, τ2)p-open set U of X containing x such that F (z) ∩ (V ∪ RV ) ̸= ∅ for each z ∈ U . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower rarely s-(τ1, τ2)p- continuous if F is lower rarely s-(τ1, τ2)p-continuous at each point x of X. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 7 of 13 Theorem 3. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) F is lower rarely s-(τ1, τ2)p-continuous at x ∈ X; (2) for every σ1σ2-open set V of Y having σ1σ2-connected complement with F (x)∩V ̸= ∅, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ (τ1, τ2)-pInt(F −(V ∪RV )); (3) for every σ1σ2-open set V of Y having σ1σ2-connected complement with F (x)∩V ̸= ∅, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ F−(V ∪RV ) ∩ τ1τ2-Int(τ1τ2-Cl(F −(V ∪RV ))). Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement such that F (x) ⊆ V . Since F is lower rarely s-(τ1, τ2)p-continuous at x ∈ X, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ and a (τ1, τ2)p-open set U of X containing x such that F (z) ∩ (V ∪RV ) ̸= ∅ for each z ∈ U . Thus, x ∈ U ⊆ F−(V ∪RV ) and hence x ∈ (τ1, τ2)-pInt(F −(V ∪RV )). (2) ⇒ (1): Let V be any σ1σ2-open set of Y having σ1σ2-connected complement with F (x) ∩ V ̸= ∅. By (2), there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ (τ1, τ2)-pInt(F −(V ∪ RV )). Let U = (τ1, τ2)-pInt(F −(V ∪ RV )). Then, U is a (τ1, τ2)p-open set U of X containing x. Furthermore, F (z)∩(V ∪RV ) ̸= ∅ for every z ∈ U . Thus, F is lower rarely s-(τ1, τ2)p-continuous at x ∈ X. (2) ⇔ (3): It follows from the fact that (τ1, τ2)-pInt(F −(V ∪RV )) = τ1τ2-Int(τ1τ2-Cl(F −(V ∪RV ))) ∩ F−(V ∪RV ). Definition 6. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called rarely s-(τ1, τ2)p-continuous at x ∈ X if for each σ1σ2-open set V of Y containing f(x) and having σ1σ2-connected complement, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV )∩V = ∅ and a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ V ∪RV . A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called rarely s-(τ1, τ2)p-continuous if f is rarely s-(τ1, τ2)p-continuous at each point x of X. Corollary 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is rarely s-(τ1, τ2)p-continuous at x ∈ X; (2) for every σ1σ2-open set V of Y containing f(x) and having σ1σ2-connected comple- ment, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ (τ1, τ2)-pInt(f −1(V ∪RV )); B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 8 of 13 (3) for every σ1σ2-open set V of Y containing f(x) and having σ1σ2-connected comple- ment, there exists a σ1σ2-rare set RV with σ1σ2-Cl(V ) ∩RV = ∅ such that x ∈ (τ1, τ2)-pInt(f −1(σ1σ2-Cl(V ) ∪RV )); (4) for every σ1σ2-open set V of Y containing f(x) and having σ1σ2-connected comple- ment, there exists a (τ1, τ2)p-open set U of X containing x such that σ1σ2-Int(f(U) ∩ (Y − V )) = ∅; (5) for every σ1σ2-open set V of Y containing f(x) and having σ1σ2-connected comple- ment, there exists a (τ1, τ2)p-open set U of X containing x such that σ1σ2-Int(f(U)) ⊆ σ1σ2-Cl(V ); (6) for every σ1σ2-open set V of Y containing f(x) and having σ1σ2-connected comple- ment, there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that x ∈ f−1(V ∪RV ) ∩ τ1τ2-Int(τ1τ2-Cl(f −1(V ∪RV ))). Theorem 4. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is rarely s-(τ1, τ2)p-continuous if and only if for every σ1σ2-open set V of Y , there exists a σ1σ2-rare set RV with σ1σ2-Cl(RV ) ∩ V = ∅ such that f−1(V ) ⊆ (τ1, τ2)-pInt(f −1(V ∪RV )). Proof. It is an immediate consequence of the above corollary. Definition 7. A bitopologcal space (X, τ1, τ2) is said to be τ1τ2-rarely separate if for every pair of distinct points x and y in X, there exist τ1τ2-open sets Vx and Vy containing x and y, respectively, and τ1τ2-rare sets RVx, RVy with τ1τ2-Cl(RVx) ∩ Vx = ∅ and τ1τ2-Cl(RVy) ∩ Vy = ∅ such that (Vx ∪RVx) ∩ (Vy ∪RVy) = ∅. Definition 8. A bitopologcal space (X, τ1, τ2) is said to be (τ1, τ2)p-Hausdorff if for any distinct pair of points x and y in X, there exist (τ1, τ2)p-open sets U and V of X containing x and y, respectively, such that U ∩ V = ∅. Theorem 5. If (Y, σ1, σ2) is σ1σ2-rarely separate and f : (X, τ1, τ2) → (Y, σ1, σ2) is a rarely s-(τ1, τ2)p-continuous injection, then (X, τ1, τ2) is (τ1, τ2)p-Hausdorff. B. Kong-ied, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5649 9 of 13 Proof. Let x and y be any distinct points in X. Then, f(x) ̸= f(y). Since (Y, σ1, σ2) is σ1σ2-rarely separate, there exist σ1σ2-open sets V and W of Y containing f(x) and f(y), respectively, and σ1σ2-rare sets RV and RW with σ1σ2-Cl(RV ) ∩ V = ∅ and σ1σ2-Cl(RW ) ∩W = ∅ such that (V ∪RV ) ∩ (W ∪RW ) = ∅. Thus, (τ1, τ2)-pInt(f −1(V ∪RV )) ∩ (τ1, τ2)-pInt(f −1(W ∪RW )) = ∅. By Theorem 4, we have x ∈ f−1(V ) ⊆ (τ1, τ2)-pInt(f −1(V ∪RV )) and y ∈ f−1(W ) ⊆ (τ1, τ2)-pInt(f −1(W ∪RW )). Since (τ1, τ2)-pInt(f −1(V ∪ RV )) and (τ1, τ2)-pInt(f −1(W ∪ RW )) are (τ1, τ2)p-open sets, (X, τ1, τ2) is a (τ1, τ2)p-Hausdorff space. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] C. Boonpok. 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