EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1023-1046 ISSN 1307-5543 – ejpam.com Published by New York Business Global On some closed sets and low separation axioms via topological ideals Chawalit Boonpok 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 Λp(⋆)-sets and (Λ, p(⋆))-closed sets which are de- fined by utilizing the notions of pre-I -open sets and pre-I -closed sets. Moreover, we investigate some properties of (Λ, p(⋆))-extremally disconnected ideal topological spaces. Several characteri- zations of (Λ, p(⋆))-continuous functions are discussed. Especially, we introduce and characterize some low separation axioms of ideal topologies constructed by the concepts of pre-I -open sets and the pre-I -closure operator. 2020 Mathematics Subject Classifications: 54A05, 54C05, 54D10, 54G05 Key Words and Phrases: pre-I -open set, Λp(⋆)-set, (Λ, p(⋆))-closed set, (Λ, p(⋆))-extremally disconnected, pre-I -T0, pre-I -T1, pre-I -R0, (Λ, p(⋆))-continuous function 1. Introduction Openness and closedness are fundamental concept for the study and investigation in topological spaces. Many mathematicians introduced and studied the various types of generalizations of open sets. In 1982, Mashhour et al. [28] introduced the notion of preopen sets which is also known under the name of locally dense sets [12] in the literature. Kar and Bhattacharya [25] introduced new separation axioms pre-T0, pre-T1 and pre-T2 by using preopen sets due to Mashhour et al. [28]. Caldas [5] and Jafari [22] introduced independently the notions of p-D-sets and a separation axiom p-D1 which is strictly between pre-T0 and pre-T1. In [7], the present authors introduced two new classes of topological spaces called pre-R0 and pre-R1 spaces in terms of concept of preopen sets and investigated some of their fundamental properties. In 1986, Maki [27] introduced the concept of Λ-sets in topological spaces as the sets that coincide with their kernel. The kernel of a set A is the intersection of all open superset A. Arenas et al. [4] introduced and investigated the concept of λ-closed sets by involving Λ-sets and closed sets. Caldas et al. [10] introduced the concept of λ-closure of a set by utilizing the notion of λ- open sets defined in [4]. In [9], the present authors introduced and studied two new low DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4343 Email address: chawalit.b@msu.ac.th (C. Boonpok) https://www.ejpam.com 1023 © 2022 EJPAM All rights reserved. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1024 separation axioms called λ-R0 and λ-R1 by utilizing the notions of λ-open sets and the λ- closure operator. Jafari et al. [23] introduced a new class of functions between topological spaces, namely almost λ-continuous functions and investigated several characterizations of almost λ-continuous functions. Ekici et al. [16] introduced a new class of generalization of continuous functions via λ-open sets called weakly λ-continuous functions and investigated some fundamental properties of such functions. Ganster et al. [19] introduced the notions of a pre-Λ-set and a pre-V -set in a topological space and investigated the topologies defined by these families of sets. Veličko [31] introduced and studied the concepts of δ-open sets, δ-closure operator and δ-closed sets. Georgiou et al. [20] by considering the notion of δ-closed sets, introduced and investigated Λδ-sets, (Λ, δ)-closed, (Λ, δ)-open sets and the (Λ, δ)-closure operator. Caldas and Jafari [6] introduced and investigated some new low separation axioms by using the notions of (Λ, δ)-open sets and the (Λ, δ)-closure operator. Cammaroto and Noiri [11] introduced and investigated three topological spaces (X,Λm), (X,Λ∗ mc) and (X,ΛgΛm) by using Λm-sets, (Λ,m)-closed sets and generalized Λm-sets, respectively. Caldas et al. [8] introduced and studied two new weak separation axioms called Λθ-R0 and Λθ-R1 spaces by using the notions of (Λ, θ)-open sets and the (Λ, θ)- closure operator. The concept of ideals in topological spaces has been introduced and studied by Kura- towski [26] and Vaidyanathswamy [30]. The topology τ of a space is enlarged to a topology τ⋆ using an ideal I whose members are disjoint with the members of τ . Every topological space is an ideal topological space and all the results of ideal topological spaces are gener- alizations of the results established in topological spaces. In 1990, Janković and Hamlett [24] introduced the notion of I -open sets in ideal topologial spaces. Abd El-Monsef et al. [18] further investigated I -open sets and I -continuous functions. Later, several au- thors studied ideal topological spaces giving several convenient definitions. Some authors obtained decompositions of continuity. For instance, Açıkgöz et al. [2] introduced and in- vestigated the notions of weakly-I -continuous and weak⋆-I -continuous functions in ideal topological spaces. Donthev [15] introduced the notion of pre-I -open sets and obtained a decomposition of I -continuity. Hatır and Noiri [21] introduced α-I -open, semi-I -open and β-I -open sets via idealization and using these sets obtained new decompositions of continuity. In [3], the present authors studied the concepts of α-I -continuity and α-I - openness in ideal topological spaces and obtained several characterizations of these func- tions. Açıkgöz et al. [1] introduced two new classes of functions called α-I -preirresolute functions and β-I -preirresolute functions in ideal topologial spaces and investigated the relationships between these classes of functions and other classes of non-continuous func- tions. In 2009, Ekici and T. Noiri [17] introduced the concept of ⋆-extremally disconnected ideal topological spaces and investigated several characterizations of ⋆-extremally discon- nected ideal topological spaces. The paper is organized as follows. In Section 3, we introduce the notions of Λp(⋆)-sets and Λp(⋆)-sets. Moreover, some properties of Λp(⋆)-sets and Λp(⋆)-sets are investigated. In Section 4, we define (Λ, p(⋆))-closed sets and investigate several characterizations of (Λ, p(⋆))-extremally disconnected. In Section 5, we introduce the concept of (Λ, p(⋆))- continuous functions and investigate some characterizations of such functions. In the last C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1025 section, we introduce and study some new low separation axioms by using the concepts of pre-I -open sets and the pre-I -closure operator. 2. Preliminaries We begin with some definitions and known results which will be used throughout this paper. In the present 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. For a subset A of a topological space (X, τ), Cl(A) and Int(A) represent the closure and the interior of A, respectively. A nonempty collection I of subsets of a set X is said to be an ideal on X if I satisfies the following two properties: (i) A ∈ I and B ⊆ A ⇒ B ∈ I ; (ii) A ∈ I and B ∈ I ⇒ A ∪ B ∈ I . For a topological space (X, τ) with an ideal I on X, a set operator (.)⋆ : P(X) → P(X) where P(X) is the set of all subsets of X, called a local function [26] of A with respect to I and τ is defined as follows: for A ⊆ X, A⋆(I , τ) = {x ∈ X | G ∩ A ̸∈ I for every G ∈ τ(x)} where τ(x) = {G ∈ τ | x ∈ G}. A Kuratowski closure operator Cl⋆(.) for a topology τ⋆(I , τ), called the ⋆-topology and finer than τ , is defined by Cl⋆(A) = A ∪ A⋆ [24]. We shall simply write A⋆ for A⋆(I , τ) and τ⋆ for τ⋆(I , τ). A basis B(I , τ) for τ⋆ can be described as follows: B(I , τ) = {V − I ′ | V ∈ τ and I ′ ∈ I }. However, B(I , τ) is not always a topology [24]. A subset A of an ideal topological space (X, τ,I ) is called ⋆-closed (τ⋆-closed) [24] if A⋆ ⊆ A. The interior of a subset A in (X, τ⋆(I , τ)) is denoted by Int⋆(A). A subset A of an ideal topological space (X, τ,I ) is said to be pre-I -open [15] if A ⊆ Int(Cl⋆(A)). The complement of a pre-I -open set is called pre-I -closed. The family of all pre-I -open sets of an ideal topological space (X, τ,I ) is denoted by pIO(X, τ). For a subset A of an ideal topological space (X, τ,I ), the intersection of all pre-I -closed sets of X containing A is called the pre-I -closure [13] of A and is denoted by pıCl(A). The union of all pre-I -open sets contained in A is called the pre-I -interior of A and is denoted by pıInt(A). Lemma 1. [13] Let A be a subset of an ideal topological space (X, τ,I ) and x ∈ X. Then, the following properties hold: (1) x ∈ pıCl(A) if and only if U ∩A ̸= ∅ for every pre-I -open set U of X containing x. (2) A is pre-I -closed if and only if A = pıCl(A). (3) X − pıCl(A) = pıInt(X −A). (4) X − pıInt(A) = pıCl(X −A). Lemma 2. [14, 29] Let A be a subset of an ideal topological space (X, τ,I ). Then, pıCl(A) = A ∪ Cl(Int⋆(A)). C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1026 3. Some properties of Λp(⋆)-sets and δp(⋆)-sets In this section, we introduce the notions of Λp(⋆)-sets and δp(⋆)-sets. Moreover, several properties of Λp(⋆)-sets and δp(⋆)-sets are discussed. Definition 1. Let A be a subset of an ideal topological space (X, τ,I ). A subset Λp(⋆)(A) is defined as follows: Λp(⋆)(A) = ∩{U | A ⊆ U ;U is pre-I -open}. Proposition 1. For subsets A, B and Cγ(γ ∈ Γ) of an ideal topological space (X, τ,I ), the following properties hold: (1) A ⊆ Λp(⋆)(A). (2) If A ⊆ B, then Λp(⋆)(A) ⊆ Λp(⋆)(B). (3) Λp(⋆)(Λp(⋆)(A)) = Λp(⋆)(A). (4) If A is a pre-I -open set, then Λp(⋆)(A) = A. (5) Λp(⋆)(∪{Cγ |γ ∈ Γ}) = ∪{Λp(⋆)(Cγ)|γ ∈ Γ}. (6) Λp(⋆)(∩{Cγ |γ ∈ Γ}) ⊆ ∩{Λp(⋆)(Cγ)|γ ∈ Γ}. Proof. We prove only properties (5) and (6) since the others are immediate conse- quences of Definition 1. (5) First for each γ ∈ Γ, Λp(⋆)(Cγ) ⊆ Λp(⋆)(∪γ∈ΓCγ). Thus, ∪γ∈ΓΛp(⋆)(Cγ) ⊆ Λp(⋆)(∪γ∈ΓCγ). On the other hand, let x ̸∈ ∪γ∈ΓΛp(⋆)(Cγ). Then, x ̸∈ Λp(⋆)(Cγ) for each γ ∈ Γ and so there exists a pre-I -open set Vγ such that Cγ ⊆ Vγ and x ̸∈ Vγ for each γ ∈ Γ. Thus, ∪γ∈ΓCγ ⊆ ∪γ∈ΓVγ and hence ∪γ∈ΓVγ is a pre-I -open set which does not contain x. This implies that x ̸∈ Λp(⋆)(∪γ∈ΓCγ). Therefore, Λp(⋆)(∪γ∈ΓCγ) ⊆ ∪γ∈ΓΛp(⋆)(Cγ). Consequently, we obtain Λp(⋆)(∪γ∈ΓCγ) = ∪γ∈ΓΛp(⋆)(Cγ). (6) Suppose that x ̸∈ ∩γ∈ΓΛp(⋆)(Cγ). There exists γ0 ∈ Γ such that x ̸∈ Λp(⋆)(Cγ0) and there exists a pre-I -open set V such that x ̸∈ V and Cγ0 ⊆ V . Therefore, ∩γ∈ΓCγ ⊆ Cγ0 ⊆ V. Thus, x ̸∈ Λp(⋆)(∩γ∈ΓCγ) and hence Λp(⋆)(∩γ∈ΓCγ) ⊆ ∩γ∈ΓΛp(⋆)(Cγ). Remark 1. In Proposition 1(6), the converse is not always true as the following example shows. Example 1. Let X = {−1, 1} with a topology τ = {∅, {−1}, X} and an ideal I = {∅, {1}}. Let A = {−1} and B = {1}. Then, Λp(⋆)(A ∩B) = Λp(⋆)(∅) = ∅ and Λp(⋆)(A) ∩ Λp(⋆)(B) = {−1}. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1027 Definition 2. A subset A of an ideal topological space (X, τ,I ) is called a Λp(⋆)-set if A = Λp(⋆)(A). The family of all Λp(⋆)-sets of an ideal topological space (X, τ,I ) is denoted by Λp(⋆)(X). Proposition 2. For subsets A and Bγ(γ ∈ Γ) of an ideal topological space (X, τ,I ), the following properties hold: (1) Λp(⋆)(A) is a Λp(⋆)-set. (2) If A is a pre-I -open set, then A is a Λp(⋆)-set. (3) If Bγ is a Λp(⋆)-set for each γ ∈ Γ, then ∪γ∈ΓBγ is a Λp(⋆)-set. (4) If Bγ is a Λp(⋆)-set for each γ ∈ Γ, then ∩γ∈ΓBγ is a Λp(⋆)-set. Proof. (1) and (2) are obvious. (3) Let Bγ be a Λp(⋆)-set for each γ ∈ Γ. Then, by Proposition 1(5), we have ∪γ∈ΓBγ = ∪γ∈ΓΛp(⋆)(Bγ) = Λp(⋆)(∪γ∈ΓBγ) ⊇ ∪γ∈ΓBγ . Thus, ∪γ∈ΓBγ = Λp(⋆)(∪γ∈ΓBγ) and hence ∪γ∈ΓBγ is a Λp(⋆)-set. (4) Let Bγ be a Λp(⋆)-set for each γ ∈ Γ. Thus, by Proposition 1(6), ∩γ∈ΓBγ = ∩γ∈ΓΛp(⋆)(Bγ) ⊇ Λp(⋆)(∩γ∈ΓBγ) ⊇ ∩γ∈ΓBγ and hence ∩γ∈ΓBγ = Λp(⋆)(∩γ∈ΓBγ). This shows that ∩γ∈ΓBγ is a Λp(⋆)-set. Proposition 3. For an ideal topological space (X, τ,I ), the pair (X,Λp(⋆)(X)) is an Alexandroff space. Proof. (1) ∅, X ∈ Λp(⋆)(X) since ∅, X ∈ pIO(X, τ) and pIO(X, τ) ⊆ Λp(⋆)(X). (2) Let Vγ ∈ Λp(⋆)(X) for each γ ∈ Γ. Then, we have ∪γ∈Γ Vγ ∈ Λp(⋆)(X) by Proposi- tion 2(3). (3) Let Vγ ∈ Λp(⋆)(X) for each γ ∈ Γ. Then, we have ∩γ∈Γ Vγ ∈ Λp(⋆)(X) by Proposi- tion 2(4). Proposition 4. Let (X, τ,I ) be an ideal topological space. Then, Λp(⋆)(X) = ΛΛp(⋆) (X). Proof. By Proposition 2(2), pI O(X) ⊆ Λp(⋆)(X). For any subset A of X, we have ΛΛp(⋆) (A) = ∩{U | A ⊆ U ;U ∈ Λp(⋆)(X)} ⊆ ∩{U | A ⊆ U ;U ∈ pI O(X)} = Λp(⋆)(A) and hence ΛΛp(⋆) (A) ⊆ Λp(⋆)(A). On the other hand, suppose that x ̸∈ ΛΛp(⋆) (A). Then, there exists U ∈ Λp(⋆)(X) such that A ⊆ U and x ̸∈ U . Since x ̸∈ U , there exists a pre-I -open set V such that U ⊆ V and x ̸∈ V . Thus, x ̸∈ Λp(⋆)(A) and hence ΛΛp(⋆) (A) ⊇ Λp(⋆)(A). Consequently, we obtain ΛΛp(⋆) (A) = Λp(⋆)(A). Definition 3. Let A be a subset of an ideal topological space (X, τ,I ). A subset δp(⋆)(A) is defined as follows: δp(⋆)(A) = ∪{F | F ⊆ A;F is pre-I -closed}. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1028 Definition 4. A subset A of an ideal topological space (X, τ,I ) is called a δp(⋆)-set if A = δp(⋆)(A). The family of all δp(⋆)-sets of an ideal topological space (X, τ,I ) is denoted by δp(⋆)(X). Proposition 5. For subsets A, B and Cγ(γ ∈ Γ) of an ideal topological space (X, τ,I ), the following properties hold: (1) δp(⋆)(A) ⊆ A. (2) If A ⊆ B, then δp(⋆)(A) ⊆ δp(⋆)(B). (3) δp(⋆)(δp(⋆)(A)) = δp(⋆)(A). (4) If A is a pre-I -closed set, then δp(⋆)(A) = A. (5) δp(⋆)(∩{Cγ |γ ∈ Γ}) = ∩{δp(⋆)(Cγ)|γ ∈ Γ}. (6) δp(⋆)(∪{Cγ |γ ∈ Γ}) ⊇ ∪{δp(⋆)(Cγ)|γ ∈ Γ}. (7) Λp(⋆)(X −A) = X − δp(⋆)(A) and δp(⋆)(X −A) = X − Λp(⋆)(A). Proposition 6. For subsets A and Bγ(γ ∈ Γ) of an ideal topological space (X, τ,I ), the following properties hold: (1) δp(⋆)(A) is a δp(⋆)-set. (2) If A is a pre-I -closed set, then A is a δp(⋆)-set. (3) If Bγ is a δp(⋆)-set for each γ ∈ Γ, then ∩γ∈ΓBγ is a δp(⋆)-set. (4) If Bγ is a δp(⋆)-set for each γ ∈ Γ, then ∪γ∈ΓBγ is a δp(⋆)-set. Proposition 7. Let A be a subset of an ideal topological space (X, τ,I ). Then, Λp(⋆)(A) = {x ∈ X | pıCl({x}) ∩A ̸= ∅}. Proof. Let x ∈ Λp(⋆)(A). Suppose that pıCl({x}) ∩ A = ∅. Then, x ̸∈ X − pıCl({x}) which is a pre-I -open set containing A. This is a contradiction. Thus, pıCl({x})∩A ̸= ∅. On the other hand, let x ∈ X such that pıCl({x})∩A ̸= ∅ and suppose that x ̸∈ Λp(⋆)(A). Then, there exists a pre-I -open set U containing A and x ̸∈ U . Let y ∈ pıCl({x}) ∩ A. Then, we have y ∈ pıCl({x}) and y ∈ A ⊆ U . Therefore, U ∩ {x} ̸= ∅. This is a contradiction. Consequently, we obtain x ∈ Λp(⋆)(A). Definition 5. Let A be a subset of an ideal topological space (X, τ,I ) and x ∈ X. Then ≺ x ≻p(⋆) is defined by ≺ x ≻p(⋆)= pıCl({x}) ∩ Λp(⋆)({x}). Proposition 8. For an ideal topological space (X, τ,I ), the following properties hold: (1) For each x ∈ X, Λp(⋆)(≺ x ≻p(⋆)) = Λp(⋆)({x}). C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1029 (2) For each x ∈ X, pıCl(≺ x ≻p(⋆)) = pıCl({x}). (3) For each pre-I -open set V and each x ∈ V , ≺ x ≻p(⋆)⊆ V . (4) For each pre-I -closed set F and each x ∈ F , ≺ x ≻p(⋆)⊆ F . Proof. (1) Let x ∈ X. Then, {x} ⊆ pıCl({x})∩Λp(⋆)({x}) =≺ x ≻p(⋆), by Proposition 1(2), Λp(⋆)(≺ x ≻p(⋆)) ⊇ Λp(⋆)({x}). On the other hand, suppose that y ̸∈ Λp(⋆)({x}). Then, there exists a pre-I open set U such that x ∈ U and y ̸∈ U . Since ≺ x ≻p(⋆)⊆ Λp(⋆)({x}) ⊆ Λp(⋆)(U) = U, we have Λp(⋆)(≺ x ≻p(⋆)) ⊆ U and hence y ̸∈ Λp(⋆)(≺ x ≻p(⋆)). Thus, Λp(⋆)(≺ x ≻p(⋆)) ⊆ Λp(⋆)({x}). Consequently, we obtain Λp(⋆)(≺ x ≻p(⋆)) = Λp(⋆)({x}). (2) Let x ∈ X. Since {x} ⊆≺ x ≻p(⋆), we have pıCl({x}) ⊆ pıCl(≺ x ≻p(⋆)). On the other hand, we have ≺ x ≻p(⋆)⊆ pıCl({x}) and so pıCl(≺ x ≻p(⋆)) ⊆ pıCl(pıCl({x})) = pıCl({x}). Thus, pıCl(≺ x ≻p(⋆)) = pıCl({x}). (3) Let V be any pre-I -open set and x ∈ V . Then, Λp(⋆)(≺ x ≻p(⋆)) ⊆ V and hence ≺ x ≻p(⋆)⊆ V . (4) Let F be any pre-I -closed set and x ∈ F . Therefore, we have ≺ x ≻p(⋆)= pıCl({x}) ∩ Λp(⋆)({x}) ⊆ pıCl({x}) ⊆ pıCl(F ) = F. 4. (Λ, p(⋆))-closed sets In this section, we introduce the concept of (Λ, p(⋆))-closed sets and investigate some properties of (Λ, p(⋆))-closed sets. Definition 6. A subset A of an ideal topological space (X, τ,I ) is said to be (Λ, p(⋆))- closed if A = T ∩C, where T is a Λp(⋆)-set and C is a pre-I -closed set. The collection of all (Λ, p(⋆))-closed sets in an ideal topological space (X, τ,I ) is denoted by (Λ, p(⋆))C(X). Theorem 1. For a subset A of an ideal topological space (X, τ,I ), the following properties are equivalent: (1) A is (Λ, p(⋆))-closed. (2) A = T ∩ pıCl(A), where T is a Λp(⋆)-set. (3) A = Λp(⋆)(A) ∩ pıCl(A). C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1030 Proof. (1) ⇒ (2): Suppose that A = T ∩ C, where T is a Λp(⋆)-set and C is a pre- I -closed set. Since A ⊆ C, we have pıCl(A) ⊆ C and A = T ∩ C ⊇ T ∩ pıCl(A) ⊇ A. Consequently, we obtain A = T ∩ pıCl(A). (2) ⇒ (3): Suppose that A = T ∩ pıCl(A), where T is a Λp(⋆)-set. Since A ⊆ T , we have Λp(⋆)(A) ⊆ Λp(⋆)(T ) = T and hence A ⊆ Λp(⋆)(A) ∩ pıCl(A) ⊆ T ∩ pıCl(A) = A. Thus, A = Λp(⋆)(A) ∩ pıCl(A). (3) ⇒ (1): Since Λp(⋆)(A) is Λp(⋆)-set, pıCl(A) is a pre-I -closed set and A = Λp(⋆)(A) ∩ pıCl(A). This shows that A is (Λ, p(⋆))-closed. Remark 2. Every Λp(⋆)-set (resp. pre-I -closed set) is (Λ, p(⋆))-closed. The converse of Remark 2 is not true in general as shown by the following example. Example 2. Let X = {1, 2} with a topology τ = {∅, {1}, X} and an ideal I = {∅, {1}}. Let A = {1}. Then, A is (Λ, p(⋆))-closed but it is not pre-I -closed. Moreover, let B = {2}, then B is (Λ, p(⋆))-closed but it is not a Λp(⋆)-set. Definition 7. A subset A of an ideal topological space (X, τ,I ) is said to be (Λ, p(⋆))- open if the complement of A is (Λ, p(⋆))-closed. The collection of all (Λ, p(⋆))-open sets in an ideal topological space (X, τ,I ) is denoted by Λp(⋆)O(X). Theorem 2. Let Aγ(γ ∈ Γ) be a subset of an ideal topological space (X, τ,I ). Then, the following properties hold: (1) If Aγ is (Λ, p(⋆))-closed for each γ ∈ Γ, then ∩{Aγ | γ ∈ Γ} is (Λ, p(⋆))-closed. (2) If Aγ is (Λ, p(⋆))-open for each γ ∈ Γ, then ∪{Aγ | γ ∈ Γ} is (Λ, p(⋆))-open. Proof. (1) Suppose that Aγ is (Λ, p(⋆))-closed for each γ ∈ Γ. Then, for each γ, there exist a Λp(⋆)-set Tγ and a pre-I -closed set Cγ such that Aγ = Tγ ∩ Cγ . Thus, ∩γ∈ΓAγ = ∩γ∈Γ(Tγ ∩ Cγ) = (∩γ∈ΓTγ) ∩ (∩γ∈ΓCγ). Since ∩γ∈ΓCγ is pre-I -closed and ∩γ∈ΓTγ is a Λp(⋆)-set by Proposition 2(4), we have ∩γ∈ΓAγ is (Λ, p(⋆))-closed. (2) Let Aγ is (Λ, p(⋆))-open for each γ ∈ Γ. Then, X − Aγ is (Λ, p(⋆))-closed. Since X − ∪γ∈ΓAγ = ∩γ∈Γ(X −Aγ) and by (1), ∪γ∈ΓAγ is (Λ, p(⋆))-open. Theorem 3. For a subset A of an ideal topological space (X, τ,I ), the following properties are equivalent: (1) A is (Λ, p(⋆))-open. (2) A = S ∪ V , where S is a δp(⋆)-set and V is a pre-I -open set. (3) A = S ∪ pıInt(A), where S is a δp(⋆)-set. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1031 (4) A = δp(⋆)(A) ∪ pıInt(A). Proof. The proof follows from Theorem 1. Definition 8. Let A be a subset of an ideal topological space (X, τ,I ). A point x ∈ X is called a (Λ, p(⋆))-cluster point of A if A∩U ̸= ∅ for every (Λ, p(⋆))-open set U containing x. The set of all (Λ, p(⋆))-cluster points of A is called the (Λ, p(⋆))-closure of A and is denoted by A(Λ,p(⋆)). Lemma 3. Let A and B be subsets of an ideal topological space (X, τ,I ). For the (Λ, p(⋆))-closure, the following properties hold: (1) A ⊆ A(Λ,p(⋆)) and [A(Λ,p(⋆))](Λ,p(⋆)) = A(Λ,p(⋆)). (2) A(Λ,p(⋆)) = ∩{F | A ⊆ F and F is (Λ, p(⋆))-closed }. (3) If A ⊆ B, then A(Λ,p(⋆)) ⊆ B(Λ,p(⋆)). (4) A is (Λ, p(⋆))-closed if and only if A(Λ,p(⋆)) = A. (5) A(Λ,p(⋆)) is (Λ, p(⋆))-closed. Definition 9. Let A be a subset of an ideal topological space (X, τ,I ). The union of all (Λ, p(⋆))-open sets contained in A is called the (Λ, p(⋆))-interior of A and is denoted by A(Λ,p(⋆)). Lemma 4. Let A and B be subsets of an ideal topological space (X, τ,I ). For the (Λ, p(⋆))-interior, the following properties hold: (1) [A(Λ,p(⋆))](Λ,p(⋆)) = A(Λ,p(⋆)). (2) If A ⊆ B, then A(Λ,p(⋆)) ⊆ B(Λ,p(⋆)). (4) A is (Λ, p(⋆))-open if and only if A(Λ,p(⋆)) = A. (5) A(Λ,p(⋆)) is (Λ, p(⋆))-open. Definition 10. A subset A of an ideal topological space (X, τ,I ) is called semi-(Λ, p(⋆))- open (resp. pre-(Λ, p(⋆))-open, α-(Λ, p(⋆))-open, β-(Λ, p(⋆))-open) if A ⊆ [A(Λ,p(⋆))] (Λ,p(⋆)) (resp. A ⊆ [A(Λ,p(⋆))](Λ,p(⋆)), A ⊆ [[A(Λ,p(⋆))] (Λ,p(⋆))](Λ,p(⋆)), A ⊆ [[A(Λ,p(⋆))](Λ,p(⋆))] (Λ,p(⋆))). The complement of a semi-(Λ, p(⋆))-open (resp. pre-(Λ, p(⋆))-open, α-(Λ, p(⋆))-open, β- (Λ, p(⋆))-open) set is called semi-(Λ, p(⋆))-closed (resp. pre-(Λ, p(⋆))-closed, α-(Λ, p(⋆))- closed, β-(Λ, p(⋆))-closed). Definition 11. An ideal topological space (X, τ,I ) is called (Λ, p(⋆))-extremally discon- nected if the (Λ, p(⋆))-closure of every (Λ, p(⋆))-open set of X is (Λ, p(⋆))-open. Example 3. Let X = {a, b, c} with a topology τ = {∅, {a}, {a, b}, {a, c}, X} and an ideal I = {∅, {c}}. Then (X, τ,I ) is a (Λ, p(⋆))-extremally disconnected space. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1032 Theorem 4. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is (Λ, p(⋆))-extremally disconnected. (2) F(Λ,p(⋆)) is (Λ, p(⋆))-closed for every (Λ, p(⋆))-closed set F of X. (3) [A(Λ,p(⋆))] (Λ,p(⋆)) ⊆ [A(Λ,p(⋆))](Λ,p(⋆)) for every subset A of X. (4) Every semi-(Λ, p(⋆))-open set is pre-(Λ, p(⋆))-open. (5) The (Λ, p(⋆))-closure of every β-(Λ, p(⋆))-open set of X is (Λ, p(⋆))-open. (6) Every β-(Λ, p(⋆))-open set is pre-(Λ, p(⋆))-open. (7) For every subset A of X, A is α-(Λ, p(⋆))-open if and only if it is semi-(Λ, p(⋆))- open. Proof. (1) ⇒ (2): Let A be any (Λ, p(⋆))-closed set. Then, X − A is (Λ, p(⋆))-open and by (1), we have (X − A)(Λ,p(⋆)) = X − A(Λ,p(⋆)) is (Λ, p(⋆))-open. Thus, A(Λ,p(⋆)) is (Λ, p(⋆))-closed. (2) ⇒ (3): Let A be any subset of X. Then, X − A(Λ,p(⋆)) is (Λ, p(⋆))-closed and by (2), [X − A(Λ,p(⋆))](Λ,p(⋆)) is (Λ, p(⋆))-closed. Therefore, we have [A(Λ,p(⋆))] (Λ,p(⋆)) is (Λ, p(⋆))-open and hence [A(Λ,p(⋆))] (Λ,p(⋆)) ⊆ [A(Λ,p(⋆))](Λ,p(⋆)). (3) ⇒ (4): Let V be any semi-(Λ, p(⋆))-open set. By (3), we have V ⊆ [V(Λ,p(⋆))] (Λ,p(⋆)) ⊆ [V (Λ,p(⋆))](Λ,p(⋆)) and hence V is pre-(Λ, p(⋆))-open. (4) ⇒ (5): Let V be any β-(Λ, p(⋆))-open set. Then, V (Λ,p(⋆)) is semi-(Λ, p(⋆))-open and by (4), we have V (Λ,p(⋆)) is pre-(Λ, p(⋆))-open. Thus, V (Λ,p(⋆)) ⊆ [V (Λ,p(⋆))](Λ,p(⋆)) and hence V (Λ,p(⋆)) is (Λ, p(⋆))-open. (5) ⇒ (6): Let V be any β-(Λ, p(⋆))-open set. By (5), V (Λ,p(⋆)) = [V (Λ,p(⋆))](Λ,p(⋆)). Therefore, V ⊆ V (Λ,p(⋆)) = [V (Λ,p(⋆))](Λ,p(⋆)) and hence V is pre-(Λ, p(⋆))-open. (6) ⇒ (7): Let V be any semi-(Λ, p(⋆))-open set. Then, V is β-(Λ, p(⋆))-open and by (6), V is pre-(Λ, p(⋆))-open. Since V is semi-(Λ, p(⋆))-open and pre-(Λ, p(⋆))-open, V is α-(Λ, p(⋆))-open. (7) ⇒ (1): Let V be any (Λ, p(⋆))-open set. Then, V (Λ,p(⋆)) is semi-(Λ, p(⋆))-open and by (7), we have V (Λ,p(⋆)) is α-(Λ, p(⋆))-open. Thus, V (Λ,p(⋆)) ⊆ [[[V (Λ,p(⋆))](Λ,p(⋆))] (Λ,p(⋆))](Λ,p(⋆)) = [V (Λ,p(⋆))](Λ,p(⋆)) and hence V (Λ,p(⋆)) = [V (Λ,p(⋆))](Λ,p(⋆)). Therefore, V (Λ,p(⋆)) is (Λ, p(⋆))-open. This shows that (X, τ,I ) is (Λ, p(⋆))-extremally disconnected. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1033 Definition 12. An ideal topological space (X, τ,I ) is said to be (Λ, p(⋆))-normal if, for any pair disjoint (Λ, p(⋆))-open sets U and V , there exist disjoint (Λ, p(⋆))-closed sets F and H such that U ⊆ F and V ⊆ H. Theorem 5. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is (Λ, p(⋆))-normal. (2) (X, τ,I ) is (Λ, p(⋆))-extremally disconnected. Proof. (1) ⇒ (2): Let U be any (Λ, p(⋆))-open set. Then, we have U and V = X − U (Λ,p(⋆)) are disjoint (Λ, p(⋆))-open sets. There exist disjoint (Λ, p(⋆))-closed sets F and H such that U ⊆ F and V ⊆ H. Since U (Λ,p(⋆)) ⊆ F (Λ,p(⋆)) = F ⊆ X −H ⊆ X − V = U (Λ,p(⋆)), we have U (Λ,p(⋆)) = F . Since V ⊆ H ⊆ X − F = V , V = H. Thus, U (Λ,p(⋆)) = X −H is (Λ, p(⋆))-open. This shows that (X, τ,I ) is (Λ, p(⋆))-extremally disconnected. (2) ⇒ (1): Let U and V be any two disjoint (Λ, p(⋆))-open sets. Then, U (Λ,p(⋆)) and X − U (Λ,p(⋆)) are disjoint (Λ, p(⋆))-closed sets containing U and V , respectively. Thus, (X, τ,I ) is (Λ, p(⋆))-normal. Definition 13. A subset A of an ideal topological space (X, τ,I ) is said to be: (1) regular (Λ, p(⋆))-open if A = [A(Λ,p(⋆))](Λ,p(⋆)); (2) regular (Λ, p(⋆))-closed if A = [A(Λ,p(⋆))] (Λ,p(⋆)). Theorem 6. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is (Λ, p(⋆))-extremally disconnected. (2) Every regular (Λ, p(⋆))-open set is (Λ, p(⋆))-closed. (3) Every regular (Λ, p(⋆))-closed set is (Λ, p(⋆))-open. Proof. (1) ⇒ (2): Suppose that (X, τ,I ) is (Λ, p(⋆))-extremally disconnected. Let V be any regular (Λ, p(⋆))-open set. Then, we have V = [V (Λ,p(⋆))](Λ,p(⋆)). Since V (Λ,p(⋆)) is (Λ, p(⋆))-open, V = [V (Λ,p(⋆))](Λ,p(⋆)) = V (Λ,p(⋆)) and hence V is (Λ, p(⋆))-closed. (2) ⇒ (1): Suppose that every regular (Λ, p(⋆))-open set is (Λ, p(⋆))-closed. Let V be any (Λ, p(⋆))-open set. Then, we have [V (Λ,p(⋆))](Λ,p(⋆)) is regular (Λ, p(⋆))-open and by (2), [V (Λ,p(⋆))](Λ,p(⋆)) is closed. Thus, V (Λ,p(⋆)) ⊆ [[V (Λ,p(⋆))](Λ,p(⋆))] (Λ,p(⋆)) = [V (Λ,p(⋆))](Λ,p(⋆)) and hence V (Λ,p(⋆)) is (Λ, p(⋆))-open. Therefore, (X, τ,I ) is (Λ, p(⋆))-extremally discon- nected. (2) ⇔ (3): This is obvious. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1034 Theorem 7. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is (Λ, p(⋆))-extremally disconnected. (2) The (Λ, p(⋆))-closure of every semi-(Λ, p(⋆))-open set is (Λ, p(⋆))-open. (3) The (Λ, p(⋆))-closure of every pre-(Λ, p(⋆))-open set is (Λ, p(⋆))-open. (4) The (Λ, p(⋆))-closure of every regular (Λ, p(⋆))-open set is (Λ, p(⋆))-open. Proof. (1) ⇒ (2): Let V be any semi-(Λ, p(⋆))-open set. Then, V is β-(Λ, p(⋆))-open and by Theorem 4, V (Λ,p(⋆)) is (Λ, p(⋆))-open. (2) ⇒ (4): Let V be any regular (Λ, p(⋆))-open set. Then, V (Λ,p(⋆)) is semi-(Λ, p(⋆))- open and by (2), we have [V (Λ,p(⋆))](Λ,p(⋆)) = V (Λ,p(⋆)) is (Λ, p(⋆))-open. (4) ⇒ (1): Let V be any (Λ, p(⋆))-open set. Then, we have [V (Λ,p(⋆))](Λ,p(⋆)) is regular (Λ, p(⋆))-open and by (4), [[V (Λ,p(⋆))](Λ,p(⋆))] (Λ,p(⋆)) is (Λ, p(⋆))-open. Thus, V (Λ,p(⋆)) ⊆ [[V (Λ,p(⋆))](Λ,p(⋆))] (Λ,p(⋆)) = [[[V (Λ,p(⋆))](Λ,p(⋆))] (Λ,p(⋆))](Λ,p(⋆)) = [V (Λ,p(⋆))](Λ,p(⋆)) and hence V (Λ,p(⋆)) is (Λ, p(⋆))-open. Therefore, (X, τ,I ) is (Λ, p(⋆))-extremally discon- nected. (1) ⇒ (3): Let V be any pre-(Λ, p(⋆))-open set. Then, V is β-(Λ, p(⋆))-open and by Theorem 4, V (Λ,p(⋆)) is (Λ, p(⋆))-open. (3) ⇒ (4): Let V be any regular (Λ, p(⋆))-open set. Then, we have V is pre-(Λ, p(⋆))- open, by (3), V (Λ,p(⋆)) is (Λ, p(⋆))-open. 5. Characterizations of (Λ, p(⋆))-continuous functions In this section, we introduce the notion of (Λ, p(⋆))-continuous functions. In particular, several characterizations of (Λ, p(⋆))-continuous functions are investigated. Definition 14. A function f : (X, τ,I ) → (Y, σ,J ) is called (Λ, p(⋆))-continuous at a point x ∈ X if, for each (Λ, p(⋆))-open set V of Y containing f(x), there exists a (Λ, p(⋆))- open set U of X containing x such that f(U) ⊆ V . A function f : (X, τ,I ) → (Y, σ,J ) is called (Λ, p(⋆))-continuous if f has this property at each point x ∈ X. Example 4. Let X = {a, b, c} with a topology τ = {∅, {a}, X} and an ideal I = {∅, {a}}. Let Y = {1, 2, 3} with a topology σ = {∅, {1}, {2}, {1, 2}, Y } and an ideal J = {∅, {2}}. A function f : (X, τ,I ) → (Y, σ,J ) is defined as follows: f(a) = 1, f(b) = 2 and f(c) = 3. Then, f is (Λ, p(⋆))-continuous. Lemma 5. Let A be a subset of an ideal topological space (X, τ,I ). Then, x ∈ A(Λ,p(⋆)) if and only if A ∩ U ̸= ∅ for every (Λ, p(⋆))-open set U of X containing x. Theorem 8. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1035 (1) f is (Λ, p(⋆))-continuous at x ∈ X. (2) x ∈ [f−1(V )](Λ,p(⋆)) for every (Λ, p(⋆))-open set V of Y containing f(x). (3) x ∈ f−1([f(A)](Λ,p(⋆))) for every subset A of X such that x ∈ A(Λ,p(⋆)). (4) x ∈ f−1(B(Λ,p(⋆))) for every subset B of Y such that x ∈ [f−1(B)](Λ,p(⋆)). (5) x ∈ [f−1(B)](Λ,p(⋆)) for every subset B of Y such that x ∈ f−1(B(Λ,p(⋆))). (6) x ∈ f−1(K) for every (Λ, p(⋆))-closed set K of Y such that x ∈ [f−1(K)](Λ,p(⋆)). Proof. (1) ⇒ (2): Let V be any (Λ, p(⋆))-open set of Y containing f(x). Then, there exists a (Λ, p(⋆))-open set U of X containing x such that f(U) ⊆ V . Thus, U ⊆ f−1(V ). Since U is (Λ, p(⋆))-open in X, we have x ∈ [f−1(V )](Λ,p(⋆)). (2) ⇒ (3): Let A be any subset of X and x ∈ A(Λ,p(⋆)). Let V be any (Λ, p(⋆))-open set of Y containing f(x). By (2), x ∈ [f−1(V )](Λ,p(⋆)) and there exists a (Λ, p(⋆))-open set U of X such that x ∈ U ⊆ f−1(V ). Since x ∈ A(Λ,p(⋆)), by Lemma 5, U ∩A ̸= ∅ and ∅ ≠ f(U ∩A) ⊆ f(U) ∩ f(A) ⊆ V ∩ f(A). Thus, f(x) ∈ [f(A)](Λ,p(⋆)) and hence x ∈ f−1([f(A)](Λ,p(⋆))). (3) ⇒ (4): Let B be any subset of Y and x ∈ [f−1(B)](Λ,p(⋆)). By (3), x ∈ f−1([f(f−1(B))](Λ,p(⋆))) ⊆ f−1(B(Λ,p(⋆))) and hence x ∈ f−1(B(Λ,p(⋆))). (4) ⇒ (5): Let B be any subset of Y such that x ̸∈ [f−1(B)](Λ,p(⋆)). Then, x ∈ X − [f−1(B)](Λ,p(⋆)) = [X − f−1(B)](Λ,p(⋆)) = [f−1(Y −B)](Λ,p(⋆)) and by (4), x ∈ f−1([Y − B](Λ,p(⋆))) = f−1(Y − B(Λ,p(⋆))) = X − f−1(B(Λ,p(⋆))). Thus, x ̸∈ f−1(B(Λ,p(⋆))). (5) ⇒ (6): Let K be any (Λ, p(⋆))-closed set of Y such that x ̸∈ f−1(K). Then, we have x ∈ X − f−1(K) = f−1(Y −K) = f−1((Y −K)(Λ,p(⋆))), by (5), x ∈ [f−1(Y −K)](Λ,p(⋆)) = [X − f−1(K)](Λ,p(⋆)) = X − [f−1(K)](Λ,p(⋆)) and hence x ̸∈ [f−1(K)](Λ,p(⋆)). (6) ⇒ (2): Let x ∈ X and let V be any (Λ, p(⋆))-open set of Y containing f(x). Suppose that x ̸∈ [f−1(V )](Λ,p(⋆)). Then, x ∈ X − [f−1(V )](Λ,p(⋆)) = [X − f−1(V )](Λ,p(⋆)) = [f−1(Y − V )](Λ,p(⋆)). C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1036 By (6), we have x ∈ f−1(Y − V ) = X − f−1(V ) and hence x ̸∈ f−1(V ). This contraries to the hypothesis. (2) ⇒ (1): Let V be any (Λ, p(⋆))-open set of Y containing f(x). By (2), we have x ∈ [f−1(V )](Λ,p(⋆)) and so there exists a (Λ, p(⋆))-open set U of X containing x such that x ∈ U ⊆ f−1(V ); hence f(U) ⊆ V . This shows that f is (Λ, p(⋆))-continuous at x. Theorem 9. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) f is (Λ, p(⋆))-continuous. (2) f−1(V ) is (Λ, p(⋆))-open in X for every (Λ, p(⋆))-open set V of Y . (3) f(A(Λ,p(⋆))) ⊆ [f(A)](Λ,p(⋆)) for every subset A of X. (4) [f−1(B)](Λ,p(⋆)) ⊆ f−1(B(Λ,p(⋆))) for every subset B of Y . (5) f−1(B(Λ,p(⋆))) ⊆ [f−1(B)](Λ,p(⋆)) for every subset B of Y . (6) f−1(K) is (Λ, p(⋆))-closed in X for every (Λ, p(⋆))-closed set K of Y . Proof. (1) ⇒ (2): Let V be any (Λ, p(⋆))-open set of Y such that x ∈ f−1(V ). Then, f(x) ∈ V and there exists a (Λ, p(⋆))-open set U of X containing x such that f(U) ⊆ V . Since U is (Λ, p(⋆))-open in X, x ∈ [f−1(V )](Λ,p(⋆)) and hence f−1(V ) ⊆ [f−1(V )](Λ,p(⋆)). Thus, f−1(V ) is (Λ, p(⋆))-open. (2) ⇒ (3): Let A be any subset of X. Let x ∈ A(Λ,p(⋆)) and let V be any (Λ, p(⋆))- open set of Y containing f(x). By (2), we have x ∈ [f−1(V )](Λ,p(⋆)) and there exists a (Λ, p(⋆))-open set U of X such that x ∈ U ⊆ f−1(V ). Since x ∈ A(Λ,p(⋆)), by Lemma 5, U ∩ A ̸= ∅ and ∅ ̸= f(U ∩ A) ⊆ f(U) ∩ f(A) ⊆ V ∩ f(A). Thus, f(x) ∈ [f(A)](Λ,p(⋆)) and hence f(A(Λ,p(⋆))) ⊆ [f(A)](Λ,p(⋆)). (3) ⇒ (4): Let B be any subset of Y . By (3), f([f−1(B)](Λ,p(⋆))) ⊆ [f(f−1(B))](Λ,p(⋆)) ⊆ B(Λ,p(⋆)). Therefore, [f−1(B)](Λ,p(⋆)) ⊆ f−1(B(Λ,p(⋆))). (4) ⇒ (5): Let B be any subset of Y . By (4), we have X − [f−1(B)](Λ,p(⋆)) = [X − f−1(B)](Λ,p(⋆)) = [f−1(Y −B)](Λ,p(⋆)) ⊆ f−1([Y −B](Λ,p(⋆))) = f−1(Y −B(Λ,p(⋆))) = X − f−1(B(Λ,p(⋆))) and hence f−1(B(Λ,p(⋆))) ⊆ [f−1(B)](Λ,p(⋆)). C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1037 (5) ⇒ (6): Let K be any (Λ, p(⋆))-closed set of Y . Then, Y −K = [Y −K](Λ,p(⋆)) and by (5), X − f−1(K) = f−1(Y −K) = f−1([Y −K](Λ,p(⋆))) ⊆ [f−1(Y −K)](Λ,p(⋆)) = [X − f−1(K)](Λ,p(⋆)) = X − [f−1(K)](Λ,p(⋆)). Thus, [f−1(K)](Λ,p(⋆)) ⊆ f−1(K) and hence f−1(K) is (Λ, p(⋆))-closed. (6) ⇒ (2): The proof is obvious. (2) ⇒ (1): Let x ∈ X and let V be any (Λ, p(⋆))-open set of Y containing f(x). By (2), x ∈ [f−1(V )](Λ,p(⋆)) and so there exists a (Λ, p(⋆))-open set U of X containing x such that x ∈ U ⊆ f−1(V ); hence f(U) ⊆ V . Thus, f is (Λ, p(⋆))-continuous at x. This shows that f is (Λ, p(⋆))-continuous. Definition 15. An ideal topological space (X, τ,I ) is said to be (Λ, p(⋆))-connected if X cannot be written as a disjoint union of two nonempty (Λ, p(⋆))-open sets of X. Example 5. Let X = {a, b, c} with a topology τ = {∅, {a, b}, X} and an ideal I = {∅, {a}, {b}, {a, b}}. Then, (X, τ,I ) is (Λ, p(⋆))-connected. Proposition 9. If f : (X, τ,I ) → (Y, σ,J ) is a (Λ, p(⋆))-continuous surjection and (X, τ,I ) is (Λ, p(⋆))-connected, then (Y, σ,J ) is (Λ, p(⋆))-connected. Proof. Suppose that (Y, σ,J ) is not (Λ, p(⋆))-connected. There exist nonempty (Λ, p(⋆))-open sets U and V of Y such that U ∩ V = ∅ and U ∪ V = Y . Then, we have f−1(U)∩ f−1(V ) = ∅ and f−1(U)∪ f−1(V ) = X. Moreover, f−1(U) and f−1(V ) are nonempty (Λ, p(⋆))-open sets of X. This shows that (X, τ,I ) is not (Λ, p(⋆))-connected. Definition 16. An ideal topological space (X, τ,I ) is said to be (Λ, p(⋆))-compact if every cover of X by (Λ, p(⋆))-open sets of X has a finite subcover. Proposition 10. If f : (X, τ,I ) → (Y, σ,J ) is a (Λ, p(⋆))-continuous surjection and (X, τ,I ) is (Λ, p(⋆))-compact, then (Y, σ,J ) is (Λ, p(⋆))-compact. Proof. Let {Vγ | γ ∈ Γ} be any cover of Y by (Λ, p(⋆))-open sets of Y . Since f is (Λ, p(⋆))-continuous, by Theorem 9, {f−1(Vγ) | γ ∈ Γ} is a cover of X by (Λ, p(⋆))-open sets of X. Thus, there exists a finite subset Γ0 of Γ such that X = ∪{f−1(Vγ) | γ ∈ Γ0}. Since f is surjective, Y = f(X) = ∪{Vγ | γ ∈ Γ0}. This shows that (Y, σ,J ) is (Λ, p(⋆))- compact. Definition 17. A subset A of an ideal topological space (X, τ,I ) is said to be a (Λ, p(⋆))- neighbourhood of x if there exists a (Λ, p(⋆))-open set U of X such that x ∈ U ⊆ A. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1038 Definition 18. Let A be a subset of an ideal topological space (X, τ,I ). A subset Λ(Λ,p(⋆))(A) is defined as follows: Λ(Λ,p(⋆))(A) = ∩{U | A ⊆ U ;U is (Λ, p(⋆))-open}. Proposition 11. For subsets A, B and Cγ(γ ∈ Γ) of an ideal topological space (X, τ,I ), the following properties hold: (1) A ⊆ Λ(Λ,p(⋆))(A). (2) If A ⊆ B, then Λ(Λ,p(⋆))(A) ⊆ Λ(Λ,p(⋆))(B). (3) Λ(Λ,p(⋆))(Λ(Λ,p(⋆))(A)) = Λ(Λ,p(⋆))(A). (4) If A is a (Λ, p(⋆))-open set, then Λ(Λ,p(⋆))(A) = A. (5) Λ(Λ,p(⋆))(∩{Cγ |γ ∈ Γ}) ⊆ ∩{Λ(Λ,p(⋆))(Cγ)|γ ∈ Γ}. (6) Λ(Λ,p(⋆))(∪{Cγ |γ ∈ Γ}) = ∪{Λ(Λ,p(⋆))(Cγ)|γ ∈ Γ}. Lemma 6. Let A be a subset of an ideal topological space (X, τ,I ) and x ∈ X. Then, x ∈ Λ(Λ,p(⋆))(A) if and only if A∩F ̸= ∅ for every (Λ, p(⋆))-closed set F of X with x ∈ F . Theorem 10. For a function f : (X, τ,I ) → (Y, σ,J ), the following properties are equivalent: (1) f is (Λ, p(⋆))-continuous. (2) For each x ∈ X and each (Λ, p(⋆))-open set V of Y such that f(x) ∈ V , f−1(V ) is a (Λ, p(⋆))-neighbourhood of x. (3) f(A(Λ,p(⋆))) ⊆ Λ(Λ,p(⋆))(f(A)) for every subset A of X. (4) [f−1(B)](Λ,p(⋆)) ⊆ f−1(Λ(Λ,p(⋆))(B)) for every subset B of Y . Proof. (1) ⇒ (2): Let x ∈ X and let V be any (Λ, p(⋆))-open set of Y such that f(x) ∈ V . Since f is (Λ, p(⋆))-continuous, there exists a (Λ, p(⋆))-open set U of X con- taining x such that f(U) ⊆ V . Thus, x ∈ U ⊆ f−1(V ) and hence f−1(V ) is a (Λ, p(⋆))- neighbourhood of x. (2) ⇒ (1): Let x ∈ X and let V be any (Λ, p(⋆))-open set of Y containing f(x). By (2), f−1(V ) is a (Λ, p(⋆))-neighbourhood of x and there exists a (Λ, p(⋆))-open set U of X such that x ∈ U ⊆ f−1(V ). Thus, f(U) ⊆ V and hence f is (Λ, p(⋆))-continuous. (1) ⇒ (3): Let A be any subset of X and let y ̸∈ Λ(Λ,p(⋆))(f(A)). By Lemma 6, there exists a (Λ, p(⋆))-closed set F of Y such that y ∈ F and f(A) ∩ F = ∅. Thus, A ∩ f−1(F ) = ∅ and hence f−1(F ) ∩ A(Λ,p(⋆)) = ∅. Therefore, f(A(Λ,p(⋆))) ∩ F = ∅. This shows that y ̸∈ f(A(Λ,p(⋆))). Consequently, we obtain f(A(Λ,p(⋆))) ⊆ Λ(Λ,p(⋆))(f(A)). (3) ⇒ (4): Let B be any subset of Y . By (3) and Proposition 11(2), we have f([f−1(B)](Λ,p(⋆))) ⊆ Λ(Λ,p(⋆))(f(f −1(B))) ⊆ Λ(Λ,p(⋆))(B) and hence [f−1(B)](Λ,p(⋆)) ⊆ f−1(Λ(Λ,p(⋆))(B)). (4) ⇒ (1): Let V be any (Λ, p(⋆))-open set of Y . By (4) and Proposition 11(4), [f−1(V )](Λ,p(⋆)) ⊆ f−1(Λ(Λ,p(⋆))(V )) = f−1(V ) and hence [f−1(V )](Λ,p(⋆)) = f−1(V ). Thus, f−1(V ) is (Λ, p(⋆))-open, by Theorem 9, f is (Λ, p(⋆))-continuous. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1039 6. Some low separation axioms We begin this section by introducing some low separation axioms. Definition 19. An ideal topological space (X, τ,I ) is said to be: (i) pre-I -T0 if, for each pair of distinct points of X, there exists a pre-I -open set containing one of the points but not the other; (ii) pre-I -T1 if, for each pair of distinct points x and y of X, there exists a pair of pre-I -open sets one containing x but not y and the other containing y but not x; (iii) pre-I -R0 if every pre-I -open set contains the pre-I -closure of each of its single- tons. Example 6. Let X = {a, b, c} with a topology τ = {∅, {b}, {b, c}, X} and an ideal I = {∅, {b}}. Then, (X, τ,I ) is a pre-T1 space. Remark 3. For an ideal topological space (X, τ,I ), the following implications hold: pre-I -R0 ⇐ pre-I -T1 ⇒ pre-I -T0. The following examples show that these implications are not reversible. Example 7. Let (X, τ,I ) be the same ideal topological space as in Example 2. Then, (X, τ,I ) is a pre-I -T0 space but (X, τ,I ) is not pre-I -T1. Example 8. Let X = {a, b} with a topology τ = {∅, X} and an ideal I = {∅, {a}, {b}, X}. Then, (X, τ,I ) is a pre-I -R0 space but (X, τ,I ) is not pre-I -T1. Theorem 11. An ideal topological space (X, τ,I ) is pre-I -T0 if and only if, for each x ∈ X, the singleton {x} is (Λ, p(⋆))-closed. Proof. Suppose that (X, τ,I ) is a pre-I -T0 space. For each x ∈ X, we have {x} ⊆ Λp(⋆)({x}) ∩ pıCl({x}). If y ̸= x, (i) there exists a pre-I -open set U such that y ̸∈ U and x ∈ U or (ii) there exists a pre-I -open set V such that x ̸∈ V and y ∈ V . In case of (i), y ̸∈ Λp(⋆)({x}) and y ̸∈ Λp(⋆)({x}) ∩ pıCl({x}). This shows that {x} ⊇ Λp(⋆)({x}) ∩ pıCl({x}). In case (ii), y ̸∈ pıCl({x}) and y ̸∈ Λp(⋆)({x}) ∩ pıCl({x}). Thus, {x} ⊇ Λp(⋆)({x}) ∩ pıCl({x}) and hence {x} = Λp(⋆)({x}) ∩ pıCl({x}). Conversely, suppose that (X, τ,I ) is not pre-I -T0. There exist two distinct points x, y of X such that (i) y ∈ U for every pre-I -open set U containing x and (ii) x ∈ V for every pre-I -open set V containing y. From (i) and (ii), we obtain y ∈ Λp(⋆)({x}) and y ∈ pıCl({x}), respectively. Thus, y ∈ Λp(⋆)({x}) ∩ pıCl({x}). By Theorem 1, {x} = Λp(⋆)({x}) ∩ pıCl({x}) since {x} is (Λ, p(⋆))-closed. This is contrary to x ̸= y. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1040 Theorem 12. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is pre-I -T1. (2) For each x ∈ X, the singleton {x} is a pre-I -closed set. (3) For each x ∈ X, the singleton {x} is a Λp(⋆)-set. Proof. (1) ⇒ (2): Let y ∈ X and x ∈ X −{y}. There exists a pre-I -open set Gx such that x ∈ Gx and y ̸∈ Gx. Therefore, we have X − {y} = ∪x∈X−{y}Gx and hence {y} is a pre-I -closed set. (2) ⇒ (3): Let x ∈ X and y ∈ X − {x}. Then, we have x ∈ X − {y} and Λp(⋆)({x}) ⊆ X − {y}. Thus, y ̸∈ Λp(⋆)({x}) and hence Λp(⋆)({x}) ⊆ {x}. This implies that Λp(⋆)({x}) = {x}. Consequently, we obtain {x} is a Λp(⋆)-set. (3) ⇒ (1): Let x and y be any distinct points of X. Then, we have y ̸∈ Λp(⋆)({x}) and so there exists a pre-I -open set U such that x ∈ U and y ̸∈ U . Similarly, x ̸∈ Λp(⋆)({y}) and there exists a pre-I -open set V such that y ∈ V and x ̸∈ V . This shows that (X, τ,I ) is a pre-I -T1 space. Corollary 1. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is pre-I -T1; (2) (X, τ,I ) is pre-I -T0 and pre-I -R0. Proof. (1) ⇒ (2): Suppose that (X, τ,I ) is pre-I -T1. By Remark 3 and Theorem 12, every pre-I -T1 space is pre-I -T0 and pre-I -R0. (2) ⇒ (1): Suppose that (X, τ,I ) is pre-I -T0 and pre-I -R0. Since (X, τ,I ) is pre-I -T0, for any distinct points x and y of X, there exists a pre-I -open set V such that x ∈ V and y ̸∈ V . Since (X, τ,I ) is pre-I -R0, we have pıCl({x}) ⊆ V . Thus, x ̸∈ X − pıCl({x}) and hence y ∈ X − V ⊆ X − pıCl({x}). Therefore, (X, τ,I ) is pre-I -T1. Lemma 7. An ideal topological space (X, τ,I ) is pre-I -R0 if and only if, for each pre- I -open set U , x ∈ U implies Cl(Int⋆({x})) ⊆ U . Proof. Let U be any pre-I -open set and x ∈ U . Then, we have pıCl({x}) ⊆ U and by Lemma 2, Cl(Int⋆({x})) ⊆ U . Conversely, let U be any pre-I -open set and x ∈ U . By the hypothesis, we have Cl(Int⋆({x})) ⊆ U and by Lemma 2, pıCl({x}) ⊆ U . This shows that (X, τ,I ) is a pre-I -R0 space. C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1041 Theorem 13. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is pre-I -R0. (2) For each pre-I -closed set F and each x ∈ X − F , there exists a pre-I -open set U such that F ⊆ U and x ̸∈ U . (3) For each pre-I -closed set F and each x ∈ X − F , pıCl({x}) ∩ F = ∅. (4) For any distinct points x and y of X, either pıCl({x}) = pıCl({y}) or pıCl({x}) ∩ pıCl({y}) = ∅. Proof. (1) ⇒ (2): Let F be any pre-I -closed set and x ∈ X − F . Then, we have x ∈ X−F and by (1), pıCl({x}) ⊆ X−F . Put U = X−pıCl({x}), then U is a pre-I -open set such that F ⊆ U and x ̸∈ U . (2) ⇒ (3): Let F be any pre-I -closed set and x ∈ X − F . Then by (2), there exists a pre-I -open set V such that F ⊆ V and x ̸∈ V . Since V is a pre-I -open set, we have pıCl({x}) ∩ V = ∅ and hence pıCl({x}) ∩ F = ∅. (3) ⇒ (4): Let x, y be any points ofX. Suppose that pıCl({x})∩pıCl({y}) ̸= ∅. By (3), x ̸∈ pıCl({y}) and y ̸∈ pıCl({x}). This implies that pıCl({x}) ⊆ pıCl({y}) ⊆ pıCl({x}). Consequently, we obtain pıCl({x}) = pıCl({y}). (4) ⇒ (1): Let U be any pre-I -open set and x ∈ U . For each y ̸∈ U , we have pıCl({y}) ∩ U = ∅ and so x ̸∈ pıCl({y}). Therefore, pıCl({x}) ̸= pıCl({y}). By (4), for each y ̸∈ U , pıCl({x})∩pıCl({y}) = ∅. SinceX−U is pre-I -closed, y ∈ pıCl({y}) ⊆ X−U and X − U = ∪y∈X−U pıCl({y}). Thus, pıCl({x}) ∩ (X − U) = pıCl({x}) ∩ [∪y∈X−V pıCl({y})] = ∪y∈X−U [pıCl({x}) ∩ pıCl({y})] = ∅ and hence pıCl({x}) ⊆ U . This shows that (X, τ,I ) is a pre-I -R0 space. Corollary 2. An ideal topological space (X, τ,I ) is pre-I -R0 if and only if, for each x, y ∈ X, pıCl({x}) ̸= pıCl({y}) implies pıCl({x}) ∩ pıCl({y}) = ∅. Proof. This is obvious by Theorem 13(4). Conversely, let U be any pre-I -open set and x ∈ U . For each y ̸∈ U , we have pıCl({y}) ∩ U = ∅. Thus, x ̸∈ pıCl({y}) and hence pıCl({x}) ̸= pıCl({y}). By the hypothesis, pıCl({x}) ∩ pıCl({y}) = ∅ and so y ̸∈ pıCl({x}). Therefore, pıCl({x}) ⊆ U . This shows that (X, τ,I ) is a pre-I -R0 space. Lemma 8. Let (X, τ,I ) be an ideal topological space and x, y ∈ X. Then, y ∈ Λp(⋆)({x}) if and only if x ∈ pıCl({y}). C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1042 Proof. Suppose that y ̸∈ Λp(⋆)({x}). Then, there exists a pre-I -open set V containing x such that y ̸∈ V . Therefore, we have x ̸∈ pıCl({y}). Conversely, suppose that x ̸∈ pıCl({y}). Then by Lemma 1, there exists a pre-I - open set V containing x such that V ∩ {y} = ∅. Therefore, we have y ̸∈ V and hence y ̸∈ Λp(⋆)({x}). Lemma 9. Let (X, τ,I ) be an ideal topological space and x, y ∈ X. Then, Λp(⋆)({x}) = Λp(⋆)({y}) if and only if pıCl({x}) = pıCl({y}). Proof. Let x, y be any points of X. Suppose that Λp(⋆)({x}) = Λp(⋆)({y}). Since x ∈ Λp(⋆)({x}), we have x ∈ Λp(⋆)({y}) and by Lemma 8, y ∈ pıCl({x}). Therefore, pıCl({y}) ⊆ pıCl({x}). Similarly, we have pıCl({x}) ⊆ pıCl({y}) and hence pıCl({y}) = pıCl({x}). Conversely, suppose that pıCl({x}) = pıCl({y}). Since x ∈ pıCl({x}), x ∈ pıCl({y}) and by Lemma 8, y ∈ Λp(⋆)({x}). Thus, Λp(⋆)({y}) ⊆ Λp(⋆)(Λp(⋆)({x})) = Λp(⋆)({x}). Similarly, we have Λp(⋆)({x}) ⊆ Λp(⋆)({y}) and hence Λp(⋆))({x}) = Λp(⋆)({y}). Theorem 14. An ideal topological space (X, τ,I ) is pre-I -R0 if and only if, for each x, y ∈ X, Λp(⋆)({x}) ̸= Λp(⋆)({y}) implies Λp(⋆)({x}) ∩ Λp(⋆)({y}) = ∅. Proof. Let x, y be any points of X. Suppose that Λp(⋆)({x}) ∩ Λp(⋆)({y}) ̸= ∅. Let z ∈ Λp(⋆)({x})∩Λp(⋆)({y}). Then, z ∈ Λp(⋆)({x}) and by Lemma 8, we have x ∈ pıCl({z}). Therefore, x ∈ pıCl({z}) ∩ pıCl({x}) and by Corollary 2, x ∈ pıCl({z}) = pıCl({x}). Similarly, we have pıCl({z}) = pıCl({y}) and hence pıCl({x}) = pıCl({y}). By Lemma 9, Λp(⋆)({x}) = Λp(⋆)({y}). Conversely, let x, y be any points of X. Suppose that pıCl({x}) ̸= pıCl({y}). By Lemma 9, Λp(⋆)({x}) ̸= Λp(⋆)({y}) and hence Λp(⋆)({x}) ∩ Λp(⋆)({y}) = ∅. Therefore, pıCl({x}) ∩ pıCl({y}) = ∅. In fact, assume that z ∈ pıCl({x}) ∩ pıCl({y}). Then, z ∈ pıCl({x}) implies x ∈ Λp(⋆)({z}) and hence x ∈ Λp(⋆)({z}) ∩ Λp(⋆)({x}). By the hypothesis, Λp(⋆)({z}) = Λp(⋆)({x}) and by Lemma 9, pıCl({z}) = pıCl({x}). Similarly, we have pıCl({z}) = pıCl({y}) and hence pıCl({x}) = pıCl({y}). This contradicts that pıCl({x}) ̸= pıCl({y}). Thus, pıCl({x}) ∩ pıCl({y}) = ∅. This shows that (X, τ,I ) is pre-I -R0. Theorem 15. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is pre-I -R0; (2) x ∈ pI Cl({y}) if and only if y ∈ pıCl({x}). Proof. (1) ⇒ (2): Suppose that (X, τ,I ) is pre-I -R0 and x ∈ pıCl({y}). By Lemma 8, we have y ∈ Λp(⋆)({x}). Thus, Λp(⋆)({x}) ∩ Λp(⋆)({y}) ̸= ∅ and by Theorem 14, C. Boonpok / Eur. J. Pure Appl. Math, 15 (3) (2022), 1023-1046 1043 Λp(⋆)({x}) = Λp(⋆)({y}). Therefore, x ∈ Λp(⋆)({y}) and by Lemma 8, y ∈ pıCl({x}). The converse is similarly shown. (2) ⇒ (1): Let V be any pre-I -open set and x ∈ V . For each y ̸∈ V , we have pıCl({x}) ∩ V = ∅. This implies that x ̸∈ pıCl({y}) and y ̸∈ pıCl({x}). Thus, pıCl({x}) ⊆ V and hence (X, τ,I ) is pre-I -R0. Theorem 16. For an ideal topological space (X, τ,I ), the following properties are equiv- alent: (1) (X, τ,I ) is pre-I -R0. (2) For each nonempty subset A of X and each pre-I -open set V such that A∩ V ̸= ∅, there exists a pre-I -closed set F such that A ∩ F ̸= ∅ and F ⊆ V . (3) F = Λp(⋆)(F ) for every pre-I -closed set F . (4) pıCl({x}) = Λp(⋆)({x}) for each x ∈ X. (5) pıCl({x}) ⊆ Λp(⋆)({x}) for each x ∈ X. Proof. (1) ⇒ (2): Let A be any nonempty subset of X and let V be any pre-I -open set such that A ∩ V ̸= ∅. Then, there exists x ∈ A ∩ V and hence pıCl({x}) ⊆ V . Put F = pıCl({x}), then F is pre-I -closed, A ∩ F ̸= ∅ and F ⊆ V . (2) ⇒ (3): Let F be any pre-I -closed set and x ̸∈ F . Then, x ∈ X − F and by (2), there exists a pre-I -closed set K such that x ∈ K and K ⊆ X−F . Now, put V = X−K. Then, V is a pre-I -open set such that F ⊆ V and x ̸∈ V . Thus, x ̸∈ Λp(⋆)(F ) and hence F ⊇ Λp(⋆)(F ). On the other hand, we have F ⊆ Λp(⋆)(F ). Consequently, we obtain F = Λp(⋆)(F ). (3) ⇒ (4): Let x ∈ X and y ̸∈ Λp(⋆)({x}). Then, there exists a pre-I -open set U such that x ∈ U and y ̸∈ U . Therefore, pıCl({y})∩U = ∅ and by (3), Λp(⋆)(pıCl({y}))∩U = ∅. Since x ̸∈ Λp(⋆)(pıCl({y})), there exists a pre-I -open set V such that pıCl({y}) ⊆ V and x ̸∈ V . Thus, pıCl({x}) ∩ V = ∅. Since y ∈ V , y ̸∈ pıCl({x}). Therefore, pıCl({x}) ⊆ Λp(⋆)({x}). Moreover, pıCl({x}) ⊆ Λp(⋆)({x}) ⊆ Λp(⋆)(pıCl({x})) = pıCl({x}). This shows that pıCl({x}) = Λp(⋆)({x}). (4) ⇒ (5): The proof is obvious. (5) ⇒ (1): Let V be any pre-I -open set and x ∈ V . Suppose that y ̸∈ V . Then, pıCl({y}) ∩ V = ∅ and x ̸∈ pıCl({y}). By Lemma 8, y ̸∈ Λp(⋆)({x}) and by (5), we have y ̸∈ pıCl({x}). Thus, pıCl({x}) ⊆ V and hence (X, τ,I ) is pre-I -R0. Corollary 3. An ideal topological space (X, τ,I ) is pre-I -R0 if and only if Λp(⋆)({x}) ⊆ pıCl({x}) for each x ∈ X. REFERENCES 1044 Proof. This is obvious by Theorem 16. Conversely, suppose that Λp(⋆)({x}) ⊆ pıCl({x}) for each x ∈ X. Let x ∈ pıCl({y}). By Lemma 8, we have y ∈ Λp(⋆)({x}) and hence y ∈ pıCl({x}). Similarly, if y ∈ pıCl({x}), then x ∈ pıCl({y}). It follows from Theorem 15 that (X, τ,I ) is pre-I -R0. Corollary 4. An ideal topological space (X, τ,I ) is pre-I -R0 if and only if ≺ x ≻p(⋆)= pıCl({x}) for each x ∈ X. Proof. Let x ∈ X. By Theorem 16, we have pıCl({x}) = Λp(⋆)({x}) and hence pıCl({x}) = Λp(⋆)({x}) ∩ pıCl({x}) =≺ x ≻p(⋆). Conversely, suppose that ≺ x ≻p(⋆)= pıCl({x}) for each x ∈ X. Let x ∈ X. By the hypothesis, we have pıCl({x}) =≺ x ≻p(⋆) ({x}) = pıCl({x}) ∩ Λp(⋆)({x}) ⊆ Λp(⋆)({x}). It follows from Theorem 16 that (X, τ,I ) is pre-I -R0. 7. Conclusion Topology plays an important role in both pure and applied sciences such as quan- tum physics, high energy physics, data mining, computational topology, digital topology and mathematical sciences. The notions of closed sets and low separation axioms are fundamental with respect to the investigation of topological spaces. Various types of gen- eralizations of closed sets and some new separation axioms have been researched by many mathematicians. This paper is concerned with the concepts of Λp(⋆)-sets and (Λ, p(⋆))- closed sets which are defined by utilizing the notions of pre-I -open sets and pre-I -open sets. Furthermore, some properties of (Λ, p(⋆))-closed sets and (Λ, p(⋆))-open sets are considered. Several characterizations of (Λ, p(⋆))-continuous functions are obtained. Ad- ditionally, some characterizations of (Λ, p(⋆))-extremally disconnected and pre-I -R0 ideal topological spaces are explored. The ideas and results of this paper may motivate further research. Acknowledgements This research project was financially supported by Mahasarakham University. References [1] A. Açıkgöz, Ş. Yüksel, and T. Noiri. α-I -preirresolute functions and β-I - preirresolute functions. Bulletin Malaysian Mathematical Sciences Society, 28:1–8, 2005. [2] A. Açıkgöz, T. Noiri, and Ş. Yüksel. A decomposition of continuity in ideal topological spaces. Acta Mathematica Hungarica, 105:285–289, 2004. [3] A. Açıkgöz, T. Noiri, and Ş. Yüksel. 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