EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6597 ISSN 1307-5543 – ejpam.com Published by New York Business Global Collectionwise Pre-Normality in Topological Spaces Sadeq Ali Thabit1,∗, Alyaa Al-Awadi2,3,∗, Rafiqa Noaman1 1 Department of Mathematics, Faculty of Applied and Health Sciences, Mahrah University, Yemen 2 Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia 3 Department of Mathematics, Faculty of Education, Mahrah University, Yemen Abstract. This paper introduces and studies a new topological property called collectionwise pre- normality. A space X is said to be collectionwise pre-normal if and only if X is T1 and for every discrete family F = {Fs}s∈S of closed subsets of X, there exists a discrete family U = {Us}s∈S of pre-open subsets of X such that Fs ⊆ Us for each s ∈ S. We investigate this property and present examples that illustrate its relationship with other known topological properties. 2020 Mathematics Subject Classifications: 54C10, 54D10, 54D20, 54D15, 54D70 Key Words and Phrases: Normal, collectionwise normal, paracompact, pre-normal, discrete family, p1-paracompact, sub-maximal 1. Introduction In this paper, we introduce and study a weak version of collectionwise normality called collectionwise pre-normality, which is a generalization of collectionwise normality. The space X means a topological space in whole paper. We need to recall that: a subset A of a space X is said to be a closed domain subset if it is the closure of its own interior [1]. The complement of a closed domain subset is called open domain. A subset A of a space X is called π-closed if it is a finite intersection of closed domain subsets [2]. The complement of a π-closed subset is called π-open. A subset A of X is said to be pre-open [3], if A ⊆ int(A). The complement of a pre-open set is called pre-closed. The intersection of all pre-closed sets containing A is called a pre-closure of A [4, 5], and denoted by p cl(A). The pre-interior of A, denoted by p int(A), is defined to be the union of all pre-open sets contained in A. A subset A is said to be a pre-neighborhood of x, [5], if there exists a ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6597 Email addresses: sthabit1975@gmail.com, s.thabit@mhru.edu.ye (S. A. Thabit), aaalawadi@uj.edu.sa (A. Al-Awadi), rafiqa7757@gmail.com (R. Noaman) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 2 of 15 pre-open set U such that x ∈ U ⊆ A. The family of all pre-open subsets of X is denoted by PO(X) and the family of all pre-closed subsets is denoted by PC(X).Observe that: closed domain =⇒ π-closed =⇒ closed =⇒ pre-closed open domain =⇒ π-open =⇒ open =⇒ pre-open A space X is called pre-normal if for every pair of disjoint closed subsets A and B, there exist disjoint pre-open subsets U and V such that A ⊆ U and B ⊆ V [6]. A space X is said to be a sub-maximal if every dense subset of X is an open [6]. A space X is called an pre-regular if for each closed set F and each x ̸∈ F , there exist disjoint pre-open sets U and V such that x ∈ U and F ⊆ V [3, 7]. A space X is called a pre-T2, if for any distinct two points x ̸= y, there exist two disjoint pre-open sets U and V in X such that x ∈ U and y ∈ V . A space X is called a pre-T1-space if for each x, y ∈ X with x ̸= y, there exist pre-open sets U and V such that x ∈ U , y ∈ V and x ̸∈ V , y ̸∈ U . A space X is called a p1-paracompact if every pre-open cover of X has a locally finite open refinement [3]. A space X is called a pre-compact space if every pre-open cover of X has a finite subcover. A space X is called a pre-Lindelöf space if every pre-open cover of X has a countable subcover. A family U = {As}s∈S of subsets of a space X is called a discrete family if every point x of X has a neighborhood that intersects at most one element of U [8]. A space X is paracompact if every open cover of X has a locally finite open refinement [8–10]. A space X is called countably paracompact if every countable open cover for X has a locally finite open-refinement, [8, 9]. A space X is called a collectionwise normal space if and only if X is a T1-space and for every discrete family F = {Fs}s∈S of closed subsets of X, there exits a discrete family U = {Us}s∈S of open subsets of X such that Fs ⊆ Us for each s ∈ S [9]. Observe that: every normal space is pre-normal. . 2. Preliminaries First, we present the main definitions of this work. Definition 1. A space X is called a collectionwise pre-normal space if and only if X is T1 and for every discrete family F = {Fs}s∈S of closed subsets of X, there exits a discrete family U = {Us}s∈S of pre-open subsets of X such that Fs ⊆ Us for each s ∈ S. From Definition 1, clearly that: every collectionwise pre-normal space is T1 and any non T1-space cannot be collectionwise pre-normal. First, we give the following basic results: Theorem 1. Every collectionwise normal space is collectionwise pre-normal. Proof. Let X be a collectionwise normal space. We show that X is collectionwise pre-normal. For that, let {Fs}s∈S be a discrete family of closed subsets of X. Since X is collectionwise normal, there exists a discrete family {Us}s∈S of open subsets of X such that Fs ⊆ Us for each s ∈ S. Since every open set is pre-open, {Us}s∈S is a discrete family of pre-open subsets of X such that Fs ⊆ Us for each s ∈ S. Therefore, X is collectionwise pre-normal. The converse of Theorem 1 is not true in general. Here is an example of a collectionwise pre-normal space which is not collectionwise normal: S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 3 of 15 Example 1. The finite complement topology: [10, Example 19], (R, CF) is a T1, compact, countably compact, Lindelöf, separable and paracompact space which is neither regular, normal, first countable nor second countable [10]. The finite complement topology is a pre-normal space which is not normal [11]. Hence, the finite complement topology is not collectionwise normal. Since X is T1 countably compact pre-normal space, by Theorem 11 the finite complement topology is collectionwise pre-normal. Theorem 2. Every collectionwise pre-normal space is pre-normal. Proof. Let Fr and Ft be any two disjoint closed subsets of X. Consider F = {Fs : s ∈ S} be a discrete family of all pairwise disjoint closed subsets of a collectionwise pre- normal space X. By collectionwise pre-normality of X, there exists a discrete family V = {Vs : s ∈ S} of pre-open subsets of X such that Fs ⊆ Vs for each s ∈ S. Thus, there exist Vr, Vt ∈ V such that Fr ⊆ Vr, Ft ⊆ Vt and Vr ∩ Vt = ∅. Hence, X is pre-normal. The converse of Theorem 2 is not true in general. Here is an example of a pre-normal space which is not collectionwise pre-normal: Example 2. The left ray topology (R,L) and the right ray topology (R,R) are normal and almost completely regular spaces. Since the two spaces are normal, we conclude (R,L) and (R,R) are pre-normal. Since the two spaces are not T1, we get (R,L) and (R,R) are not collectionwise pre-normal. Therefore, (R,L) and (R,R) are examples of pre-normal spaces which are not collectionwise pre-normal. Since every Hausdorff paracompact space is collectionwise normal [9], and every col- lectionwise normal is collectionwise pre-normal, we conclude the next corollary: Corollary 1. Every Hausdorff paracompact space is collectionwise pre-normal. Observe that: every p1-paracompact space is paracompact [11, 12], we get: Corollary 2. Every regular p1-paracompact T1-space is collectionwise pre-normal. Theorem 3. Every T1 pre-regular p1-paracompact space is pre-normal. Proof. Let X be a pre-regular paracompact space. We show that X is pre-normal. Let A and B be any disjoint closed sets in X, i.e. A ∩ B = ∅. Then for each x ∈ A, we have x ̸∈ B. Therefore, X \ B is an open containing x and hence X \ B is pre-open. By pre-regularity of X, there exists a pre-open set Ux such that x ∈ Ux and cl(Ux) ∩ B = ∅. So, the family {Ux : x ∈ A}∪{X \B} is pre-open cover of X. Since X is p1-paracompact, there exists a locally finite pre-open refinement of it. Let U = {Uα : α ∈ Λ} denotes to the members of the family which have a non-empty intersection with A. Let V1 = ∪α∈ΛUα. Then, V1 is pre-open such that A ⊆ V1. Let V2 = X \ ∪α∈Λcl(Uα). Then, V2 is pre-open because {Uα : α ∈ Λ} is locally finite and cl(∪α∈ΛUα) = ∪α∈Λcl(Uα). Thus, V1 ∩ V2 = ∅. Since U is refinement and each member of it intersects A, for each Uα ∈ U there exists x ∈ A such that Uα ⊆ cl(Ux). Now, cl(Uα) ⊆ X \ B. Thus, B ⊆ X \ cl(Uα) for each Uα ∈ U . So, B ⊆ ∩α∈Λ(X \ cl(Uα)) = X \ ∪α∈Λcl(Uα) = V2. Thus, B ⊆ V2. Therefore, V1 and V2 are disjoint pre-open subsets of X such that A ⊆ V1 and B ⊆ V2. Hence, X is pre-normal. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 4 of 15 Theorem 4. Every T1 pre-regular space is pre-T2. Proof. Let X be a T1 pre-regular space. Let x, y ∈ X such that x ̸= y. Since X is T1, {x} and {y} are closed sets in X such that x ̸∈ {y}. By pre-regularity of X, there exist two pre-open sets U and V in X such that x ∈ U , {y} ⊆ V and U ∩ V = ∅. Thus, there exist two pre-open sets U and V in X such that x ∈ U , y ∈ V and U ∩ V = ∅. Therefore, X is pre-T2. Theorem 5. Every pre-T2 p1-paracompact space is collectionwise pre-normal. Proof. Let F = {Bs : s ∈ S} be a discrete family of closed subsets of a p1-paracompact space X. Then, for each x ∈ X, choose a pre-open neighborhood Hx of a point x whose closure meets at most one set Bs. Thus, {Hx : x ∈ X} is a pre-open cover for X. By p1- paracompactness ofX, there exists a locally finite pre-open refinementW of {Hx : x ∈ X}. Now, for each s ∈ S, let Vs = X \ ⋃ {cl(W ) : W ∈ W and cl(W ) ∩ Bs = ∅}, which is pre-open in X for each s ∈ S such that Bs ⊆ Vs. Since for each W ∈ W, cl(W ) meets at most one set Bs. Then, W meets at most one set Bs. So, {Vs : s ∈ S} is a discrete family of pre-open subsets of X such that Bs ⊆ Vs for each s ∈ S. Since X is T1, we get X is collectionwise pre-normal. Since every T2-space is pre-T2-space, we conclude: Corollary 3. Every T2 p1-paracompact space is collectionwise pre-normal. Since every pre-compact space is p1-paracompact, we get: Corollary 4. Every pre-T2 pre-compact space is collectionwise pre-normal. Corollary 5. Every pre-regular pre-compact T1-space is collectionwise pre-normal. The proofs of the next results is similar to that of the corresponding results for nor- mality. Theorem 6. Every T1-pre-normal space is pre-regular. Proof. Let X be a T1 pre-normal space. Let x ∈ X and F be any closed set in X such that x ̸∈ F . Since X is T1, we have {x} is closed set in X and {x} ∩ F = ∅. By pre-normality of X, there exist two disjoint pre-open sets U and V in X such that {x} ⊆ U and F ⊆ V . Hence, x ∈ U , F ⊆ V and U ∩ V = ∅. Therefore, X is pre-regular. Since every collectionwise pre-normal space is T1, we get: Corollary 6. Every collectionwise pre-normal space is pre-regular. Theorem 7. Every pre-regular pre-Lindelöf space is pre-normal. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 5 of 15 Proof. Let X be a pre-regular pre-Lindelöf space. Let A and B be any disjoint closed subsets of X, i.e. A ∩ B = ∅. Then for each x ∈ A, we have x ̸∈ B. Therefore, X \ B is an open containing x and hence B is pre-open. By pre-regularity of X, there exists a pre-open set Ux such that x ∈ Ux, Ux ∩ B = ∅ and cl(Ux) ∩ B = ∅. So, the family {Ux : x ∈ A}∪{X \B} is pre-open cover of X. Since X is pre-Lindelöf, X has a countable subcover say {Uxi : i ∈ N}. Observe that A ⊆ ∞ ∪ i=1 Uxi and B ⊆ X \ cl( ∞ ∪ i=1 Uxi). Let U = ∞ ∪ i=1 Uxi and V = X \ cl( ∞ ∪ i=1 Uxi). Then, U and V are disjoint pre-open sets in X such that A ⊆ U and B ⊆ V . Therefore, X is pre-normal. Theorem 8. Every T1 pre-regular pre-Lindelöf space is collectionwise pre-normal. Proof. Let X be a pre-regular pre-Lindelöf space. By Theorem 7 X is pre-normal. Let F = {Fs : s ∈ S} be a discrete family of pairwise disjoint closed subsets of X. Then, Fs ∩ Ft = ∅ for each s ̸= t. By pre-normality of X, there exist two disjoint pre-open sets Us and Ut in X such that Fs ⊆ Us, Ft ⊆ Ut and cl(Us) ∩ cl(Ut) = ∅ where s ̸= t. Then, the family {Us}s∈S is a family of pre-open sets in X. Now, we show that {Us}s∈S is discrete. If not, there exists x ∈ X such that for any pre-open neighborhood Wx of x we have Wx ∩ Us ̸= ∅ ≠ Wx ∩ Ut with s ̸= t. Thus, x ∈ cl(Us) and x ∈ cl(Ut). Hence, x ∈ cl(Us)∩cl(Ut), which is a contradiction. Hence, the family {Us}s∈S must be a discrete family of pre-open sets in X such that Fs ⊆ Us for each s ∈ S. Since X is T1, we obtain X is collectionwise pre-normal. Recall that: a space X is called collectionwise Hausdorff if X is T1 and for every discrete collection {xs}s∈S of points of X, there exists a disjoint collection {Vs}s∈S of open subsets of X such that xs ∈ Vs for each s ∈ S [13]. Since every collectionwise Hausdorff p1-paracompact space is Hausdorff paracompact, and every pre-compact space is p1-paracompact, we conclude: Corollary 7. Every collectionwise Hausdorff p1-paracompact space is collectionwise pre- normal. Observe that: every p1-paracompact space is paracompact, every pre-Lindelöf space is Lindelöf, every pre-compact space is compact, every p1-paracompact space is sub- maximal [12], every sub-maximal pre-regular space is regular [12], int(A) ⊆ p int(A) ⊆ A ⊆ p cl(A) ⊆ A for each A ⊆ X [11], and if X is sub-maximal space, then p cl(A) = A for each A ⊆ X. Lemma 1. [11], Let X be a space. Then: (1) Any dense subset of X is pre-open. If D is dense subset of X and A is closed subset of X, then D ∪A and D \A are pre-open. (2) Let D be a dense subset of X. For any two disjoint closed subsets A and B, the sets U = (D \A) ⋃ B and V = (D \B) ⋃ A are pre-open subsets. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 6 of 15 (3) If X has two disjoint dense subsets, then X is pre-normal. Theorem 9. [11], Let X be a sub-maximal space. Fix a point p ∈ X and let M = X \{p}. Then, M is a sub-maximal subspace of X. Observe that: the product space ω1 × ω1 + 1 is not pre-normal, the product space X = (ω0 + 1) × (ω1 + 1) is pre-normal sub-maximal space and hence X is collectionwise pre-normal, the Tychonoff plank M = (ω0+1)×(ω1+1)\{(ω0, ω1)} is dense sub-maximal subspace of X, which is not collectionwise pre-normal, every pre-normal sub-maximal space is normal, the product of two sub-maximal spaces is sub-maximal [11] and every p1-paracompact space is sub-maximal and paracompact [12]. Theorem 10. Every collectionwise pre-normal sub-maximal space is collectionwise nor- mal. Proof. Let {Fs}s∈S be a discrete family of closed subsets ofX. SinceX is collectionwise pre-normal, there exists a discrete family {Us}s∈S of pre-open subsets of X such that Fs ⊆ Us for each s ∈ S. Since X is sub-maximal, every pre-open set in X is open. Therefore, {Us}s∈S is a discrete family of open subsets of X such that Fs ⊆ Us for each s ∈ S. Hence, X is collectionwise normal. Since every Hausdorff p1-paracompact space is Hausdorff paracompact, we get: Corollary 8. Every Hausdorff p1-paracompact space is collectionwise normal and hence collectionwise pre-normal. Note that: every Hausdorff countably compact normal space is collectionwise normal [9], thus we get the following results: Theorem 11. Every T1 countably compact pre-normal space is collectionwise pre-normal. Proof. Let {Fs}s∈S be any discrete family of closed subsets of X. Since every discrete family is locally finite family, the family {Fs}s∈S is locally finite family of closed subsets of X. Since X is countably compact, the family {Fs}s∈S is finite family of pairwise disjoint closed subsets of X. Then, the family can be rewritten as {Fsi}ni=1, for some n ∈ N. By pre-normality of X, for any disjoint closed sets Fsi and Fsj , there exist two disjoint pre-open sets Usi and Usj in X such that Fsi ⊆ Usi and Fsj ⊆ Usj , Usi ∩Usj = ∅ and thus cl(Usi)∩ cl(Usj ) = ∅ for each i ̸= j. Then, the family {Usi}ni=1 is a family of pre-open sets in X such that Fsi ⊆ Usi for each i = 1, 2, 3, . . . , n. It can be observed that the family {Usi}ni=1 is discrete. Therefore, the family {Usi}ni=1 is discrete family of pre-open sets in X such that Fsi ⊆ Usi for each i = 1, 2, 3, . . . , n. Hence, X is collectionwise pre-normal. Since every countable countably-compact space is separable compact [9], we obtain: Corollary 9. Every Countable Hausdorff countably compact space is collectionwise pre- normal. Since every collectionwise pre-normal space is T1 and every finite T1-space is discrete, we get the following corollary: S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 7 of 15 Corollary 10. Every finite collectionwise pre-normal space is discrete and hence it is collectionwise normal. Theorem 12. Collectionwise pre-normality is a topological property. Proof. Let X ∼= Y and X be a collectionwise pre-normal space. Then, there exists a function f : X → Y such that f is 1-1, onto, continuous and f−1 is continuous. We show that Y is collectionwise pre-normal. Let F = {Fs : s ∈ S} be any discrete family of closed subsets of Y . Then, Fs is closed in Y for each s ∈ S. Since f is continuous, f−1(Fs) is a closed subset of X for each s ∈ S. Note that: {f−1(Fs) : s ∈ S} is a discrete family of closed subsets of X. Since X is collectionwise pre-normal, there is a discrete family {Vs : s ∈ S} of pre-open subsets of X such that f−1(Fs) ⊆ Vs for each s ∈ S. So, Fs ⊆ f(Vs) for each s ∈ S. Since f is homeomorphism, we have f(Vs) is a pre-open subset of Y for each s ∈ S. Thus, we have {f(Vs)}s∈S is a discrete family of pre-open subsets of Y such that Fs ⊆ f(Vs) for each s ∈ S. Therefore, Y is collectionwise pre-normal. Theorem 13. The sum X = ⊕s∈SXs, Xs ̸= ∅ for each s ∈ S, is collectionwise pre-normal if and only if each Xs is collectionwise pre-normal. Proof. Let X = ⊕ s∈S Xs be a collectionwise pre-normal space. Since Xs ⊆ X is a clopen subspace of a collectionwise pre-normal space X and a clopen subspace of a collectionwise pre-normal space is collectionwise pre-normal (Corollary 12), we have Xs is collectionwise pre-normal for each s ∈ S. Now, let Xs be a collectionwise pre-normal space for each s ∈ S. We show that X = ⊕ s∈S Xs is collectionwise pre-normal. Let {Fi : i ∈ I} be a discrete family of closed subsets of X. Then, {Fi ∩ Xs : i ∈ I} is a discrete family of closed subsets of Xs for each s ∈ S. By collectionwise pre-normality of Xs, there exists a discrete family {Uis : i ∈ I} of pre-open subsets of Xs such that Fi ∩Xs ⊆ Uis for each s ∈ S. Thus, ∪ s∈S (Fi ∩Xs) ⊆ ∪ s∈S Uis. Put Ui = ∪ s∈S Uis, which is a pre-open set in X for each i ∈ I. So, we have Fi ⊆ Ui for each i ∈ I. Hence, {Ui : i ∈ I} is a discrete family of pre-open subsets of X such that Fi ⊆ Ui for each i ∈ I. Therefore, X = ⊕ s∈S Xs is collectionwise pre-normal. Corollary 11. Collectionwise pre-normality is an additive property. 3. Characterizations of collectionwise pre-normality Now, we give some characterizations of collectionwise pre-normal spaces. First, we need to recall the next definitions: Definition 2. A subset A of X is called: • generalized closed (briefly; g-closed) if A ⊆ U whenever A ⊆ U and U is open [14]. • generalized pre-open (briefly; g-pre-open) if F ⊆ p int(A) whenever F ⊆ A and F is closed[15]. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 8 of 15 • strongly generalized pre-open (briefly; g∗-pre-open) if F ⊆ p int(A) whenever F ⊆ A and F is g-closed [16]. • π-generalized pre-open, (briefly; πg-pre-open) if F ⊆ p int(A) whenever F ⊆ A and F is π-closed.[17] Observe that: every open set is pre-open and every closed set is pre-closed. From the Definition 2, we have: pre-open =⇒ g∗-pre-open =⇒ g-pre-open =⇒ πg-pre-open g∗-closed (g-closed, πg-closed) =⇒ g∗-pre-closed (g-pre-closed, πg-pre-closed) Now, we give the following theorem, which is useful for giving some characterizations of collectionwise pre-normal spaces. Theorem 14. Let X be a space. The following statements are equivalent: (1) X is collectionwise pre-normal. (2) for any discrete family {Fs}s∈S of closed sets in X, there exists a discrete family {Us}s∈S of g⋆-pre-open sets in X such that Fs ⊆ p int(Us) for each s ∈ S. (3) for any discrete family {Fs}s∈S of closed sets in X, there exists a discrete family {Us}s∈S of g-pre-open sets in X such that Fs ⊆ p int(Us) for each s ∈ S. (4) for any discrete family {Fs}s∈S of closed sets in X, there exists a discrete family {Us}s∈S of πg-pre-open sets in X such that Fs ⊆ p int(Us) for each s ∈ S. Proof. (1) =⇒ (2): Let X be collectionwise pre-normal. Let {Fs}s∈S be a discrete family of closed subsets of X. By collectionwise pre-normality of X, there exists a discrete family {Us}s∈S of pre-open sets in X such that Fs ⊆ Us for each s ∈ S. Since every pre-open set is g⋆-pre-open, we have {Us}s∈S is a discrete family of g⋆-pre-open sets in X such that Fs ⊆ Us for each s ∈ S. Since Us is g-pre-open as every pre-open set is g-pre-open, and Fs ⊆ Us we have Fs ⊆ p int(Us) for each s ∈ S. (2) =⇒ (3) =⇒ (4) are obvious. (4) =⇒ (1): Suppose (4) holds. We show that X is collectionwise pre-normal. Let {Fs}s∈S be a discrete family of closed subsets of X. By (4), there exists a discrete family {Us}s∈S of πg-pre-open sets in X such that Fs ⊆ p int(Us) for each s ∈ S. Put Vs = p int(Us) for each s ∈ S. Then, Vs is pre-open subset of X for each s ∈ S. Since {Us}s∈S is discrete family and Vs ⊆ Us for each s ∈ S, we obtain {Vs}s∈S is a discrete family of pre-open subsets of X such that Fs ⊆ Vs for each s ∈ S. Therefore, X is collectionwise pre-normal. Theorem 15. A space X is collectionwise pre-normal if one of the next equivalent state- ments holds: (1) for any discrete family {Fs}s∈S of g-closed sets in X, there exists a discrete family {Us}s∈S of π-pre-open sets in X such that p cl(Fs) ⊆ Us for each s ∈ S. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 9 of 15 (2) for any discrete family {Fs}s∈S of g-closed sets in X, there exists a discrete family {Us}s∈S of pre-open sets in X such that p cl(Fs) ⊆ Us for each s ∈ S. (3) for any discrete family {Fs}s∈S of g-closed sets in X, there exists a discrete family {Us}s∈S of g⋆-pre-open sets in X such that p cl(Fs) ⊆ p int(Us) for each s ∈ S. (4) for any discrete family {Fs}s∈S of g-closed sets in X, there exists a discrete family {Us}s∈S of g-pre-open sets in X such that p cl(Fs) ⊆ p int(Us) for each s ∈ S. (5) for any discrete family {Fs}s∈S of g-closed sets in X, there exists a discrete family {Us}s∈S of πg-pre-open sets in X such that p cl(Fs) ⊆ p int(Us) for each s ∈ S. Proof. (1) =⇒ (2) =⇒ (3) =⇒ (4) =⇒ (5) are obvious. Now, we show that: (5) =⇒ collectionwise pre − normality : Suppose (5) holds. Let {Fs}s∈S be a discrete family of closed subsets of X. Since every closed set is g-closed, the family {Fs}s∈S is a discrete family of g-closed subsets of X. By (5), there exists a discrete family {Us}s∈S of πg-pre-open sets in X such that p cl(Fs) ⊆ p int(Us) for each s ∈ S. Since Fs is pre-closed for each s ∈ S, we get Fs ⊆ p int(Us) for each s ∈ S. Let Vs = p int(Us) for each s ∈ S. Then, Vs is pre-open set in X for each s ∈ S. Since Vs ⊆ Us for each s ∈ S and {Us}s∈S is discrete, we conclude that {Vs}s∈S is a discrete family of pre-open sets in X such that Fs ⊆ Vs for each s ∈ S. Therefore, X is collectionwise pre-normal. 4. Collectionwise pre-normality in subspaces Now, we study collectionwise pre-normality in subspaces. The next example shows that collectionwise pre-normality is not a hereditary property in general. Example 3. Consider the product space X = (ω0+1)×(ω1+1), which is pre-normal, but the subspace M = X \ {⟨ω0, ω1⟩} is not pre-normal [11]. Since X = (ω0 + 1)× (ω1 + 1) is normal, it is pre-normal. The Tychonoff plank M = X \{⟨ω0, ω1⟩} is dense subspace of X. Since both ω0+1 and ω1+1 are sub-maximal spaces and the product of two sub-maximal spaces is sub-maximal, we obtain the space X = (ω0 + 1)× (ω1 + 1) is sub-maximal. By Theorem 9, M = X \ {⟨ω0, ω1⟩} is sub-maximal subspace of X. Since the subspace M is not normal, we obtain M is not pre-normal. Since M = X \{⟨ω0, ω1⟩} is not collectionwise normal, we get M = X \ {⟨ω0, ω1⟩} is not collectionwise pre-normal. Lemma 2. [11], Let M be a closed domain subspace of X and A ⊆ M . A is pre-closed (pre-open) in M , if and only if A is pre-closed (pre-open) in X. Theorem 16. Let M be a closed domain subspace of X. Then: (1) A family {Fs}s∈S is discrete family of closed sets in M if and only if {Fs}s∈S is discrete family of closed sets in X, where Fs ⊆ M for each s ∈ S. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 10 of 15 (2) A family {Fs}s∈S is discrete family of pre-open sets in M if and only if {Fs}s∈S is discrete family of pre-open sets in X, where Fs ⊆ M for each s ∈ S. Proof. Let M be a closed domain subspace of X. Then: (1): Let {Fs}s∈S be a discrete family of closed sets in M . Then, Fs is closed subset of M for each s ∈ S. Since M is closed subset of X, we have Fs is closed set in X for each s ∈ S. Hence, {Fs}s∈S is discrete family of closed sets in X, where Fs ⊆ M for each s ∈ S. Conversely, let {Fs}s∈S be a discrete family of closed sets in X, where Fs ⊆ M for each s ∈ S. Then, Fs is closed in X for each s ∈ S. Since M is closed in X, we have Fs∩M = Fs is closed set in M for each s ∈ S. Then, {Fs}s∈S is a discrete family of closed sets in M . (2): Let {Fs}s∈S be a discrete family of pre-open sets in M . Then, Fs is pre-open set in M for each s ∈ S. Since M is closed domain set in X, by Lemma 2 Fs is pre-open set in X for each s ∈ S. Hence, {Fs}s∈S is a discrete family of pre-open sets in X, where Fs ⊆ M for each s ∈ S. Conversely, let {Fs}s∈S be a discrete family of pre-open sets in X, where Fs ⊆ M for each s ∈ S. Then, Fs is pre-open set in X for each s ∈ S. Since M is closed domain in X, by Lemma 2 we have Fs is pre-open set in M for each s ∈ S. Then, {Fs}s∈S is a discrete family of pre-open sets in M . Lemma 3. [11], Let M be a closed domain subspace of X and A ⊆ M . Then: (1) A is pre-closed (pre-open) in M if and only if A is pre-closed (pre-open) in X. (2) If A ⊆ X and A is pre-closed (pre-open) in X, then A ∩M is pre-closed (pre-open) in M . Theorem 17. A closed domain subspace of a collectionwise pre-normal space is collec- tionwise pre-normal. Proof. Let {Fs : s ∈ S} be a discrete family of closed subsets of M . By Theorem 16, {Fs : s ∈ S} is a discrete family of closed subsets of X. Since X is collectionwise pre-normal, there exists a family {Us : s ∈ S} of pre-open subsets of X such that Fs ⊆ Us for each s ∈ S. Thus, Fs∩M ⊆ Us∩M and so Fs ⊆ Us∩M for each s ∈ S. By Lemma 3, we have Us∩M is pre-open set in M for each s ∈ S. Hence, {Us∩M : s ∈ S} is a discrete family of pre-open subsets of M such that Fs ⊆ Us ∩M for each s ∈ S. Therefore, M is collectionwise pre-normal. Since every clopen subset of a spaceX is closed domain, we conclude the next corollary: Corollary 12. A clopen subspace of a collectionwise pre-normal space is collectionwise pre-normal. 5. The product of collectionwise pre-normality In this section, we study the product of collectionwise pre-normality as follows: S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 11 of 15 Theorem 18. Let (Xi, Ti) be a topological space for each i ∈ {1, 2, 3, ..., n}, n ∈ N. Let T be the product topology on X = ∏n i=1Xi. If (X, T ) is collectionwise pre-normal, then (Xi, Ti) is collectionwise pre-normal for each i ∈ {1, 2, 3, ..., n}. Proof. Let X = n∏ i=1 Xi be a collectionwise pre-normal space. Let m ∈ {1, 2, 3, ..., n} be arbitrary. Let {Fsm}s∈S be any discrete family of closed subsets of Xm. Let πm : n∏ i=1 Xi −→ Xm be the natural projection map from X onto Xm. Now, π−1 m (Fsm) = n∏ i=1 Wi, (where Wi = Xi for each i ̸= m) is closed in X. Then, {π−1 m (Fsm)}s∈S is a discrete family of closed sets in X. Since X is collectionwise pre-normal, there exists a discrete family {Us}s∈S of pre-open sets in X such that π−1 m (Fsm) ⊆ Us for each s ∈ S. Then, we have Fsm ⊆ πm(Us) for each s ∈ S. Since πm is a clopen onto continuous function, then πm(Us) is pre-open set in Xm for each s ∈ S. Thus, {πm(Us)}s∈S is a discrete family of pre- open sets in Xm such that Fsm ⊆ πm(Us) for each s ∈ S. Hence, Xm is collectionwise pre-normal. Since m was arbitrary, then (Xi, Ti) is collectionwise pre-normal for each i ∈ {1, 2, 3, ..., n}. Corollary 13. • If the product space X × Y is collectionwise pre-normal, then both X and Y are collectionwise pre-normal. • If X × I is collectionwise pre-normal, then X is collectionwise pre-normal. • A space X is collectionwise pre-normal if and only if X × {0} is collectionwise pre- normal. Note that: collectionwise pre-normality is not productive in general. Here is an exam- ple: Example 4. The space ω1×(ω1+1), [10], is Tychonoff, mildly normal, locally compact and countably compact space which is neither almost normal, normal, compact nor Lindelöf. Since X is not normal, the space ω1 × (ω1 + 1) is not collectionwise normal. Since ω1 and ω1+1 are sub-maximal spaces [11], we get ω1×(ω1+1) is sub-maximal. Since ω1×(ω1+1) is not normal, we conclude that ω1× (ω1+1) is not pre-normal. Therefore, ω1× (ω1+1) is not collectionwise pre-normal. This example shows that the product of two collectionwise pre-normal spaces cannot be collectionwise pre-normal. Observe that: any Tychonoff space Y has a one-point compactification X = Y ∪ {p}, p ̸∈ Y and X is a Hausdorff compact space [18], we get: Corollary 14. Any compactification X of a Tychonoff space Y is collectionwise pre- normal. In particular, any Tychonoff space Y has a one-point compactification X = Y ∪ {p}, p ̸∈ Y and X is collectionwise pre-normal. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 12 of 15 6. The closed extension and the discrete extension spaces of collectionwise pre-normality Now, we study the closed extension and the discrete extension spaces of collection- wise pre-normality. In fact, collectionwise pre-normality is not preserved by the discrete extension space XM in general. Here is a counterexample: Example 5. [18, Example 8], The rational sequence topology [10, Example 65], is a first countable, zero-dimensional, Tychonoff, locally compact, separable space which is neither paracompact, normal nor Lindelöf [10]. By Corollary 14, R with the rational sequence topology has a one-point compactification. Let X = R ∪ {p}, p ̸∈ R, be a one-point com- pactification of R. By Corollary 14, X is Hausdorff compact. Hence, X is collectionwise pre-normal. Now, let XR = R∪{p}. Then, XR is first countable, separable and Tychonoff space which is not normal and {p} is clopen subset [18]. Since R is clopen subspace in XR and R is not pre-normal, we conclude that XR is not pre-normal. Therefore, XR is neither collectionwise normal nor collectionwise pre-normal because R with the rational sequence topology is sub-maximal [11]. Hence, XR is a discrete extension space of a collectionwise pre-normal space X = R ∪ {p} which is not collectionwise pre-normal. Now, we give the following results: Lemma 4. [11], Let M be a closed subspace of X and A ⊆ M . Then: if A is pre-closed (pre-open) in M , then A is pre-closed (pre-open) in X. Lemma 5. Let M be a closed subspace of X. Then: (1) If A is a pre-closed (pre-open) set in X, then A is pre-closed (pre-open) set in XM . (2) If A is a closed set in XM , then A ∩M is closed subset of a subspace M in X. (3) If A is a closed set in XM , then A1 = A ∩M is a closed set in X. Proof. Let M be a closed subspace of X. (1) Let A be a pre-closed (pre-open) set in X. Since X ⊂ XM is a closed subspace of XM [18], and A is pre-closed (pre-open) in X, by Lemma 4 we conclude that A is pre-closed (pre-open) in XM . (2) Let A be a closed set in XM . Since M ⊂ XM is closed in both X and XM and its topology coincides with the topology on M by the topology on X, i.e. TM = T(M)M [18], we get A ∩M is a closed subset of a subspace M in XM . Since TM = T(M)M , A ∩M is a closed subset of a subspace M in X. (3) Let A be a closed set in XM . By part (2), A1 = A ∩ M is a closed subset of a subspace M in X. Since M is closed subspace of X and A ∩M is closed subset of M in X, we have A1 = A ∩M is a closed set in X. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 13 of 15 Lemma 6. Let M be a closed subspace of a space X. Then: If {Fs}s∈S is a discrete family of closed sets in XM , then {Fs ∩M}s∈S is a discrete family of closed sets in X. Proof. Let M be a closed subspace of X and {Fs}s∈S be a discrete family of closed sets in XM . Then, Fs is closed set in XM for each s ∈ S. By Lemma 5, we get Fs∩M is closed subset of a subspace M in X for each s ∈ S. Put Gs = Fs ∩ M for each s ∈ S. Then, {Gs}s∈S is a family of closed subsets of X. Since {Fs}s∈S is a discrete family, X = XM , T ⊆ T(M) and Gs ⊆ Fs for each s ∈ S, we obtain {Gs}s∈S is a discrete family of closed subsets of X. Hence, {Fs ∩M}s∈S is a discrete family of closed sets in X. Theorem 19. If X is collectionwise pre-normal and M is a closed subspace of X, then XM is collectionwise pre-normal. Proof. Let {Fs}s∈S be a discrete family of closed sets in XM . Since M is closed in X, by Lemma 6 we get {Fs∩M}s∈S is a discrete family of closed sets in X. By collectionwise pre-normality of X, there exists a discrete family {Us}s∈S of pre-open sets in X such that Fs∩M ⊆ Us for each s ∈ S. Let Vs = Us∪Fs \M for each s ∈ S. Then, Vs is pre-open set in XM and Fs ⊆ Vs for each s ∈ S. Observe that {Vs : s ∈ S} is discrete. Hence, {Vs}s∈S is a discrete family of pre-open sets in XM such that Fs ⊆ Vs for each s ∈ S. Therefore, XM is collectionwise pre-normal. Since the closed extension space (Xp, T ∗) of a space (X, T ) is separable, first countable, second countable and T0-space which is neither T1, Hausdorff, regular nor normal [19], we get the next corollary: Corollary 15. Any closed extension space (Xp, T ∗) of a collectionwise pre-normal space (X, T ) cannot be collectionwise pre-normal. That is: collectionwise pre-normality is not preserved by the closed extension spaces. Proof. Since the closed extension space (Xp, T ∗) of a space (X, T ) is not T1-space, and every collectionwise pre-normal space is T1, we conclude that any closed extension space (Xp, T ∗) of a collectionwise pre-normal space (X, T ) is not collectionwise pre-normal. Now, we present the next examples. Here is a Tychonoff space which is not collection- wise pre-normal: Example 6. The rational sequence topology [10, Example 65], (R,RS) is a Tychonoff first countable, zero-dimensional, locally compact, separable and almost normal space which is neither paracompact, normal, extremally disconnected, π-normal nor Lindelöf [10, 11]. Observe that: the rational sequence topology is an example of a Tychonoff space which is neither pre-normal nor normal being sub-maximal space [11]. Therefore, the rational sequence topology is neither collectionwise pre-normal nor collectionwise normal The following problems are still open in this research: is there an example of a T1 pre- normal space which is not collectionwise pre-normal?, is there a Tychonoff collectionwise pre-normal space which is not collectionwise normal?, is a closed subspace of a collection- wise pre-normal space, collectionwise pre-normal?, are the Niemytzki plane topology and the countable complement topology (R, CC), collectionwise pre-normal?, and is a quotient space of a collectionwise pre-normal space, collectionwise pre-normal?. S. A. Thabit, A. Al-Awadi, R. Noaman / Eur. J. Pure Appl. Math, 18 (4) (2025), 6597 14 of 15 7. Conclusion New topological property, called collectionwise pre-normality has been studied in this work. Some results, properties, relationships, characterizations and counterexamples were given and discussed. 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