EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 1161-1168 ISSN 1307-5543 – ejpam.com Published by New York Business Global New Topologies between the usual topology and the half-disc Nadiah Alghamdi1,2, Lutfi Kalantan1,∗ 1 King Abdulaziz University, Department of Mathematics, P.O.Box 80203, Jeddah 21589, Saudi Arabia 2 Umm Al-Qura University. Department of Mathematics, Saudi Arabia Abstract. We generate new topologies on the closed upper half plane which lie between the usual topology and the half-disc topology. We study some of their fundamental properties and weaker versions of normality. 2020 Mathematics Subject Classifications: 54A10, 54G20 Key Words and Phrases: H-space, Half-disc, usual metric topology, mildly normal, κ-normal, κ-metrizable, submetrizable. We generate new topologies on the closed upper half plane which lie between the usual metric topology and the half-disc topology. These new spaces may work as counterexam- ples in Topology and help in study of some advances topological properties. We study some of their fundamental properties and weaker versions of normality. Throughout this paper, we denote an ordered pair by ⟨x, y⟩, the set of positive integers by N, the rationals by Q, the irrationals by P, and the set of real numbers by R. A T4 space is a T1 normal space and a Tychonoff space (T3 1 2 ) is a T1 completely regular space. We do not assume T2 in the definition of compactness and countable compactness. We do not assume regularity in the definition of Lindelöfness. For a subset A of a space X, intA and A denote the interior and the closure of A, respectively. If two topologies τ and τ ′ on a set X are considered, we denote the interior of A in (X , τ ) by int τA and the closure of A in (X , τ ′ ) by A τ ′ . We start by state some definitions and fix some notations. Let X = { ⟨x, y⟩ ∈ R2 : y ≥ 0 } be the closed upper half plane. K = { ⟨x, y⟩ ∈ R2 : y > 0 }, so the x-axis is L = X \K. Denote the usual metric topology on X by U and the half-disc topology on X be H. For every ⟨a, b⟩ ∈ X and r > 0 where r ∈ R, let Ur(⟨a, b⟩) be the set of all points in X inside the circle of radius r centered at ⟨a, b⟩. So, Ur(⟨a, b⟩) = { ⟨x, y⟩ ∈ X : ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4061 Email addresses: nalghamdi0454@stu.kau.edu.sa, namghamdi@uqu.edu.sa (N. Alghamdi), lnkalantan@hotmail.com, lkalantan@kau.edu.sa (L. Kalantan) http://www.ejpam.com 1161 © 2021 EJPAM All rights reserved. N. Alghamdi, L. Kalantan / Eur. J. Pure Appl. Math, 14 (4) (2021), 1161-1168 1162√ (x− a)2 + (y − b)2 < r }. For every ⟨a, 0⟩ ∈ L, let C(⟨a, 0⟩, r) be the set of all points of K inside the circle of radius r centered at ⟨a, 0⟩. So, C(⟨a, 0⟩, r) = Ur(⟨a, 0⟩) ∩ K. Let Cr(⟨a, 0⟩) = C(⟨a, 0⟩, r) ∪ {⟨a, 0⟩}. Recall that the half-disc topology H on X [8, Example 78] is generated by the following neighborhood system: For every ⟨a, 0⟩ ∈ L, let B(⟨a, 0⟩) = {Cr(⟨a, 0⟩) : r > 0 } and for every ⟨a, b⟩ ∈ K, let B(⟨a, b⟩) = {Ur(⟨a, b⟩) : r > 0 }. Observe that K as a subspace of X with the usual metric topology coincides with K as a subspace of X with the half-disc topology. ⟨a, b⟩• ⟨a, b⟩ ∈ K, Ur(⟨a, b⟩) ⟨a, 0⟩ •◦ ◦ ⟨a, 0⟩ ∈ L, Ur(⟨a, 0⟩) ⟨a, 0⟩ •◦ ◦ ⟨a, 0⟩ ∈ L, Cr(⟨a, 0⟩) 1. H-generated topology. Definition 1. Let A be a non-empty proper subset of the x-axis L. For each ⟨a, b⟩ ∈ K∪A, let B(⟨a, b⟩) = {Ur(⟨a, b⟩) : r > 0 }. For each ⟨a, 0⟩ ∈ L \ A, let B(⟨a, 0⟩) = {Cr(⟨a, 0⟩) : r > 0 }. So, the points in K ∪ A will have the same local base as in (X , U ). The points in L \A will have the same local base as in (X , H ). We call the topology on X generated by the neighborhood system {B(⟨a, b⟩) : ⟨a, b⟩ ∈ X } the H-generated topology on X from U and H, shortly H-topology, and denote it by UAH. We call X with this H-generated topology an H-space and denote it by (X , UAH ). Observe that if A = ∅, then the H-generated topology on X,UAH is just the half-disc topology H and if A = L, then the H-generated topology on X, UAH is just the usual metric topology U . So, from now on, when we consider an H-space (X , UAH ), we are assuming that A is a non-empty proper subset of the x-axis L. In Definition 1, if we interchange the local bases as follows: The points in K ∪ (L \A) will have the same local base as in (X , U ). The points in A will have the same local base as in (X , H ). Then we get the H-space (X , HAU ) and it is easy to see that (X , HAU ) ∼= (X , UL\AH ). So, in this paper, we will study the H-spaces of Definition 1, (X , UAH ). Observe that K as a subspace of X with the usual metric topology coincides with K as a subspace of X with the H-topology. 2. Basic Properties of an H-space. Since any basic open set in (X , U ) is also open in (X , UAH ), we get the following fact. N. Alghamdi, L. Kalantan / Eur. J. Pure Appl. Math, 14 (4) (2021), 1161-1168 1163 Theorem 1. The usual topology U on X is coarser than the H-topology UAH and the H-topology UAH is coarser than the half-disc topology H. That is, U ⊂ UAH ⊂ H. By Theorem 1, we conclude that any H-space (X , UAH ) is T0, T1, T2 Hausdorff, and T2 1 2 Urysohn (completely Hausdorff) [1]. Now, take any ⟨x, 0⟩ ∈ L \A. For any 0 < ϵ < r, we have that Cϵ(⟨x, 0⟩) ̸⊂ Cr(⟨x, 0⟩), thus any H-space (X , UAH ) is not regular nor zero-dimensional, hence neither normal, T4, nor metrizable. We conclude also that any H-space (X , UAH ) cannot be Tychonoff T3 1 2 hence has no compactification and is neither paracompact, as any T2 paracompact space is T4, nor locally compact, as any T2 locally compact space is Tychonoff. The subset D = { ⟨x, y⟩ ∈ X : x, y ∈ Q } is a countable dense subset, thus any H-space (X , UAH ) is separable. For ⟨x, y⟩ ∈ K ∪ A, the family B(⟨x, y⟩) = {U 1 n (⟨x, y⟩) : n ∈ N } is a countable local base for (X , UAH ) at ⟨x, y⟩. For ⟨x, 0⟩ ∈ L \ A, the family B(⟨x, 0⟩) = {C 1 n (⟨x, 0⟩) : n ∈ N } is a countable local base for (X , UAH ) at ⟨x, 0⟩. Therefore, any H-space (X , UAH ) is first countable. Theorem 2. An H-space (X , UAH ) is second countable if and only if L\A is countable. Proof. If L \ A is uncountable, then L \ A is an uncountable discrete subspace of the H-space (X , UAH ), thus cannot be second countable. Assume that L \ A is countable. Since X = K ∪ A ∪ (L \ A) and K ∪ A is a separable metrizable subspace from (X , U), then K ∪A is second countable. Let B′ be a countable base for K ∪ A. Let B⋆ = {B(⟨x, 0⟩) = {C 1 n (⟨x, 0⟩) : n ∈ N } : ⟨x, 0⟩ ∈ L \ A } . We show that B = B′⋃B⋆ is a countable base for the H- space (X , UAH ). Let W be an arbitrary non-empty open set in (X , UAH ) and pick an arbitrary ⟨x, y⟩ ∈ W . Case 1: If ⟨x, y⟩ ∈ K ∪A, then there exists r > 0 such that Ur(⟨x, y⟩) ⊆ W . Since Ur(⟨x, y⟩) is open in K ∪A containing ⟨x, y⟩, then there exists B ∈ B′ ⊂ B such that ⟨x, y⟩ ∈ B ⊆ Ur(⟨x, y⟩) ⊆ W . Case 2: If ⟨x, y⟩ ∈ L\A, so y = 0, then there exists r > 0 such that Cr(⟨x, 0⟩) ⊆ W , thus there exists an n ∈ N such that 0 < 1 n < r, thus ⟨x, 0⟩ ∈ C 1 n (⟨x, 0⟩) ⊆ Cr(⟨x, 0⟩) ⊆ W , where C 1 n (⟨x, 0⟩) ∈ B⋆. Therefore, B is a base for the H-space (X , UAH ) . For each n ∈ N, let Gn = R × [0, n), then the family {Gn : n ∈ N } is a countable open cover for X which has no finite subcover. Thus any H-space (X , UAH ) is neither compact nor countably compact. Since any second countable is Lindelöf, by Theorem 2, we conclude the following. Theorem 3. The H-spaces (X , UAH ) is Lindelöf if L \A is countable. N. Alghamdi, L. Kalantan / Eur. J. Pure Appl. Math, 14 (4) (2021), 1161-1168 1164 3. Other Properties of an H-space. Definition 2. A subset A of a space X is called a closed domain of X [1, 1.1.C] (also called regularly closed, κ-closed) if A = intA. A space X is called mildly normal [7] (also called κ-normal [10]) if for any two disjoint closed domains A and B of X there exist two disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V , see also [2, 5]. A space X is called almost normal [6] if for any two disjoint closed subsets A and B of X one of which is closed domain, there exist two disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V , see also [4]. It is clear from the definitions that normal ⇒ almost normal ⇒ mildly normal. Each implication above is not reversible, see [2, 4]. Lemma 1. Let D be any non-empty closed domain in (X , UAH ), then D is a closed set in (X , U ). Proof. If D = X, we are done. Assume X \D ̸= ∅. Let ⟨x, y⟩ ∈ X \D be arbitrary. Case 1: ⟨x, y⟩ ∈ K ∪A. Since D is closed in (X , UAH ), then X \D is open in (X , UAH ), thus there exists r > 0 such that Ur(⟨x, y⟩) ⊆ X \D. But Ur(⟨x, y⟩) is also a basic open set in (X , U ). Case 2: y = 0 and ⟨x, 0⟩ ∈ L \A. There exists r > 0 such that Cr(⟨x, 0⟩) ⊆ X \D . . . ( ⋆ ) Suppose that for all 0 < ε ≤ r we have Uε(⟨x, 0⟩) ̸⊆ X \ D. That is, Uε(⟨x, 0⟩) ∩ D ̸= ∅. Fix such an ε, then there exists z ∈ (x − ε, x + ε ); z ̸= x such that ⟨z, 0⟩ ∈ D = intUAHD UAH . For all 0 < δ < ε, we have Cδ(⟨z, 0⟩) ∩ intUAHD ̸= ∅ if ⟨z, 0⟩ ∈ L \ A or Uδ(⟨z, 0⟩) ∩ intUAHD ̸= ∅ if ⟨z, 0⟩ ∈ A. If ⟨z, 0⟩ ∈ L \ A, then⟨z, 0⟩ ∈ intUAHD , because Cδ(⟨z, 0⟩)∩L = {⟨z, 0⟩} and Cδ(⟨z, 0⟩)∩K ⊆ X \D. Now, ⟨z, 0⟩ ∈ intUAHD means that there exists δ′ < δ such that Cδ′(⟨z, 0⟩) ⊆ D which contradicts ( ⋆ ), because Cδ′(⟨z, 0⟩) ∩ K ⊆ Cr(⟨x, 0⟩) ⊆ X \ D. If ⟨z, 0⟩ ∈ A, and Uδ(⟨z, 0⟩) ∩ intUAHD ̸= ∅, then pick ⟨u, 0⟩ ∈ Uδ(⟨z, 0⟩) ∩ intUAHD , thus ⟨u, 0⟩ ∈ intUAHD . Then there exists δ′ < δ such that Uδ′(⟨u, 0⟩) ⊆ D if ⟨u, 0⟩ ∈ A or Cδ′(⟨u, 0⟩) ⊆ D, if ⟨u, 0⟩ ∈ L \ A. In both cases, we get a contradiction to ( ⋆ ), because Uδ′(⟨u, 0⟩)∩K ⊆ Cr(⟨x, 0⟩) ⊆ X\D and Cδ′(⟨u, 0⟩)∩K ⊆ Cr(⟨x, 0⟩) ⊆ X\D. Thus X \D is open in the usual metric topology. Therefore, D is a closed set in (X , U ). Lemma 2. Let D be any non-empty closed domain in (X , UAH ), then (intUAHD )∩(K∪ A) = (intUD ) ∩ (K ∪A). Proof. Let ⟨u, v⟩ ∈ K ∪A. ⟨u, v⟩ ∈ intUAHD if and only if there exists r > 0 such that Ur(⟨u, v⟩) ⊆ D if and only if ⟨u, v⟩ ∈ intUD N. Alghamdi, L. Kalantan / Eur. J. Pure Appl. Math, 14 (4) (2021), 1161-1168 1165 Lemma 3. Let D be any non-empty closed domain in (X , UAH ), and ⟨x, 0⟩ ∈ (L \A)∩ (intUAHD ), then there exists r > 0 such that Ur(⟨x, 0⟩) ⊆ D. Proof. Since ⟨x, 0⟩ ∈ L \ A and ⟨x, 0⟩ ∈ intUAHD , then there exists r > 0 such that Cr(⟨x, 0⟩) ⊆ D ⊆ D ⟨x, 0⟩ • ◦◦ Now Cr(⟨x, 0⟩) UAH ⊆ D UAH = D as D is closed. ⊆ D ⟨x, 0⟩ • ◦◦ But Ur(⟨x, 0⟩) ⊆ Cr(⟨x, 0⟩) UAH ⊆ D. Theorem 4. Let D be any non-empty closed domain in (X , UAH ), then D is a closed domain in (X , U ). Proof. Assume D = intUAHD UAH ̸= ∅, we show D = intUD U .Since intUD ⊆ D, then intUD U ⊆ D U = D, by Lemma 1. Now, we show D ⊆ intUD U . Let ⟨x, y⟩ ∈ D arbitrary. If ⟨x, y⟩ ∈ intUD , then clearly ⟨x, y⟩ ∈ intUD U . So, assume that ⟨x, y⟩ ∈ D \ intUD . To show ⟨x, y⟩ ∈ intUD U we have to show that for all r > 0, we have Ur(⟨x, y⟩) ∩ intUD ̸= ∅ Case 1: ⟨x, y⟩ ∈ K ∪A. Let r > 0 be arbitrary, we have Ur(⟨x, y⟩) ∩ intUAHD ̸= ∅. By Lemma 2, we have (intUAHD ) ∩ (K ∪ A) = (intUD ) ∩ (K ∪ A), since ⟨x, y⟩ ∈ K ∪ A, then Ur(⟨x, y⟩) ∩ intUD ̸= ∅. Case 2: ⟨x, y⟩ ∈ L \A, then y = 0. We want to show that for any r > 0 we have Ur(⟨x, 0⟩)∩ intUD ̸= ∅. Suppose that there exists r > 0 such that Ur(⟨x, 0⟩) ∩ intUD = ∅ . . . (⋆). Since Cr(⟨x, 0⟩) ⊆ Ur(⟨x, 0⟩), then Cr(⟨x, 0⟩) ∩ intUD = ∅. Claim: Cr(⟨x, 0⟩) ∩ intUAHD = ∅. If we prove the claim, we get ⟨x, 0⟩ ∈ D \ intUAHD , but ⟨x, 0⟩ ̸∈ intUAHD UAH , thus D is not a closed domain in (X , UAH ), which is a contradiction. N. Alghamdi, L. Kalantan / Eur. J. Pure Appl. Math, 14 (4) (2021), 1161-1168 1166 Proof of Claim: Suppose Cr(⟨x, 0⟩)∩ intUAHD ̸= ∅. Pick ⟨u, v⟩ ∈ Cr(⟨x, 0⟩)∩ intUAHD . If v > 0, then ⟨u, v⟩ ∈ K, since Cr(⟨x, 0⟩) ⊆ Ur(⟨x, 0⟩), and (intUAHD )∩K = (intUD )∩K, then ⟨u, v⟩ ∈ Ur(⟨x, 0⟩) ∩ intUD , which is a contradicts (⋆). Then v = 0, so ⟨u, v⟩ = ⟨x, 0⟩, hence ⟨x, 0⟩ ∈ intUAHD , therefore there exists 0 < s < r such that Cs(⟨x, 0⟩) ⊆ intUAHD ⊂ D . By Lemma 3 Us(⟨x, 0⟩) ⊆ intUAHD , then ⟨x, 0⟩ ∈ intUD , but ⟨x, 0⟩ ∈ D \ intUD , which is a contradiction. So, claim is proved. Theorem 5. Any H-spaces (X , UAH ) is mildly normal. Proof. Let E and F be any tow disjoint closed domains in (X , UAH ), by Theorem 4, E and F are closed domains in (X,U), and (X,U) is mildly normal, then there exists U and V in U such that E ⊂ U , F ⊂ V and U ∩ V = ∅. Since U ⊆ UAH, then U and V are both open in H-space, thus (X,UAH) is mildly normal. Theorem 6. Any closed domain in usual metric space (X , U ) is closed domain in H- spaces (X,UAH). Proof. Let D be a closed domain in usual metric space, we want to show that intUAHD UAH = intUD U . We have intUAHD UAH ⊆ intUD U . Claim : intUD U ⊆ intUAHD UAH . Proof of claim: Let ⟨x, y⟩ ∈ intUD U be arbitrary, then for all r > 0 we have Ur(⟨x, y⟩) ∩ intUD ̸= ∅, by Lemma 2 we have Ur(⟨x, y⟩)∩intUAHD ̸= ∅, therefore ⟨x, y⟩ ∈ intUAHD UAH . Recall that a space X is semiregular if it has a base consisting of open domains, [1, 1.7.8 (a)], see also [9]. Now, let (X , τ ) be a T2 space. Generate a coarser topology τ ′ ⊆ τ on X by the base of all open domains in (X , τ ). Then (X , τ ′ ) is semiregular and the two spaces (X , τ ) and (X , τ ′ ) have the same open domains. (X , τ ′ ) is called the semiregularization of (X , τ ) [1, 1.7.8 (b)], see also [9]. Since any closed domain in an H-space (X , UAH ) is a closed domain in the usual metric space (X , U ), see Theorem 4 and Theorem 6, we conclude that any open domain in an H-space (X,UAH) is an open domain in the usual metric space (X , U ). Thus the semiregularization of an H-space (X , UAH ) is (X , U ). We can conclude more interesting result from Theorem 4. Since anH-space (X , UAH ) and the usual metric space (X , U ) are having the same closed domain, then any H-space (X , UAH ) is κ-metrizable. Let us recall the definitions. Denote the family of all closed domains in X by R[X]. A κ-metric on a T3 space is a non-negative real-valued function ϕ(x,C) of two variables, x ∈ X and C ∈ R[X], with the requirements: (i) (K1) (membership axiom) For every x ∈ X and C ∈ R[X], ϕ(x,C) = 0 ⇔ x ∈ C. (ii) (K2) (monotonicity) If C,C ′ ∈ R[X] and C ⊂ C ′, then ϕ(x,C) ≥ ϕ(x,C ′), for all x ∈ X. (iii) (K3) (continuity) For every C ∈ R[x], ϕ(x,C) is continuous in x. 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