EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 3, 2020, 697-700 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Quasi-normality of Mrówka spaces Ibtesam Alshammari1,∗, Lutfi Kalantan2 1 Department of Mathematics, Faculty of Science, University of Hafr Al Batin, P.O.Box 1803, Hafr Al Batin 31991, Saudi Arabia 2 Department of Mathematics, Faculty of Science, King Abdulaziz University Abstract. A topological space X is called quasi-normal if X is regular and any two disjoint π- closed subsets A and B of X are separated. We give a Mrówka space which is not quasi-normal and use the continuum hypothesis (CH) and truly cardinality c to present Mrówka spaces which are quasi-normal. 2020 Mathematics Subject Classifications: 54D15, 54D10 Key Words and Phrases: Normal, π-normal, mildly normal, quasi-normal, closed domain, π-closed, Mrówka space, continuum hypothesis (CH), truly cardinality c 1. Introduction In this paper, we give a Mrówka space which is not quasi-normal and use the continuum hypothesis (CH) and truly cardinality c to present Mrówka spaces which are quasi-normal. Throughout this paper, we denote an ordered pair by 〈x, y〉 and the set of positive integers by N. A T4 space is a T1 normal space, a Tychonoff (T3 1 2 ) space is a T1 completely regular space, and a T3 space is a T1 regular space. For a subset A of a space X, intA and A denote the interior and the closure of A, respectively. An ordinal γ is the set of all ordinal α such that α < γ. The first infinite ordinal is ω and the first uncountable ordinal is ω1. Definition 1. Two disjoint subsets E and F of a space X are called separated if there exist two disjoint open sets U and V such that E ⊆ U and F ⊆ V . A subset A of a space X is called closed domain [1], called also regularly closed, κ-closed, if A = intA. A space X is called mildly normal [6], called also κ-normal [5], if any two disjoint closed domains A and B of X are separated. In [5], Stchepin required regularity in his definition of κ-normality. A subset A of a space X is called π-closed [8] if A is a finite intersection of closed domains. A space X is called π-normal [3] if any two disjoint closed subsets A and B of X one of which is π-closed are separated. A space X is called quasi-normal [8] if X is regular and any two disjoint π-closed subsets A and B of X are separated, see also [3]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3770 Email addresses: lkalantan@hotmail.com (L. Kalantan), iealshamri@hotmail.com and iealshamri@uhb.edu.sa (I. Alshammari) https://www.ejpam.com 697 c© 2020 EJPAM All rights reserved. I. Alshammari, L. Kalantan / Eur. J. Pure Appl. Math, 13 (3) (2020), 697-700 698 Since any closed domain is π-closed and any π-closed is closed, then it is clear from the definitions that normal =⇒ π-normal =⇒ quasi-normal =⇒ mildly normal. Recall that two countably infinite sets are said to be almost disjoint [7] if their inter- section is finite. Call a subfamily of [ω]ω = {A ⊂ ω : A is infinite } a mad family [7] on ω if it is a maximal (with respect to inclusion) pairwise almost disjoint subfamily. Let A be a pairwise almost disjoint subfamily of [ω]ω. The Mrówka space Ψ(A) is defined as follows: The underlying set is ω ∪ A, each point of ω is isolated, and a basic open neighborhood of W ∈ A has the form {W} ∪ (W \ F ), with F ∈ [ω]<ω = {B ⊆ ω : B is finite}. 2. Main Results It is well known that there exists an almost disjoint family A ⊂ [ω]ω such that |A| > ω and the Mrówka space Ψ(A) is a Tychonoff, separable, first countable, and locally compact space which is neither countably compact nor normal. And A is a mad family if and only if Ψ(A) is pseudo compact [4]. The interesting thing about Mrówka spaces is that some Mrówka spaces are quasi- normal and some are not. In [2, 1.3], a mad family R ⊂ [ω]ω was constructed such that the Mrówka space Ψ(R) is not mildly normal. So, such a Mrówka space cannot be quasi- normal. Now, we use the continuum hypothesis (CH) to produce a mad family A ⊂ [ω]ω such that its Mrówka space Ψ(A) is quasi-normal. The existence of such a mad family in ZFC is still unsettled. Proposition 1. Under CH, there exists a mad family A such that Ψ(A) is quasi-normal. Proof. Let P = {Pi : i < ω} be a partition of ω such that for each i < ω, Pi is infinite. We will use P to build our mad family. Let E = [[ω]ω]<ω. That is, the family of all finite subsets of [ω]ω. Consider the family B = {〈C,D〉 : C,D ∈ E , (∩C ) ∩ (∩D ) = ∅}. Using CH, we can write B = {〈Cα, Dα〉 : α < ω1}. We will build our mad family recursively on α < ω1. For α = 0, C0 = {A0,1, ..., A0,n} and D0 = {B0,1, ..., B0,m} for some n,m ∈ N. If for each i ≤ n and each j ≤ m there exist G0,i ∈ [A0,i] ω and H0,j ∈ [B0,j ] ω such that P ∪{G0,i} and P ∪{H0,j} are almost disjoint, let E0 = ( ⋃n i=1G0,i)∪ ( ⋃m j=1H0,j) and put A0 = P ∪ {E0}, which is almost disjoint. Otherwise let A0 = P. Now, for each 0 < α < ω1, assume we have built Aβ for each β < α. If α is a limit ordinal, let A′α = ⋃ β<αAβ. It is clear that A′α is an almost disjoint family. Now consider 〈Cα, Dα〉, we write Cα = {Aα,1, ..., Aα,n} and Dα = {Bα,1, ..., Bα,m} for some n,m ∈ N. We proceed as before, if for each i ≤ n and each j ≤ m there exist Gα,i ∈ [Aα,i] ω and Hα,j ∈ [Bα,j ] ω such that A′α ∪ {Gα,i} and A′α ∪ {Hα,j} are almost disjoint, let Eα = I. Alshammari, L. Kalantan / Eur. J. Pure Appl. Math, 13 (3) (2020), 697-700 699 ( ⋃n i=1Gα,i)∪ ( ⋃m j=1Hα,j) and put Aα = A′α∪{Eα}. Otherwise let Aα = A′α. If α = β+1, let A′α = Aβ and consider 〈Cα, Dα〉. Construct Aα by doing the process as before. Finally, let A = ⋃ α<ω1 Aα. Clearly, A is almost disjoint. In order to show that A is maximal, let M be any infinite subset of ω. We need to show that there exists E ∈ A such that E ∩M is infinite. Suppose that for each E ∈ A, |E ∩M | < ω. Partition M into two infinite subsets M1 and M2. Pick the least α < ω1 such that 〈Cα, Dα〉 = 〈{M1}, {M2}〉 = 〈{Aα,1}, {Bα,1}〉. Since for each E ∈ A, E ∩M is finite, we have that for each E ∈ A′α, |E ∩M | < ω. Thus, for each E ∈ A′α we have |E ∩M1| < ω and |E ∩M2| < ω. That is, M1 ∈ [Aα,1] ω and M2 ∈ [Bα,1] ω satisfy that A′α∪{M1} and A′α∪{M2} are almost disjoint, hence there exists Eα ∈ Aα ⊂ A such that Eα ∩M is infinite which is a contradiction. So, A is mad. Claim: Ψ(A) is quasi-normal. Proof of Claim: Let A and B be non-empty disjoint π-closed subsets of Ψ(A). Write A =⋂n i=1Ai and B = ⋂m j=1Bj where each Ai and Bj are closed domains for each i ∈ {1, ..., n} and j ∈ {1, ...,m}. Observe that if there exists i ∈ {1, ..., n} such that |Ai ∩ ω| < ω, then Ai ∩ A = ∅ because for each a ∈ A we have {a} ∪ (a \ Ai) is an open neighborhood of a disjoint from Ai. Hence, A is a finite closed-and-open subset of Ψ(A) which can be separated from B. Similarly, if there exists j ∈ {1, ...,m} such that |Bj ∩ ω| < ω, then B can be separated from A. So, assume that for each i ∈ {1, ..., n} and j ∈ {1, ...,m}, |Ai ∩ ω| = ω = |Bj ∩ ω|. Take the least α < ω1 such that for each i ∈ {1, ..., n} and j ∈ {1, ...,m} we have Aα,i = Ai ∩ ω and Bα,j = Bj ∩ ω. Recalling our construction, at stage α, either Aα = A′α or Aα = A′α ∪ {Eα}. But, Aα = A′α ∪ {Eα} is not possible since Eα = ( ⋃n i=1Gα,i) ∪ ( ⋃m j=1Hα,j) for some Gα,i ∈ [Aα,i] ω and Hα,j ∈ [Bα,j ] ω and that implies Eα is in the closure of each Aα,i and each Bα,j , hence Eα ∈ A ∩ B and this is a contradiction as A ∩ B = ∅. Thus, Aα = A′α and this means that for some i ∈ {1, ..., n}, it is the case that for each Gα,i ∈ [Aα,i] ω, A′α ∪ {Gα,i} is not almost disjoint or, for some j ∈ {1, ...,m}, every infinite Hα,j ⊆ Bα,j , is so that A′α ∪ {Hα,j} is not almost disjoint. Without loss of generality, assume that there exists such i ∈ {1, ..., n}. Observe that A′α is countable, hence A′α �Aα,i= {a ∈ A′α : |a ∩ Aα,i| = ω} is either finite or countably infinite. But, it cannot be countably infinite because {a ∩ Aα,i : a ∈ A′α �Aα,i} would be a countably infinite almost disjoint family on the set Aα,i. Hence, it is not maximal and there is Gα,i ∈ [Aα,i] ω such that {a ∩ Aα,i : a ∈ A′α} ∪ {Gα,i} is almost disjoint, contradicts that for each Gα,i ∈ [Aα,i] ω, A′α ∪ {Gα,i} is not almost disjoint. Therefore, F = {a ∈ A′α : |a ∩Aα,i| = ω} is finite. Claim: |Aα,i \ ⋃ F | < ω. Assume |Aα,i \ ⋃ F | = ω, then Aα,i \ ⋃ F ∈ [Aα,i] ω, and by our hypothesis, A′α ∪ {Aα,i \ ⋃ F} is not almost disjoint. 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