EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 62-70 ISSN 1307-5543 – ejpam.com Published by New York Business Global Results about C-κ-normality and C-mild normality Lutfi Kalantan1, Alya’a Al-Awadi1,2,∗, Sadeq Thabit3 1 King Abdulaziz University, Department of Mathematics, P.O.Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah, 21589, Saudi Arabia 3 Hadhramout University, Department of Mathematics, Yemen Abstract. A topological space X is C-κ-normal (C-mildly normal ) if there exist a κ-normal (mildly normal) space Y and a bijective function f : X −→ Y such that the restriction f|A : A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X. We present new results about those two topological properties and use a discrete extension space to solve open problems regarding C2-paracompactness and α-normality. 2020 Mathematics Subject Classifications: 54D15, 54C10 Key Words and Phrases: κ-normal, normal, mildly normal, compact, C-normal, C−κ-normal, C-mildly normal, minimal Hausdorff, discrete extension, epinormal, epi-mildly normal, α-normal, C2-paracompact 1. Preliminaries In the present work, we give some new results about C-κ-normality and C-mild nor- mality [2] and use the discrete extension space to answer the open problems “Is C2- paracompactness hereditary with respect to closed subspaces?” [5] And “Is α-normality preserved by the discrete extension?” [3]. Throughout this paper, we denote the set of positive integers by N, the rationals by Q, the irrationals by P, and the set of real numbers by R. Two subsets A and B of a space X are called separated if there are two disjoint open subsets U and V such that A ⊆ U and B ⊆ V . A space X is regular if for any closed subset E of X and for any element x ∈ X \E we have {x} and E can be separated. A T3 space is a T1 regular space, a normal space is a space where any two disjoint closed subsets can be separated, a T4 space is a T1 normal space, and a Tychonoff space (T3 1 2 ) is a T1 completely regular space. We do not ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4607 Email addresses: lkalantan@kau.edu.sa, lnkalantan@hotmail.com (L. Kalantan), amohammedalawadi@stu.kau.edu.sa, aaalawadi@uj.edu.sa (A. Alawadi), sthabit@hu.edu.ye,sthabit1975@gmail.com. (S. Thabit) https://www.ejpam.com 62 © 2023 EJPAM All rights reserved. L. Kalantan, A.Alawadi, S.Thabit / Eur. J. Pure Appl. Math, 16 (1) (2023), 62-70 63 assume T2 in the definition of compactness, countable compactness, local compactness, and paracompactness. 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. An ordinal γ is the set of all ordinal α such that α < γ . The first infinite ordinal is ω0 and the first uncountable ordinal is ω1. A subset A of a space X is called a closed domain [12], called also regularly closed [23], κ-closed [14], if A = intA. On 1972, S̆c̆epin introduced the notion of κ-normality [22]. A space X is κ-normal if X is regular and any two disjoint closed domains can be separated. About the same time, Singal defined the notion of mild normality [23]. A space X is mildly normal if any two disjoint closed domains can be separated. We begin by recalling the following definitions. Definition 1. [15] A space (X , τ ) is called epi-mildly normal if there exists a coarser topology τ ′ on X such that (X , τ ′ ) is Hausdorff (T2) mildly normal . Definition 2. [4] A topological space (X , τ ) is called epi-normal if there is a topology τ ′ on X coarser than τ such that (X , τ ′ ) is T4. In a personal contact, Arhangel’skii intoduced in 2012 to Kalantan the following defi- nition: Definition 3. (Arhangel’skii) A topological space X is called C-κ-normal if there exist a κ-normal space Y and a bijective function f : X −→ Y such that the restriction f|A : A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X. Definition 4. [2] A topological space X is called C-mildly normal if there exist a mildly normal space Y and a bijective function f : X −→ Y such that the restriction f|A : A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X . In [2], the following theorem was proved. Theorem 1. If X is C-mildly normal (C-κ-normal) Fréchet space and f : X −→ Y is a witness of the C-mild normality (C-κ-normality ) of X, then f is continuous. 2. Main Results and Examples Recall that a topological space X is called almost compact [19] if each open cover of X has a finite subfamily such that the closures of whose members covers X. A space X is said to be almost regular [23] if for any closed domain subset A and any x ̸∈ A, there exist two disjoint open sets U and V such that x ∈ U and A ⊆ V . A technique which is useful in the theory of coarser topologies is the semiregularization. The topology on X generated by the family of all open domains is denoted by τ s. The space (X, τ s) is called the semiregularization of X. A space (X , τ ) is semi-regular if τ=τ s. L. Kalantan, A.Alawadi, S.Thabit / Eur. J. Pure Appl. Math, 16 (1) (2023), 62-70 64 Theorem 2. Let X be an almost regular Hausdorff space. If X is mildly normal then X is C-κ-normal. Proof. Since (X ,τ ) is an almost regular Hausdorff space, then (X,τ s) is a Hausdorff regular space [20]. Since X is mildly normal space we get (X,τ s) is mildly normal [15]. Then the identity function idX : (X ,τ ) −→ (X,τ s) is a continuous bijective function. If C is any compact subspace of (X ,τ ), then the restriction of the identity function from C onto idX(C) is continuous and “every continuous one-to-one mapping of a compact space onto a Hausdorff space is a homeomorphism.” [12, Theorem 3.1.13]. So X is C-κ-normal space. Theorem 3. If (X , τ ) is almost regular almost compact space and τ s is T1, then (X , τ ) is C-κ-normal (C-mildly normal). Proof. Since (X , τ ) is an almost regular space, (X,τ s) is regular space [20]. Hence (X , τ s) is T3. Moreover, the coarser topology of an almost compact space is an almost compact space. So τ s is almost compact. But every almost regular almost compact space is mildly normal [23]. Thus τ s is regular mildly normal. Therefore by using the same argument of the proof of theorem 2 we conclude that (X, τ ) is C-κ-normal (C-mildly normal). Theorem 4. C-κ-normality (C-mild normality) is an additive property. Proof. Let Xα be a C-κ-normal (C-mildly normal) space for each α ∈ Λ. We show that their sum ⊕α∈ΛXα is C-κ-normal (C-mildly normal). For each α ∈ Λ, pick a κ- normal (mildly normal) space Yα and a bijective function fα : Xα −→ Yα such that fα|Cα : Cα −→ fα(Cα) is a homeomorphism for each compact subspace Cα of Xα. Since regularity is additive [12, Theorem 2.2.7], then (Yα,⊕α∈Λτ ′ α) is a regular space. On the other hand, mild normality is an additive property because each factor is open-and-closed in ⊕α∈ΛXα and the intersection of any closed domain in ⊕α∈ΛXα with each factor Xα will be a closed domain in Xα. Then the sum ⊕α∈ΛYα is κ-normal (mildly normal). Consider the function sum [12, Exercises 2.2.E], ⊕α∈Λfα : ⊕α∈ΛXα −→ ⊕α∈ΛYα defined by ⊕α∈Λfα(x) = fβ(x) if x ∈ Xβ, β ∈ Λ. Now, a subspace C ⊆ ⊕α∈ΛXα is compact if and only if the set Λ0 = {α ∈ Λ : C ∩Xα ̸= ∅} is finite and C ∩Xα is compact in Xα for each α ∈ Λ0. If C ⊆ ⊕α∈ΛXα is compact, then (⊕α∈Λfα)|C is a homeomorphism because fα|C∩Xα is a homeomorphism for each α ∈ Λ0. Recall that a topology τ on a non-empty set X is said to be minimal Hausdorff if (X , τ ) is Hausdorff and there is no Hausdorff topology on X strictly coarser than τ , see [7, 8]. It was proved that “if the product space is minimal Hausdorff, then each factor is minimal Hausdorff” [7], In [13] the converse of the previous statement was proved. Namely, “the product of minimal Hausdorff spaces is minimal Hausdorff”. In the next theorem we will use the following theorem: “A minimal Hausdorff space is compact if and only if it is completely Hausdorff (T2 1 2 )”[21, Theorem 1.4]. We conclude the following theorems. L. Kalantan, A.Alawadi, S.Thabit / Eur. J. Pure Appl. Math, 16 (1) (2023), 62-70 65 Theorem 5. Let X and Y be minimal Hausdorff spaces, if X and Y are κ-normal spaces then X × Y is κ-normal. Proof. Since X and Y are κ-normal, they are regular Hausdorff spaces, which implies T3, hence T2 1 2 . Since the T2 1 2 is multiplicative, the product space is T2 1 2 . So X × Y is T2 1 2 minimal Hausdorff space which implies that X × Y is T2 compact, hence T4 and thus κ-normal. Now, we give the following characterization in the class of minimal Hausdorff spaces. Theorem 6. Let X be a minimal Hausdorff Fréchet space. The following are equivalent. (i) X is C-κ-normal. (ii) X is locally compact. (iii) X is compact (iv) X is T4. (v) X is epinormal, hence epi-mildly normal. Proof. (1) ⇒ (2) Since X is C-κ-normal Fréchet space, X is T2 1 2 see [2], By Theorem “A minimal Hausdorff space is compact if and only if it is completely Hausdorff (T2 1 2 )” [21, Theorem 1.4], gives that X is T2 compact, hence locally compact. (2) ⇒ (3) Since any T2 locally compact space is Tychonoff and hence T2 1 2 , we obtain X is compact. (3) ⇒ (4) Any T2 compact space is T4. (4) ⇒ (5) Any T4 is epinormal, hence epi-mildly normal. (5) ⇒ (1) Any epinormal space is C-κ-normal [2]. From the above theorem, we conclude the following corollary Corollary 1. In class of minimal Hausdorff, any Frėchet C-κ-normal space is κ-normal. Since κ-normality is not hereditary [16], it seems to us that both C-mild normality and C-κ-normality are not hereditary, but we still could not find a counterexample. The question “Is there a Tychonoff space which is not C-κ-normal (C-mildly normal) ?¨ We answer this in the class of minimal Tychonoff spaces by using theorem “All minimal completely regular spaces are compact”, [7], hence T4. So we get the following corollary. Corollary 2. Any minimal Tychonoff space is C-κ-normal. We know Tychonoff spaces which are not κ-normal (mildly normal). This spaces turn out to be C-κ-normal (C-mildly normal) see [2, example 3], and also see example 5 below. Let M be a non-empty proper subset of a topological space (X , τ ). Define a new topology τ (M) on X as follows: τ (M) = {U ∪K : U ∈ τ and K ⊆ X \M }. (X , τ (M) ) is called a discrete extension of (X , τ ) and we denote it by XM see [12, Example 5.1.22]. In general, C-κ-normality is not preserved by a discrete extension space. Here is an example of C-κ-normal space whose a discrete extension space is not C-κ-normal. L. Kalantan, A.Alawadi, S.Thabit / Eur. J. Pure Appl. Math, 16 (1) (2023), 62-70 66 Example 1. Consider (R , I ) where I is the indiscrete topology. Let M = R \ {1, 2, 3}. We have 1 /∈ M and M is closed in RM . The only open set in RM containing M is R. But R ∩ {1} ≠ ∅. Thus RM is not regular. So, RM is not C-κ-normal because it is a compact non-regular space [2]. Recall that a topological space X is called C2-paracompact if there exist a Hausdorff paracompact space Y and a bijective function f : X −→ Y such that the restriction f|A : A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X [17]. From definition since any T2 paracompact space is T4, any C2-paracompact space is C-κ-normal. The converse is not true, we did show in [2] that example 2 below is a C-κ-normal and it was shown in [5, example 4] it is not C2-paracompact. By using the discrete extension space, we answer the following open problem : “Is C2- paracompactness hereditary with respect to closed subspaces?” [5]. The answer is negative even for open subspaces and here is a counterexample. Example 2. Consider the infinite Tychonoff product space G = Dω1 = ∏ α∈ω1 D, where D = {0, 1} considered with the discrete topology. Let H be the subspace of G consisting of all points of G with at most countably many non-zero coordinates. Put M = G × H. Raushan Buzyakova proved that M cannot be mapped onto a normal space Z by a bijective continuous function [9, example 4] result and the fact that M is a k-space, we conclude that M is a Tychonoff space which is not C2-paracompact [5, example 4] . Let X be any compactification of M and consider the discrete extension space XM of X. By Theorem “Every lower compact space is C2-paracompact” [17, theorem 2.20] , XM is C2- paracompact. Since M as a subspace of XM is the same as a subspace of X and M is closed-and-open in XM , we get that C2-paracompactness is not hereditary with respect to both closed and open subspaces . Recall that a space X is called α-normal if for any two disjoint closed subsets A and B of X there exist disjoint open subsets U and V of X such that A∩U is dense in A and B ∩ V is dense in B [6]. We answer the following open problem : “Is α-normality preserved by the discrete extension?” [3]. The answer is no and here is an example of an α-normal space whose a discrete extension space is not α-normal. Example 3. Let M=((ω1 + 1) × (ω0 + 1)) \ {⟨ω1, ω0⟩} is a Tychonoff Plank space see [24, example 87] we know that M is a Tychonoff non α-normal space [6], take the com- pactification X of M then it is α-normal being T2 compact space. Consider the discrete extension XM . Observe that M is closed in XM . Since α-normality is hereditary with respect to closed subspaces [6], we conclude that XM cannot be α-normal. . The following example answers three kinds of invariants. We used two well-known spaces, the Alexandroff duplicate space and the closed extension space. Let X be any T1 topological space. Let X ′ = X × {1}. Note that X ∩ X ′ = ∅. Let A(X) = X ∪X ′. For simplicity, for an element x ∈ X, we will denote the element ⟨x, 1⟩ L. Kalantan, A.Alawadi, S.Thabit / Eur. J. Pure Appl. Math, 16 (1) (2023), 62-70 67 in X ′ by x′ and for a subset B ⊆ X let B′ = {x′ : x ∈ B} = B × {1} ⊆ X ′. For each x′ ∈ X ′, let B(x′) = {{x′}}. For each x ∈ X, let B(x) = {U ∪ (U ′ \ {x′}) : U is open in X with x ∈ U }. Let τ denote the unique topology on A(X) which has {B(x) : x ∈ X} ∪ {B(x′) : x′ ∈ X ′} as its neighborhood system. A(X) with this topology is called the Alexandroff Duplicate of X [11]. Example 4. Consider the Alexandroff duplicate space A(R) of R with its usual metric topology. It is C2-paracompact [5], hence C-κ-normal. Now, let i = √ −1 ̸∈ R and put X = R∪{i}. Let τ be the closed extension topology on X generated from R with its usual metric topology and i. So, τ= {∅} ∪ {W ∪ {i} : W ⊆ R;W is open in the usual metric topology }. (X , τ ) is not C-normal see [1]. So it is not not C-κ-normal because it is Fréchet being first countable, Lindelöf space, which is not C-normal [2, theorem0.10]. Define g : A(R) −→ X by g(x) = { i ; if x ∈ R′ x ; if x ∈ R g is an open onto function. Thus C-κ-normality is neither invariant, open invariant, nor quotient invariant. . Since κ-normality is not multiplicative, it seems to us that both C-mild normality and C-κ-normality are not multiplicative, but we still could not find a counterexample. We know the example of two linearly ordered topological spaces whose product is not κ-normal (mildly normal) was given in [14]. This space turns out to be C-κ-normal. Here is an example. Example 5. We will define a Hausdorff compact linearly ordered space Y such that ω1×Y is C-κ-normal. Let {yn : n < ω0} be a countably infinite set such that {yn : n < ω0} ∩ (ω1 + 1) = ∅. Let Y = {yn : n < ω0} ∪ (ω1 + 1). Let τ be the topology on Y generated by the following neighborhood system: For an α ∈ ω1, a basic open neighborhood of α is the same as in ω1 with its usual order topology. For n ∈ ω0, a basic open neighborhood of yn is {yn}. A basic open neighborhood of ω1 is of the form (α, ω1] ∪ {yn : n ≥ k} where α < ω1 and k ∈ ω0. In other words, {yn : n < ω0} is a sequence of isolated points which converges to ω1. Note that if we define an order < on Y as follows: For each n ∈ ω0, ω1 < yn+1 < yn, and < on ω1 + 1 is the same as the usual order on ω1 + 1, then (Y , τ ) is a linearly ordered topological space. It was shown in [14] that (Y , τ ) is a Hausdorff compact space, hence it is mildly normal. Also, it is well known that ω1 is a Hausdorff normal space and hence mildly normal. But ω1 × Y is not mildly normal [14]. A similar proof as in [15] shows that ω1 × Y is C-κ-normal. Here are cases when the product of two C-κ-normal spaces will be C-κ-normal. Since the product of ordinals is always κ-normal (mildly normal) [18], we conclude the following theorem. Theorem 7. The product of ordinals is C-κ-normal (C-mildly normal). L. Kalantan, A.Alawadi, S.Thabit / Eur. J. Pure Appl. Math, 16 (1) (2023), 62-70 68 Theorem 8. If X and Y are minimal Hausdorff Fréchet C-κ-normal, then X × Y is C-κ-normal. Proof. Since X and Y are minimal Hausdorff Fréchet C-κ-normal, X and Y are T2 1 2 [2]. Since the T2 1 2 is multiplicative, the product space is T2 1 2 . So X×Y is T2 1 2 minimal Hausdorff space implies that X × Y is T2 compact, hence C-κ-normal. Theorem 9. If X is Fréchet and countably compact C-κ-normal space, and Z is T2 paracompact first countable space then X × Z is C-κ-normal. Proof. Let Y be a κ-normal space, f : X −→ Y be a bijective function such that the restriction on any compact subspace is a homeomorphism. Now, X is Fréchet gives that f is continuous, see Theorem 1. Since X is countably compact and f continuous surjective, we have Y is countably compact κ-normal . Since a product of a countably compact κ-normal space with a paracompact first countable space is κ-normal , Y × Z is κ-normal [14]. Now, define g : X × Z −→ Y × Z by g(⟨x, i⟩) = ⟨f(x), i⟩. Then g is a bijective function and g = f × idZ , where idZ is the identity function on Z. Let C be any compact subspace of X × Z. Then C ⊆ p1(C) × p2(C), where p1 and p2 are the usual projection functions. p1(C) is a compact subspace of X and p2(C) is a compact subspace of Z, thus p1(C) × p2(C) is a compact subspace of X × Z. Now, f|p1(C) : p1(C) −→ f(p1(C)) is a homeomorphism and idZ|p2(C) : p2(C) −→ p2(C) is a homeomorphism. Thus (f × idZ)|(p1(C)×p2(C)) : p1(C) × p2(C) −→ fp1(C)× p2(C) is a homeomorphism. We conclude that g|C : C −→ g(C) is a homeomorphism because g|C = ((f × idZ)|p1(C)×p2(C) )|C . Recall that a space X is Dowker if X is T4 and X × I is not normal, where I is the closed unit interval considered with its usual metric topology, [12]. Dowker, in [10], stated the following theorem: “A space X is normal and countably paracompact if and only if X× I is normal”. Here is a C-κ-normal version, one direction of the Dowker’s theorem. If C-κ-normality is hereditary with respect to closed spaces, then the converse will be true. Theorem 10. If X is T1 Fréchet C-κ-normal Lindelöf space , then X × I is C-κ-normal space. Proof. Let X be a T1 Fréchet C-κ-normal Lindelöf space. Pick a witness function f and a κ-normal space Y . Then by Theorem 1 the witness function f : X −→ Y is continuous and the witness space Y is T3, [2]. Since X is Lindelöf and T3, we get Y is paracompact, and hence T4. So, by Dowker’s theorem Y × I is T4. By a similar argument as in the proof of Theorem9, we can prove that X × I is a C-κ-normal space. Recall that a space X is called nearly compact [24] if each open cover of X has a finite subfamily the interiors of the closures of whose members covers X. REFERENCES 69 Theorem 11. Let X, Y are Hausdorff nearly compact C-κ-normal space, then X × Y is a C-κ-normal space. Proof. Since (X ,τ ) and (Y ,τ ′ ) are nearly compact Hausdorff spaces, we get (X,τ s) and (Y ,τ ′s) are Hausdorff compact spaces [20], and (X,τ s) × (Y ,τ ′s) is a T2 compact topological space which is coarser than the topology on X ×Y . Thus, X ×Y is epinormal and hence it is C-κ-normal [2]. The following problems are still open: (i) Does there exist a Tychonoff space which is not C-κ-normal? Observe that such a space is not in the class of minimal Hausdorff space, not in the class of minimal T3 spaces, not locally compact, not submetrizable, not C-normal, a space can not be ordinal, can not epinormal, not Lindelöf. Observe also that the existence of such a space, will show that C-κ-normal is not hereditary just by taking a compactification of it. (ii) Is C-κ-normality (C-mild normality) multiplicative? References [1] D Abuzaid, S Al-Qarhi, and L Kalantani. Closed extension topological spaces. Eu- ropean Journal of Pure and Applied Mathematics (EJPAM), 15(2):672–680, 2022. [2] A Al-Awadi, L Kalantan, and S Thabit. c-κ-normal and c-mildly normal topological properties. to appear. [3] A Alawadi, L Kalantan, and M Saeed. On the discrete extension spaces. Journal of Mathematical Analysis, 9(2):150–157, 2018. [4] S AlZahrani and L Kalantan. Epinormality. Journal of Nonlinear Sciences & Appli- cations, 9(9):5398–5402, 2016. [5] H Alzumi, L Kalantan, and M M Saeed. Results on c2-paracompactness. European Journal of Pure and Applied Mathematics., 14(2):351–357, 2021. [6] A V Arhangel’skii and L D Ludwig. On α-normal and β-normal spaces. Commenta- tiones Mathematicae Universitatis Carolinae., 42(3):507–519, 2001. [7] M P Berri. Minimal topological spaces. Transactions of the American Mathematical Society., 108(1):97–105, 1963. [8] N Bourbaki. General Topology. Springer-Verlag, 1995. [9] R Z Buzyakova. An example of a product of two normal groups that cannot be con- densed onto a normal space. Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika., 3:59–59, 1997. REFERENCES 70 [10] C H Dowker. On countably paracompact spaces. Canadian Journal of Mathematics., 3:219–224, 1951. [11] R Engelking. On the double circumference of alexandroff. Bull. Acad. Pol. Sci. Ser. Astron. Math. Phys., 16(8):629–634, 1968. [12] R Engelking. General Topology. PWN, Warszawa, 1977. [13] S Ikenage. Product of minimal topological spaces. Proceedings of the Japan Academy., 40(5):329–331, 1964. [14] L Kalantan. Results about κ-normality. Topology and its Applications, 125(1):47–62, 2002. [15] L Kalantan and I Alshammari. Epi-mild normality. Open Mathematics., 16(1):1170– 1175, 2018. [16] L Kalantan and N Kemoto. Mild normality in products of ordinals. Houston Journal of Mathematics., 29(4):937–947, 2003. [17] L Kalantan, M M Saeed, and H Alzumi. c-paracompactness and c2-paracompactness. Turkish Journal of Mathematics., 43(1):9–20, 2019. [18] L Kalantan and P Szeptycki. κ-normality and products of ordinals. Topology and its Applications, 123(3):537–545, 2002. [19] P T Lambrinos. On almost compact and nearly compact spaces. Rendiconti del Circolo Matematico di Palermo, 24(1):14–18, 1975. [20] M Mršević, I L Reilly, and M K Vamanamurthy. On semi-regularization topologies. Journal of the Australian Mathematical Society, 38(1):40–54, 1985. [21] J R Porter and R M Stephenson. Minimal hausdorff spaces—then and now. In Handbook of the History of General Topology, pages 669–687, 1998. [22] E V Shchepin. Real functions and spaces that are nearly normal. Sibirskii Matem- aticheskii Zhurnal, 13(5):1182–1196, 1972. [23] M K Singal and A R Singal. Mildly normal spaces. Kyungpook Mathematical Journal, 13(1):29–31, 1973. [24] L Steen and J A Seebach. Counterexamples in Topology. Dover Publications INC, USA, 1995.