Relatively Strongly Normal and Relatively Normal Subspaces - m-hikari

International Mathematical Forum, 5, 2010, no. 12, 579 - 586

Relatively Strongly Normal and Relatively Normal Subspaces M. S. Sarsak Department of Mathematics, Faculty of Science The Hashemite University P.O. Box 150459, Zarqa 13115, Jordan [email protected] H. Z. Hdeib Department of Mathematics, Faculty of Science Jordan University, Jordan [email protected]

Abstract. The primary purpose of this paper is to obtain some results concerning some questions in [1]. Mathematics Subject Classification: Primary 54D15; Secondary 54D20, 54D30 Keywords: relatively regular, normal, relatively normal, relatively strongly normal, compact, Lindel¨ of, paracompact, α-paracompact, σ-paracompact 1. Introduction A subspace Y of a space X is said to be strongly normal (resp. normal) in X if for every two disjoint subsets A and B of Y that are closed in Y (resp. in X), there are disjoint open subsets U and V of X such that A ⊂ U, B ⊂ V . It is clear that if Y is strongly normal in a space X, then Y is normal in X and nomal as a subspace. It is also easy to see that if X is normal, and Y is a subspace of X, then Y is normal in X. In [2], Archangel’skii and Genedi showed that every compact subspace Y of a T2 -space X is strongly normal in X. In [1], Archangel’skii and Tartir posed the following questions:

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Question 1. Characterize regular spaces Y , that are strongly normal in every larger regular space X. Question 2. Characterize regular spaces Y , that are normal in every larger regular space X. In this note, we obtain some results concerning these questions. To proceed, we need the following definitions and facts. Definition 1.1. Let A and B be two collections of subsets of a space X. We say that A is a refinement of B provided that for each A ∈ A, there exists B ∈ B such that A ⊂ B. Observe that the refinement is usually used in literature for covers. Definition 1.2. If A = {Aα : α ∈ Λ} and B = {Bα : α ∈ Λ} are covers of a space X, then A is called a shrinking of B provided that for each α ∈ Λ, Aα ⊂ Bα . Observe that the shinking is usually used in literature for open covers. Definition 1.3. Let X be a space. A collection A of subsets of X is called point finite, if each point of X belongs to only finitely many members of A. Definition 1.4. Let X be a space. A collection A of subsets of X is called locally finite, if for each x ∈ X, there is an open subset U of X such that U intersects only finitely many members of A; A is called σ-locally finite if it is the countable union of locally finite collections. Definition 1.5. [3] A subset Y of a space X is called α-paracompact (resp. σ-paracompact), if every open cover of Y in X has a refinement which is open and locally finite (resp. σ-locally finite) in X, and covers Y . Theorem 1.6. [3] Let Y be an α-paracompact subset of a T2 -space X. Then Y is closed in X. Theorem 1.7. [3] (i) Let A and B be two disjoint α-paracompact subsets of a T2 -space X. Then there are disjoint open sets U, V in X such that A ⊂ U, B ⊂ V. (ii) Let A and B be two disjoint closed σ-paracompact subsets of a regular space X. Then there are disjoint open sets U, V in X such that A ⊂ U, B ⊂ V . In [3], it was shown that if X is paracompact and A is closed in X, then A is α-paracompact. The following theorem contains an improvement of this result, the straightforward proof is omitted. Theorem 1.8. (i) Let Y be an α-paracompact subset of a space X and A ⊂ Y . If A is closed in Y , then A is α-paracompact. (ii) Let Y be a σ-paracompact subset of a space X and A ⊂ Y . If A is closed in Y , then A is σ-paracompact.

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Throughout this paper, a space stands for a topological space with no separation axioms are assumed unless stated explicitly. If A is a subset of a space X, Int (A), Bd (A) and A denote, respectively, the interior of A in X, the boundary of A in X and the closure of A in X. For the concepts and terminology not defined here, we refer the reader to [4]. 2. strongly normal and normal subspaces In this section, we obtain some results related to strongly normal and normal subspaces in larger spaces. We first begin with the following observation, the easy proof is omitted. Proposition 2.1. Let Y be a closed subset of a space X. Then the following are equivalent: (i) Y is strongly normal in X. (ii) Y is normal in X. The following theorem provides with several characterizations of normal subspaces in larger spaces. Theorem 2.2. Let Y be a subspace of a space X. Then the following are equivalent: (i) Y is normal in X. (ii) For each subset A of Y which is closed in X, and for each open subset U of X such that A ∪ (X\Y ) ⊂ U, there is an open subset V of X such that A ⊂ V ⊂ V ⊂ U. (iii) Any point finite open cover U of X, each of whose members contains X\Y , has an open shrinking. (iv) Any locally finite open cover U of X, each of whose members contains X\Y , has an open shrinking. (v) Any finite open cover U of X, each of whose members contains X\Y , has an open shrinking. Proof. (i)→(ii): Let A be a subset of Y which is closed in X, and let U be an open subset of X such that A ∪ (X\Y ) ⊂ U. Then X\U is a subset of Y which is closed in X and disjoint from A. Since Y is normal in X, there are disjoint open subsets V and W of X such that A ⊂ V, X\U ⊂ W , so V ∩ W = φ, and thus V ∩ X\U = φ, that is, V ⊂ U. (ii)→(iii): Let U = {Uα ⊃ X\Y : α ∈ Λ} be a point finite open cover of X. Well order the set Λ, for convenience, suppose Λ = {1, 2, ..., α, ...}. Now construct V = {Vα : α ∈ Λ} by transfinite induction as follows: let A1 = X\ (∪α>1 Uα ). Then A1 is closed in X, A1 ⊂ Y and A1 ∪ (X\Y ) ⊂ U1 , so by (ii), there is an open subset V1 of X such that A1 ⊂ V1 ⊂ V1 ⊂ U1 . Suppose that Vβ has

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been defined for each β < α, and let Aα = X\ [(∪βα Uγ )]. Then Aα ⊂ Y , Aα is closed in X and Aα ∪ (X\Y ) ⊂ Uα . Thus by (ii), there is an open subset Vα of X such that Aα ⊂ Vα ⊂ Vα ⊂ Uα . Now if x ∈ X, then x belongs to only finitely many elements of U, say Uα1 , Uα2 , ..., Uαn . Let / Uγ for any γ > α and hence, if x ∈ / Vβ for α = max {α1 , α2 , ..., αn }. Now x ∈ any β < α, then x ∈ Aα ⊂ Vα . Hence, in any case, x ∈ Vβ for some β ≤ α. Thus, V is an open shrinking of U. The implications (iii)→(iv) and (iv)→(v) follow since every locally finite family is point finite, and since every finite family is locally finite. (v)→(i): Let A and B be two disjoint subsets of Y that are closed in X. Then U = {X\A, X\B} is a finite open cover of X, each of whose members contains X\Y . Thus by (v), U has an open shrinking V = {V1 , V2 }. Suppose that V1 ⊂ X\A and V2 ⊂ X\B. Then A ⊂ X\V 1 and B ⊂X\V2 , X\V 1 and X\V2 are open subsets of X and X\V1 ∩ X\V2 = X\ V1 ∪ V2 = X\X = φ. Hence, Y is normal in X. The following theorem provides with several characterizations of strongly normal subspaces in larger spaces, although the proof is similar to that of Theorem 2.2, we will show it for the convenience of the reader. Theorem 2.3. Let Y be a subspace of a space X. Then the following are equivalent: (i) Y is strongly normal in X. (ii) For each closed subset A of Y , and for each open subset U of Y such that A ⊂ U, there is an open subset V of X such that A ⊂ V ⊂ V ⊂ U ∪ (X\Y ). (iii) For any point finite (in X) open (in Y ) cover U of Y , the cover U ∗ = {U ∪ (X\Y ) : U ∈ U} of X has an open shrinking. (iv) For any locally finite (in X) open (in Y ) cover U of Y , the cover U ∗ = {U ∪ (X\Y ) : U ∈ U} of X has an open shrinking. (v) For any finite open (in Y ) cover U of Y , the cover U ∗ = {U ∪ (X\Y ) : U ∈ U} of X has an open shrinking. Proof. (i)→(ii): Let A be a closed subset of Y , and let U be an open subset of Y such that A ⊂ U. Then Y \U is a closed subset of Y and disjoint from A. Since Y is normal in X, there are disjoint open subsets V and W of X such that A ⊂ V, Y \U ⊂ W , so V ∩ W = φ, and thus V ∩ Y \U = φ, that is, V ⊂ U ∪ (X\Y ). (ii)→(iii): Let U = {Uα : α ∈ Λ} be a point finite (in X) open (in Y ) cover of Y . Well order the set Λ, for convenience, suppose Λ = {1, 2, ..., α, ...}. Now construct V = {Vα : α ∈ Λ} by transfinite induction as follows: let A1 = Y \ (∪α>1 Uα ). Then A1 is closed in Y and A1 ⊂ U1 , so by (ii), there is an open subset V1 of X such that A1 ⊂ V1 ⊂ V1 ⊂ U1 ∪ (X\Y ). Suppose that Vβ


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has been defined for each β < α, and let Aα = Y \ [(∪βα Uγ )] = Y \ [(∪βα Uγ )]. Then Aα ⊂ Uα and Aα is closed in Y . Thus by (ii), there is an open subset Vα of X such that Aα ⊂ Vα ⊂ Vα ⊂ Uα ∪ (X\Y ). Now if x ∈ X, then x belongs to only finitely many elements of U, say / Uγ for any γ > α and Uα1 , Uα2 , ..., Uαn . Let α = max {α1 , α2 , ..., αn }. Now x ∈ hence, if x ∈ / Vβ for any β < α, then x ∈ Aα ⊂ Vα . Hence, in any case, x ∈ Vβ for some β ≤ α. Thus, V is an open shrinking of U ∗ = {U ∪ (X\Y ) : U ∈ U}. The implications (iii)→(iv) and (iv)→(v) follow since every locally finite family is point finite, and since every finite family is locally finite. (v)→(i): Let A and B be two disjoint closed subsets of Y . Then U = {Y \A, Y \B} is a finite open (in Y ) cover of Y . Thus by (v), U ∗ = {X\A, X\B} has an open shrinking V = {V1 , V2 }. Suppose that V1 ⊂ X\A and V2 ⊂ X\B. Then A ⊂X\V1 and B ⊂ X\V2 ,X\V1 and X\V2 are open subsets of X and X\V1 ∩ X\V2 = X\ V1 ∪ V2 = X\X = φ. Hence, Y is strongly normal in X. The first part of the following theorem generalizes the result of Archangel’skii and Genedi which states that every compact subspace Y of a T2 -space X is strongly normal in X. Theorem 2.4. (i) Let Y be an α-paracompact subset of a T2 -space X. Then Y is strongly normal in X. (ii) Let Y be a σ-paracompact subset of a regular space X. Then Y is normal in X. Proof. (i) Let A, B be disjoint subsets of Y that are closed in Y . Then it follows from Theorem 1.8 (i), that A, B are α-paracompact. Thus, the result follows from Theorem 1.7 (i). (ii) Let A, B be disjoint subsets of Y that are closed in X. Then it follows from Theorem 1.8 (ii), that A, B are σ-paracompact. Thus, the result follows from Theorem 1.7 (ii). Corollary 2.5. [2] Let Y be a compact subspace of a T2 -space X. Then Y is strongly normal in X. Proof. Follows from Theorem 2.4 (i) since every compact subspace is α-paracompact. The following theorem provides with a sufficient condition for a normal subspace to be normal in a larger space. Theorem 2.6. Let Y be a normal subspace of a space X. If Bd(Y ) is normal in X, then Y is normal in X.

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Proof. Let A, B be disjoint subsets of Y that are closed in X. Then A, B are disjoint closed subsets of Y , and thus by normality of Y , there are open subsets U, V of X such that A ⊂ Y ∩ U, B ⊂ Y ∩ V and Y ∩ U ∩ V = φ. Since Bd(Y ) is normal in X, there are open subsets Uo , Vo of X such that Bd(Y ) ∩ A ⊂ Uo , Bd(Y ) ∩ B ⊂ Vo and Uo ∩ Vo = φ. Now let G = (Int (Y ) ∩ U) ∪ (Uo ∩ U) and H = (Int (Y ) ∩ V ) ∪ (Vo ∩ V ). Then A ⊂ G, B ⊂ H and G ∩ H = φ. Hence, Y is normal in X. The following theorem provides with a sufficient condition for a normal subspace of a T2 -space X to be normal in X. Theorem 2.7. Let Y be a normal subspace of a T2 -space X. If Bd(Y ) is α-paracompact, then Y is normal in X. Proof. It follows from Theorem 2.4 (i) that Bd(Y ) is normal in X, but Y is normal, so it follows from Theorem 2.6 that Y is normal in X. Corollary 2.8. Let Y be a paracompact subspace of a T2 -space X. If Bd(Y ) is α-paracompact, then Y is normal in X. Proof. Since every paracompact T2 -space is normal, it follows that Y is normal, and thus by Theorem 2.7, Y is normal in X. The following theorem provides with a sufficient condition for a normal subspace of a regular space X to be normal in X. Theorem 2.9. Let Y be a normal subspace of a regular space X. If Bd(Y ) is σ-paracompact, then Y is normal in X. Proof. Follows immediately from Theorem 2.4 (ii) and Theorem 2.6. Corollary 2.10. Let Y be a normal subspace of a regular space X. If Bd(Y ) is Lindelöf, then Y is normal in X. Recall that a subspace Y of a space X is said to be regular in X [2], if for each y ∈ Y and for every closed subset A of X which does not contain y, there are open sets U, V in X such that y ∈ U, A ∩ Y ⊂ V and U ∩ V = φ. Clearly, if Y is regular in a space X, then Y is regular. The following theorem provides with several characterizations of regular subspaces in larger spaces, the proof follows using standard techniques. However, we will show it for the convenience of the reader. Theorem 2.11. Let Y be a subspace of a space X. Then the following are equivalent: (i) Y is regular in X. (ii) For each y ∈ Y and for every closed subset A of Y which does not contain


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y, there are open sets U, V in X such that y ∈ U, A ⊂ V and U ∩ V = φ. (iii) For each y ∈ Y and for every closed subset A of Y which does not contain y, there is an open set U in X such that y ∈ U and U ∩ A = φ. (iv) For each y ∈ Y and for every open set V in X containing y, there is an open set U in X such that y ∈ U ⊂ U ⊂ V ∪ (X\Y ). Proof. The implications (i)→(ii) and (ii)→(iii) are clear. (iii)→(iv): Suppose that y ∈ Y and V is an open set in X containing y. Then y ∈ / (X\V ) ∩ Y = Y \V and Y \V is closed in Y . Thus by (iii), there is an open set U in X such that y ∈ U and U ∩ (Y \V ) = φ. Hence, y ∈ U ⊂ U ⊂ V ∪ (X\Y ). (iv)→(i): Suppose that y ∈ Y and that A is a closed subset of X which does not contain y. Then y ∈ X\A and X\A is an open subset of X. Thus by (iv), there is an open set U in X such that y ∈ U ⊂ U ⊂ (X\A) ∪ (X\Y ) = X\ (A ∩ Y ). Therefore, A ∩ Y ⊂ X\U and X\U is an open subset of X such that U ∩ X\U = φ. Hence, Y is regular in X. In [2], Archangel’skii and Genedi proved the following theorem: Theorem 2.12. Let Y be a subspace of a space X such that Y is regular in X and Y is Lindelöf. Then Y is strongly normal in X. We now generalize Theorem 2.12 by using a σ-paracompact subspace instead of a Lindelöf one, to proceed, we introduce the following lemma whose proof is similar to that of Theorem 1.7 (ii). Lemma 2.13. Let Y be regular in a space X , and let A, B be disjoint closed (in Y ) σ-paracompact subsets. Then there are disjoint open sets U, V in X such that A ⊂ U, B ⊂ V . Theorem 2.14. Let Y be a subspace of a space X such that Y is regular in X and Y is σ-paracompact. Then Y is strongly normal in X. Proof. Let A, B be disjoint closed subsets of Y . Since Y is σ-paracompact, it follows from Theorem 1.8 (ii) that A, B are σ-paracompact, but Y is regular in X, so the result follows from Lemma 2.13. References [1] A. V. Archangel’skii and J. Tartir, A Characterization of compactness by a relative separation property, Questions Answers Gen. Topology 14 (1996), 49-52. [2] A. V. Archangel’skii and H. M. M. Genedi, Begining of the theory of relative topological properties, General Topology, Spaces and Mappings-MGU, Moscow, (1989), 3-48. [3] C. E. Aull, Paracompact subsets, Proceedings of the second prague topological symposium (1966), 45-51.

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[4] R. Engelking, General Topology, Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989.

Received: September, 2009