() @ Appl. Gen. Topol. 18, no. 2 (2017), 231-240doi:10.4995/agt.2017.4220 c© AGT, UPV, 2017 A decomposition of normality via a generalization of κ-normality Ananga Kumar Das and Pratibha Bhat Department of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir- 182320, INDIA (ak.das@smvdu.ac.in, akdasdu@yahoo.co.in and pratibha87bhat@gmail.com) Communicated by T. Nogura Abstract A simultaneous generalization of κ-normality and weak θ-normality is introduced. Interrelation of this generalization of normality with existing variants of normality is studied. In the process of investigation a new decomposition of normality is obtained. 2010 MSC: Primary 54D10; 54D15; Secondary 54D20. Keywords: regularly open set; regularly closed set; θ-open set; θ-closed set; κ-normal (mildly) normal space; almost normal space; (weakly) (functionally) θ-normal space; weakly κ-normal space; ∆-normal space; strongly seminormal space. 1. Introduction and Preliminaries Several generalized notions of normality such as almost normal, κ-normal, ∆- normal, θ-normal, semi-normal, Quasi-normal, π-normal, densely normal etc. exist in the literature. Recently, Interrelation among some of these variants of normality was studied in [4] and factorizations of normality are obtained in [4, 5, 12, 14]. In this paper, we tried to exhibit the interrelations that exist among these generalized notions of normality and introduced a simultaneous generalization of κ-normality and weak θ-normality called weak κ-normality. Intrestingly, the class of weakly κ-normal spaces contains the class of almost compact spaces whereas the class of κ-normal spaces does not contain the class of almost compact spaces. This newly introduced notion of weak normality Received 29 September 2015 – Accepted 10 March 2017 http://dx.doi.org/10.4995/agt.2017.4220 A. K. Das and P. Bhat is utilized to obtain a factorization of normality. Moreover, it is verified that some covering properties which need not imply κ-normality implies weak κ- normality. Let X be a topological space and let A ⊂ X. Throughout the present paper, the closure and interior of a set A will be denoted by A (or clA) and intA (or Ao) respectively. A set U ⊂ X is said to be regularly open [15] if U = intU. The complement of a regularly open set is called regularly closed. It is observed that an intersection of two regularly closed sets need not be regularly closed. A finite union of regular open sets is called π-open set and a finite intersection of regular closed sets is called π-closed set. It is obvious that the complement of a π-open set is π-closed and the complement of a π-closed set is π-open, the finite union (intersection) of π-closed sets is π closed, but the infinite union (intersection) of π-closed sets need not be π-closed (See [11]). A point x ∈ X is called a θ-limit point (respectively δ-limit point) [21] of A if every closed (respectively regularly open) neighbourhood of x intersects A. Let clθA (respectively clδA) denotes the set of all θ-limit point (respectively δ-limit point) of A. The set A is called θ-closed (respectively δ-closed) if A = clθA (respectively A = clδA). The complement of a θ-closed (respectively δ-closed) set will be referred to as a θ-open (respectively δ-open) set. The family of θ-open sets as well as the family of δ open sets form topologies on X. The topology formed by the set of δ-open sets is the semiregularization topology whose basis is the family of regularly open sets. Let Y be a subspace of X. A subset A of X is concentrated on Y [2] if A is contained in the closure of A ∩ Y in X. A subset A of Y is said to be strongly concentrated on Y [6] if A ⊂ (A ∩ Y )o. It is obvious that every strongly concentrated set is concentrated. We say that X is normal on Y if every two disjoint closed subsets of X concentrated on Y can be separated by disjoint open neighbourhoods in X [2]. Similarly, X is said to be weakly normal on Y [6] if for every disjoint closed subsets A and B of X strongly concentrated on Y , there exist disjoint open sets in X separating A and B respectively. A space X is called densely normal if there exists a dense subspace Y of X such that X is normal on Y [2]. A topological space X is said to be weakly densely normal [6] if there exist a proper dense subspace Y of X such that X is weakly normal on Y . It is easy to see that every densely normal space is weakly densely normal and every weakly densely normal space is κ-normal. On the other hand, the converses are not true, as were shown in [10] and [6]. Lemma 1.1. A subset A of a topological space X is θ-open if and only if for each x ∈ A, there is an open set U such that x ∈ U ⊂ U ⊂ A. Definition 1.2. A topological space X is said to be (i) quasi-normal [23] if any two disjoint π-closed subsets A and B of X there exist two open disjoint subsets U and V of X such that A ⊂ U and B ⊂ V . c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 232 A decomposition of normality via a generalization of κ-normality (ii) π-normal [11] if for any two disjoint closed subsets A and B of X one of which is π-closed, there exist two open disjoint subsets U and V of X such that A ⊂ U and B ⊂ V . (iii) ∆-normal [9] if every pair of disjoint closed sets one of which is δ-closed are contained in disjoint open sets. (iv) weakly ∆-normal [9] if every pair of disjoint δ-closed sets are contained in disjoint open sets. (v) weakly functionally ∆-normal (wf ∆-normal) [9] if for every pair of disjoint δ-closed sets A and B there exists a continuous function f : X → [0,1] such that f(A) = 0 and f(B)= 1. (vi) θ-normal [12] if every pair of disjoint closed sets one of which is θ-closed are contained in disjoint open sets; (vii) weakly θ-normal [12] if every pair of disjoint θ-closed sets are contained in disjoint open sets; (viii) functionally θ-normal [12] if for every pair of disjoint closed sets A and B one of which is θ-closed there exists a continuous function f : X →[0,1] such that f(A) = 0 and f(B)=1; (ix) weakly functionally θ-normal (wf θ-normal) [12] if for every pair of disjoint θ-closed sets A and B there exists a continuous function f : X → [0,1] such that f(A) = 0 and f(B)= 1. (x) β-normal [1] if for any two disjoint closed subsets A and B of X, there exist open sets U and V of X such that A ∩ U is dense in A, B ∩ V is dense in B and U ∩ V = φ. (xi) almost β-normal [3] if for every pair of disjoint closed sets A and B, one of which is regularly closed, there exist open sets U and V such that A ∩ U = A, B ∩ V = B and U ∩ V = φ. (xii) θ-regular [12] if for each closed set F and each open set U containing F , there exists a θ-open set V such that F ⊂ V ⊂ U. (xiii) semi-normal [22] if for every closed set F and each open set U contain- ing F, there exists a regular open set V such that F ⊂ V ⊂ U. (xiv) almost normal [18] if every pair of disjoint closed sets one of which is regularly closed are contained in disjoint open sets. (xv) mildly normal [19] ( or κ-normal [20]) if every pair of disjoint regularly closed sets are contained in disjoint open sets. (xvi) ∆-regular [9] if for every closed set F and each open set U containing F, there exists a δ-open set V such that F ⊂ V ⊂ U. 2. Weakly κ-normal spaces Definition 2.1. A θ-closed set A is said to be a regularly θ-closed set if intA = A. The complement of a regularly θ-closed set will be regularly θ open. Clearly every regularly θ-closed set is regularly closed as well as θ-closed but the converse need not be true. Example 2.2. Let X be the set of positive integers. Define a topology on X by taking every odd integer to be open and a set U ⊂ X is open if for every c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 233 A. K. Das and P. Bhat even integer p ∈ U, the predecessor and the successor of p are also in U. Here the set {2k, 2k + 1, 2k + 2 : k ∈ Z+} is a regularly closed set which is not θ-closed. Example 2.3. Let X denote the interior of the unit square S in the plane together with the points (0, 0) and (1, 0), i.e. X = So ∪ {(0, 0), (1, 0)}. Every point in So has the usual Euclidean neighourhoods. The points (0, 0) and (1, 0) have neighbourhoods of the form Un and Vn respectively, where, Un = {(0, 0)} ∪ {(x, y) : 0 < x < 1/2, 0 < y < 1/n} and Vn = {(1, 0)} ∪ {(x, y) : 1/2 < x < 1, 0 < y < 1/n}. Clearly, the sets {(0, 0)} and {(1, 0)} are θ-closed but not regularly θ-closed. Definition 2.4. A topological space X is said to be weakly κ-normal if for every pair of disjoint regularly θ-closed sets A and B there exist disjoint open sets U and V such that A ⊂ U and B ⊂ V . From the definitions it is obvious that every κ-normal space is weakly κ- normal and every weakly θ-normal space is weakly κ-normal. The following diagram illustrates the interrelations that exist between weakly κ-normal spaces and variants of normality that exist in literature. But none of the implications below is reversible (See [7], [9], [11], [12], [14], [18] and examples below ). β-normal ))❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚ normal 66♥♥♥♥♥♥♥♥♥♥♥♥ �� ((PP PP PP PP PP PP P // densely normal ))❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚ almost β-normal ∆-normal // �� ,,❨❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨ π-normal ,,❨❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨❨ ❨❨ $$■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ weak densely normal $$❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ fθ-normal // !!❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ wfθ-normal $$■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ wf∆-normal uu❥❥❥ ❥❥ ❥❥ ❥❥ ❥❥ ❥❥ ❥❥ ❥ almost normal �� dd❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ w∆-normal ))❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ // quasi normal // κ-normal �� θ-normal // wθ-normal // wκ-normal Example 2.5. The space defined in Example 2.2 is weakly κ-normal but not κ-normal. Example 2.6. The example of a Tychonoff κ-normal space which is not densely normal was given by Just and Tartir [10]. Since every regular space is θ-regular [12], this space is θ-regular but not normal. Thus the space is not weakly θ- normal as every θ-regular, weakly θ-normal space is normal [12]. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 234 A decomposition of normality via a generalization of κ-normality Theorem 2.7. A topological space X is weakly κ-normal if and only if for every regularly θ-closed set A and a regularly θ-open set U containing A there is an open set V such that A ⊂ V ⊂ V ⊂ U. Proof. Let X be a weakly κ-normal space and U be a regularly θ-open set containing a regularly θ-closed set A. Then A and X − U are disjoint regularly θ-closed sets in X. Since X is weakly κ-normal, there are disjoint open sets V and W containing A and X − U, respectively. Then A ⊂ V ⊂ X − W ⊂ U. Since X − W is closed, A ⊂ V ⊂ V ⊂ U. Conversely, let A and B be two disjoint regularly θ-closed sets in X. Then U = X −B is a regularly θ-open set containing the regularly θ-closed set A. Thus by the hypothesis there exists an open set V such that A ⊂ V ⊂ V ⊂ U. Then V and X − V are disjoint open sets containing A and B, respectively. Hence X is weakly κ-normal. � Theorem 2.8. Let X be a finite topological space. For a subset A of X, the following statements are equivalent. (a) A is clopen. (b) A is θ-closed. (c) A is θ-open. Proof. The implication (a) =⇒ (b) is obvious. To prove (b) =⇒ (a), let A be a closed subset of X. Then (X − A) is θ-open in X. By Lemma 1.3.4, for each x ∈ X − A there exists an open set Ux containing x such that x ∈ Ux ⊂ Ux ⊂ X − A. Since X is finite, ⋃ x∈X−A Ux = X − A, is the union of finitely many closed sets and hence closed. Thus A is open. By hypothesis A is θ-closed and hence closed. Consequently, A is clopen. The proofs of (a) =⇒ (c) and (c) =⇒ (a) are similar and hence omitted. � From the above result the following observation is obvious. Remark 2.9. Every finite topological space is weakly κ-normal whereas finite topological spaces need not be κ-normal. Theorem 2.10 ([13]). A space X is almost regular if and only if for every open set U in X, intU is θ-open. Theorem 2.11. In an almost regular space, the following statements are equiv- alent (a) X is κ-normal. (b) X is weakly κ-normal. Proof. The proof of (a) =⇒ (b) directly follows from definitions. To prove (b) =⇒ (a), let X be an almost regular, weakly κ-normal space. Let A and B be two disjoint regularly closed sets in X. By Theorem 2.10, A and B are disjoint regularly θ-closed sets in X. Thus by weak κ-normality of X, there exist disjoint open sets separating A and B. Hence X is κ-normal. � Theorem 2.12. In an almost regular space, every π-closed set is θ-closed. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 235 A. K. Das and P. Bhat Proof. Let X be an almost regular space and let A ⊂ X be π-closed in X. Thus A is finite intersection of π-closed sets in X. Since in an almost regular space every regularly closed set is θ-closed [16] and finite intersection of θ-closed sets is θ-closed [21], A is θ-closed. � Theorem 2.13. Every almost regular, weakly θ-normal space is quasi normal. Proof. Let X be an almost regular, weakly θ-normal space. Let A and B be two disjoint π-closed sets in X. By Theorem 2.12, A and B are disjoint θ-closed sets which can be separated by disjoint open set as X is weakly θ-normal. � Theorem 2.14. Every almost regular, θ-normal space is π-normal. It is well known that every compact Hausdorff space is normal. However, in the absence of Hausdorffness or regularity a compact space may fail to be normal. Thus it is useful to know which topological property weaker than Hausdorffness with compactness implies normality. The property of being a T1 space fails to do the job since the cofinite topology on an infinite set is a compact T1 space which is not normal. However, it is well known that Every compact R1-space is normal ( See [17]). In [12], it is shown that every compact space in particular every paracompact space in absence of any separation axioms is θ-normal. It is also known that every Lindelöf spaces need not be κ-normal. However, by the following theorem of [12] it follows that every Lindelöf space is weakly κ-normal. Similarly, almost compactness need not implies κ-normality, but by Theorem of [12], every almost compact space is weakly κ-normal. Theorem 2.15 ([12]). Every Lindelöf space is weakly θ-normal. Corollary 2.16. Every Lindelöf space is weakly κ-normal. Corollary 2.17. Every almost regular, Lindelöf space is κ-normal. Proof. The prove immediately follows from Theorem 2.11, since in an almost regular space every weakly κ-normal space is κ-normal. � Theorem 2.18 ([12]). Every almost compact space is weakly θ-normal. Corollary 2.19. Every almost compact space is weakly κ-normal. Corollary 2.20. Every almost regular, almost compact space is κ-normal. Proof. The prove immidiately follows from Theorem 2.11, since in an almost regular space every weakly κ-normal space is κ-normal. � Remark 2.21. Corollary 2.17 and Corollary 2.20 were independently prooved in [18]. In contrary to the above results the following example establishes that Lindelöf spaces need not be κ-normal and almost compactness need not imply κ-normality. Example 2.22. Let X be the set of positive integers with the topology as defined in Example 2.2 and Y = {1, 2, 3, ..., 11}. Then the subspace topology on Y is compact but not κ-normal as disjoint regularly closed sets {2, 3, 4} and {6, 7, 8} can not be separated by disjoint open sets. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 236 A decomposition of normality via a generalization of κ-normality Definition 2.23 ([13]). A space X is said to be θ-compact if every open covering of X by θ-open sets has a finite subcollection that covers X. The following result is useful to show that every almost regular, θ-compact space is κ-normal as well as weakly θ-normal. Theorem 2.24 ([16]). Let A ⊂ X be θ-closed and let x /∈ A. Then there exist regular open sets which separate x and A. Theorem 2.25. In an almost regular space, every θ-compact space is weakly θ-normal. Proof. Let X be an almost regular θ-compact space. Let A and B be any two disjoint θ-closed subsets of X. By Theorem 2.24, for every a ∈ A, there exist disjoint regularly open sets Ua and Va containing a and B respectively. Since X is almost regular, Ua and Va are disjoint θ-open sets containing a and B. Now the collection {Ua : a ∈ A} is a θ-open cover of A. Then A ⊂ ⋃ a∈A Ua = O. Since arbitrary union of θ-open sets is θ-open, X − O = D is θ-closed. Since A is a θ-closed set disjoint from D, by Theorem 2.24, for every d ∈ D, there exist disjoint regularly open sets Sd and Td containing A and d respectively. Again by almost regularity of X, Td is a θ-open set which is disjoint from A. Now the collection U = {Ua : a ∈ A} ∪ {Td : d ∈ D} is a θ-open covering of X. By θ-compactness of X, mathcalU has a finite subcollection V which covers X. Let the members of V which intersects A be W. Each member of W is of the form Ua for some a ∈ A as for each d ∈ D, Td ∩ A = φ. Suppose W = {Uai : i = 1, 2, 3, ..., n}. Then n⋃ i=1 Uai = U and n⋂ i=1 Vai = V are disjoint open sets containing A and B respectively. Hence X is weakly θ-normal. � Corollary 2.26. In an almost regular space, every θ-compact space is weakly κ-normal. Proof. The proof immidiately follows from the fact that every θ normal space is weakly κ-normal. � Corollary 2.27. In an almost regular space, every θ-compact space is κ- normal. Proof. The proof immidiately follows from Theorem 2.11. � Corollary 2.28. In an almost regular space, every almost compact space is weakly κ-normal. Proof. The proof is immediate as every almost compact space is θ-compact [13]. � 3. Decompositions of normality Theorem 3.1. An T1-space is almost normal if and only if it is almost β- normal and weakly κ-normal. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 237 A. K. Das and P. Bhat Proof. Necessary part is obvious. Conversely, let X be a T1-amlost β normal, weakly κ-normal space. Since X is T1-almost β-normal, by Theorem 2.9 of [3], X is almost regular. So by Theorem 2.11, X is κ- normal. Hence X is almost normal as every almost β-normal, κ-normal space is almost normal [3]. � Corollary 3.2. An T1 space is normal if and only if it is almost β-normal, weakly κ-normal and semi-normal. Proof. The prove follows from the result that every almost normal, semi normal space is normal [18]. � Definition 3.3. A space X is said to be strongly seminormal if for every closed set A contained in an open set U there exists a regularly θ-open set V such that A ⊂ V ⊂ U. Theorem 3.4. Every normal space is strongly seminormal. Proof. Let A be a closed set and U be an open set containing A. Let B = X−U. Then A and B are disjoint closed sets in X. By Urysohn’s lemma there exists a continuous function f : X → [0, 1] such that f(A) = 0 and f(B) = 1. Let V = f−1[0, 1/2) and W = f−1(1/2, 1]. Then A ⊂ V ⊂ X − W ⊂ U. Thus A ⊂ V ⊂ V o ⊂ X − W ⊂ U. We claim that V o is a regularly θ-open set. V o is regularly open, only we have to show that V o is θ-open. let x ∈ V o . Then f(x) ∈ [0, 1/2). So there is a closed neighbourhood N of f(x) contained in [0, 1/2). Let Ux = (f −1(N))o. Then x ∈ Ux ⊂ f −1(N) ⊂ V o . By Lemma 1.1, V o is θ-open. Hence X is strongly seminormal. � Theorem 3.5. Every strongly seminormal space is seminormal. Theorem 3.6. Every strongly seminormal space is θ-regular. The following implications are obvious but none of these is reversible. normal // strongly semi normal ��uu❦❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦ regular uu❦❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦ �� seminormal ))❙❙ ❙❙ ❙❙ ❙❙ ❙❙ ❙❙ ❙❙ ❙ θ-regular �� semi regular uu❦❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦❦ ❦ ∆-regular Example 3.7. Let X be the set of positive integers with the topology as defined in Example 2.2, then X is seminormal but not strongly seminormal. Example 3.8. The space given in [10] by Just and Tartir is an example of a Tychonoff κ-normal space which is not densely normal. Since every seminormal κ-normal space is normal [18], thus becoming densely normal, this space is not seminormal but is θ-regular as every regular space is θ-regular. Theorem 3.9. A space X is normal if and only if it is strongly seminormal and weakly κ-normal. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 238 A decomposition of normality via a generalization of κ-normality Proof. The necessary part i.e., a normal space is strongly seminormal as well as weakly κ-normal directly follows from the definition. Conversely, let X be a strongly seminormal and weakly κ-normal space. let A and B be two disjoint closed sets in X. Thus A is a closed set contained in an open set U = X − B. Since X is strongly seminormal, there exists an regularly θ-open set V such that A ⊂ V ⊂ U. Now X − V is a regularly θ-closed set contained in an open set X −A. Again by strong seminormality of X, there exists a regularly θ-open set W such that X − V ⊂ W ⊂ X − A. Thus X − V and X − W are two disjoint regularly θ-closed sets in X containing B and A respectively. By weak κ-normality of X, there exist two disjoint open sets O and P separating X −W and X − V . Hence X is normal. � Corollary 3.10. In the class of strongly seminormal spaces, the following statements are equivalent. (a) X is normal. (b) X is ∆-normal. (c) X is wf∆-normal. (d) X is weakly ∆-normal. 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