4_678_das.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 1, 2011, 34-41 ISSN 1307-5543 – www.ejpam.com Simultaneous Generalizations of Regularity and Normality A. K. Das School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India Abstract. A generalization of regularity called θ -regularity was earlier introduced to decompose nor- mality and also utilised to factorize regularity. Every normal space need not be regular, but every normal space is θ -regular. In this paper three variants of θ -regular spaces is introduced and studied. 2000 Mathematics Subject Classifications: 54D10, 54D15 Key Words and Phrases: θ -open sets, θ -closed sets, almost normal space, (weakly)(functionally) θ -normal space, (weakly) θ -regular, point (weakly) θ -regular. 1. Introduction and Preliminaries Many generalizations of regularity that exists in the mathematical literature fails to be a generalization of normality. But in order to obtain a decomposition of normality, the notion of θ -regularity was introduced in [6] which is a simultaneous generalization of regularity as well as normality. It is obvious from the definition that every regular space is θ -regular as in a regular space every closed set is θ -closed [14]. In general a normal space need not be regular, but in contrast every normal space is θ -regular [6]. Also it is observed in [5] that the notion of θ -regularity serves as a decomposition of regularity in terms of R0 and R1 spaces. In this paper we introduced three more variants of θ -regular spaces and studied their properties. Let X be a topological space and let A⊂ X . Throughout the present paper, the closure of a set A will be denoted by A or clA and the interior by intA. A set U ⊂ X is said to be regularly open if U = intU . The complement of a regularly open set is called regularly closed. A point x ∈ X is called a θ -limit point [14] of A if every closed neighbourhood of x intersects A. Let clθA denotes the set of all θ -limit point of A. The set A is called θ -closed if A = clθA . The complement of a θ -closed set will be referred to as a θ -open set. The family of θ -open sets forms a topology on X . A space X is said to be almost regular [9] if every regularly closed set and a point not in it are contained in disjoint open sets. A space is called almost normal [10] if every pair of disjoint closed sets, one of which is regularly closed, are contained in disjoint open sets and a space X is said to be mildly normal [12] ( or κ-normal [13]) if every pair of disjoint regularly closed sets are contained in disjoint open sets. A space X is said to be Email addresses: ak.das�smvdu.a .in, akdasdu�yahoo. o.in http://www.ejpam.com 34 c© 2010 EJPAM All rights reserved. A. Das / Eur. J. Pure Appl. Math, 4 (2011), 34-41 35 nearly compact[11] if every open covering of X admits a finite subcollection the interiors of the closures of whose members cover X . Definition 1. A topological space X is said to be (i) θ -normal[6] if every pair of disjoint closed sets one of which is θ -closed are contained in disjoint open sets; (ii) weakly θ -normal[6] if every pair of disjoint θ -closed sets are contained in disjoint open sets; (iii) functionally θ -normal( [4, 6]) 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; (iv) weakly functionally θ -normal (wf θ -normal)([4, 6]) 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; and (v) Σ-normal[7] if for each closed set F and each open set U containing F, there exists a regular Fσ set V such that F ⊂ V ⊂ U. 2. Variants of θ -regular Spaces Definition 2. A topological space X is said to be (i) θ -regular[6] 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. (ii) weakly θ -regular if for each θ -closed set F and each open set U containing F, there exists a θ -open set V such that F ⊂ V ⊂ U. (iii) point θ -regular if for each closed singleton {x} and each open set U containing x, there exists a θ -open set V such that x ∈ V ⊂ U. (iv) point weakly θ -regular if for each θ -closed singleton {x} and each open set U containing x, there exists a θ -open set V such that x ∈ V ⊂ U. The above notion of θ -regularity is exclusively different from the concept of θ -regularity introduced by Jankovic [3] which was utilized by Kovar [8] to study covering axioms in- cluding compactness and paracompactness. In [8], Kovar proved that Jankovic’s θ -regularity coincides with the notion of point paracompactness introduced by Boyte [1]. From here on- ward the term “θ -regularity” will always be meant in the sense of Definition 2. The following implications are obvious, but none of them are reversible. A. Das / Eur. J. Pure Appl. Math, 4 (2011), 34-41 36 Example 1 (A point θ -regular space which is not θ -regular.). Let X = {a, b, c, d , e} and T = {{a, b, c}, {c, d , e}, {c},ϕ, X }. Here X is vacuously point θ -regular, but not θ -regular as {a, b} ⊂ {a, b, c} but there is no θ -open set containing {a, b} and contained in {a, b, c}. Example 2 (A point weakly θ -regular space which is not point θ -regular.). Co-finite topology is point weakly θ -regular but not point θ -regular. Example 3 (A point weakly θ -regular space which is not point θ -regular.). Let X = {a, b, c} and T = {{a, b}, {b, c}, {b},ϕ, X }. Here X is vacuously point weakly θ -regular, but not point θ -regular as {a} ⊂ {a, b} but there is no θ -open set containing {a} and contained in {a, b}. Question 1. Does there exists a point weakly θ -regular space which is not weakly θ -regular? It is obvious from the definitions that, a R0-space is regular if and only if it is θ -regular and a T1-space is T3 if and only if it is point θ -regular. Similarly, a Hausdroff space is T3 if and only if it is point weakly θ -regular. Theorem 1. For a point θ -regular space, the following statements are equivalent. (i) For every pair of distinct points x and y in X, there exist θ -open sets P and Q such that x ∈ U , y ∈ V and P ∩Q = ϕ. (ii) X is θT2. (iii) X is Urysohn. (iv) X is T2. (v) X is T1. Proof. Let x and y be two disjoint points in X . Since X is T1, the closed set {x} is contained in an open set X − {y}. Thus by point θ -regularity of X , there exists a θ -open set V such that x ∈ V ⊂ X − {y}. Since V is θ -open there exists a open set U such that x ∈ U ⊂ U ⊂ V ⊂ X −{y}. i.e.; x ∈ U and y ∈ X − U . Again by point θ -regularity, there exist θ -open sets P and Q such that x ∈ P, y ∈ Q and P ∩Q = ϕ. A. Das / Eur. J. Pure Appl. Math, 4 (2011), 34-41 37 Theorem 2. For a T1 space, the following statements are equivalent. (i) X is T3. (ii) X is regular. (iii) X is θ -regular. (iv) X is point θ -regular. Proof. Let X be a T1 point θ -regular space. Let x /∈ A, where A is a closed set in X . Since X is a T1 space, the singleton {x} is closed and contained in X − A. By Point θ -regularity of X , there exists a θ -open set V such that x ∈ V ⊂ X − A. Since V is θ -open there exists an open set U such that x ∈ U ⊂ U ⊂ V ⊂ X − A. Therefore X is regular and thus T3. Theorem 3. Every T1 point θ -regular space is Hausdorff. Proof. Let X be a T1 point θ -regular space and let x , y be two distinct points in X . Since X is T1, {x} is a closed singleton contained in the open set X − {y}. By point θ -regularity of X , there exists a θ -open set U such that x ∈ U ⊂ X − {y}. Thus there exists an open set V such that x ∈ V ⊂ V ⊂ U ⊂ X − {y}. So V and X − V are two disjoint open sets containing x and y respectively. Theorem 4. For a T2 space, the following statements are equivalent. (i) X is T3. (ii) X is regular (iii) X is θ -regular (iv) X is weakly θ -regular (v) X is point θ -regular (vi) X is point weakly θ -regular Proof. Obvious. Theorem 5. Every functionally θ -normal space is weakly θ -regular. Proof. Let A be a θ -closed set contained in an open set U . Let B = X − U . Then A and B are disjoint closed sets in X . By functional θ -normality of X , 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). Then A⊂ V ⊂ U . We claim that V is a θ -open set. Let x ∈ V . Then f (x) ∈ [0,1/2). So there is a closed neighbourhood N of f (x) contained in [0,1/2) ⊂ [0,1]. Let Ux = int f −1(N). Then x ∈ Ux ⊂ U x ⊂ f −1(N)⊂ V . Hence V is θ -open. Therefore X is θ -regular. A. Das / Eur. J. Pure Appl. Math, 4 (2011), 34-41 38 Remark 1. Functionally θ -normal spaces need not be θ -regular. i.e.; Let X = {a, b, c}, τ = {{a, b}, {b}, {b, c},φ, X } is a functionally θ -normal space which is not θ -regular. Theorem 6. Every nearly compact weakly θ -regular space is θ -normal. Proof. Let A and B be two disjoint closed sets of X where A is θ -closed. Then A⊂ X − B. Thus by θ -regularity of X there exist an θ -open set V such that A ⊂ V ⊂ X − B. Since V is θ -open, for every x ∈ A there exist an open set Ux such that x ∈ Ux ⊂ U x ⊂ V ⊂ X − B. Then U = {ux : x ∈ A} is an open cover of A. Since A is θ -closed, by [2, Proposition 2.1], A is N -closed relative to X . Hence U has finite subcollection such that A ⊂ n⋃ i=1 intUxi . Thus B ⊂ n⋂ i=1 (X − Uxi ). therefore X is θ -normal. Corollary 1. Every nearly compact θ -regular space is normal. Proof. The above result is obvious, since every θ -regular θ -normal space is normal. Remark 2. The following example shows that the hypothesis of θ -regularity in the above Corol- lary cannot be weakened to “weak θ -regularity” as nearly compact weakly θ -regular spaces need not be almost normal. e.g.; The set X = {a, b, c, d} with topology τ = {{a, b}, {b}, {b, c}, {c}, {b, c, d}, {a, b, c}, X ,;} is compact and weakly θ -regular but not al- most normal as the regularly closed set {c, d} and closed set {a} cannot be separated by disjoint open sets. 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 The following result of [6] is an improvement of well known results such as every compact Hausdorff space is normal and every compact (or Lindelöf) regular space is normal. Theorem 7. Every paracompact θ -regular space is normal. Theorem 8. Every Lindelöf θ -regular space is normal. Remark 3. The condition of θ -regularity in the above theorem cannot be weakened as the exam- ple cited in Remark 2 is a paracompact weakly θ -regular space which fails to be almost normal. Although every compact θ -regular space is normal, but it is in the absence of T1 property, as every T1 θ -regular space is regular. Thus it is very natural to ask the following Question. Question 2. Which non-regular, non-Hausdorff, T1-compact spaces are normal ? A. Das / Eur. J. Pure Appl. Math, 4 (2011), 34-41 39 Let us recall that a space X is seminormal if for every closed set F contained in an open set U there exists a regularly open set V such that F ⊂ V ⊂ U . A space is said to be θ -seminormal [15] if for every θ -closed set F contained in an open set U there exists a regularly open set V such that F ⊂ V ⊂ U . Example 4. A seminormal space which is not θ -regular. 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 even integer p ∈ U, the predecessor and the successor of p are also in U. Since every open set is regularly open in this topology, the space is seminormal but the space is not θ -regular. Theorem 9. Every almost regular seminormal space is θ -regular. Proof. Let F be a closed set contained in an open set U . Since X is seminormal there exists a regularly open set V such that F ⊂ V ⊂ U . Since in an almost regular space every regularly open set is θ -open, the space is θ -regular. Corollary 2. An almost regular space is normal if and only if it is seminormal and weakly θ - normal. Proof. Proof is obvious, since every θ -regular weakly θ -normal space is normal. Theorem 10. Every almost regular θ -seminormal space is weakly θ -regular. 3. Subspaces Lemma 1. If Y ⊂ X and A is any θ -open set in X then A∩ Y is θ -open in Y . Theorem 11. If Y is a closed subspace of X and X is θ -regular then Y is θ -regular. Proof. Let X be a θ -regular space and Y ⊂ X . Let F be a closed set in Y which is contained in an open set U of Y . Since F is closed in Y and Y is a closed subspace of X , F is closed in X . Since U is open in Y , there exists an open set V in X such that U = V ∩ Y . Thus F ⊂ V . By θ -regularity of X , there exists a θ -open set W in X such that F ⊂W ⊂ V , i.e,; F ∩ Y ⊂W ∩ Y ⊂ V ∩ Y ⇒ F ⊂W ∩ Y ⊂ U . By the previous lemma W ∩ Y is θ -open in Y . Hence Y is θ -regular. Theorem 12. If Y is a closed subspace of X and X is point θ -regular, then Y is point θ -regular. Lemma 2. If Y is θ -open in X and A is θ -open in Y , then A is θ -open in X . Lemma 3. If Y is θ -open in X and A is θ -closed in Y then A is θ -closed in X . Proof. Let Y be a θ -open set in X and let A be θ -closed in Y . Then (Y − A) is θ -open in Y .Thus by previous lemma (Y − A) is θ -open in X . Therefore X − (Y − A) is θ -closed in X . Hence A is θ -closed in X . REFERENCES 40 Theorem 13. If Y is a θ -open subspace of X and X is weakly θ -regular, then Y is weakly θ -regular. Proof. Let Y be a θ -open subspace of X and X is weakly θ -regular. Let F be a θ -closed set in Y and contained in an open set U of Y . Since Y is θ -open in X , F is θ -closed in X . Since U is open in Y , there exists a open set V in X such that U = V ∩ Y . So F ⊂ V . By weak θ -regularity of X , there exists a θ -open set W in X such that F ⊂W ⊂ V . 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