Semi Star Generalized w - Closed Sets in Bitopological ... - m-hikari

Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 12, 587 - 595

Semi Star Generalized w - Closed Sets in Bitopological Spaces K. Chandrasekhara Rao and D. Narasimhan Department of Mathematics, Srinivasa Ramanujan Centre SASTRA University, Kumbakonam - 612 001, India k.chandrasekhara@rediffmail.com dnsastra@rediffmail.com Abstract K. Chandrasekhara Rao and K. Joseph [3] introduced the concepts of semi star generalized open sets and semi star generalized closed sets in a topological space. The aim of this paper is to introduce the concepts of semi star generalized w - closed sets , semi star generalized w - open sets and study their basic properties in Bitopological spaces.

Mathematics Subject Classification: 54E55 Keywords : τ1 τ2 - Semi star generalized w closed sets , τ1 τ2 - semi star generalized w - open sets , τ1 τ2 - generalized w - closed sets

1

Introduction

Let (X, τ1 , τ2 ) or simply X denote a bitopological space. For any subset A ⊆ X , τi − int(A) and τi − cl(A) denote the interior and closure of a set A with respect to the topology τi . The closure and interior with respect to the topology τi of B relative to A is written as τi − clB (A) and τi − intB (A) respectively. A point x ∈ X is called a condensation point of A if for each U ∈ τ with x ∈ U, the set U ∩ A is uncountable. A is called w - closed if it contains all its condensation points. The complement of an w - closed set is called w - open. It is well known that a subset A of a space (X, τ ) is w open if and only if for each x ∈ A, there exists U ∈ τ such that x ∈ U and U ∩ W is countable. The family of all w - open subsets of a space (X, τ ) , by τw or wO(X) , forms a topology on X finer than τ . The w - closure and w - interior with respect to the topology τi , that can be defined in a manner similar to τi −cl(A) and τi −int(A) , respectively, will be denoted by τi −clw (A) and τi − intw (A) , respectively. AC denotes the complement of A in X unless

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explicitly stated. We shall require the following known definitions. Definition 1.1 A set A of a bitopological space (X, τ1 , tau2 ) is called (a) τ1 τ2 - semi open if there exists an τ1 - open set U such that U ⊆ A ⊆ τ2 − cl(U), (b) τ1 τ2 - semi closed if X − A is τ1 τ2 - semi open. equivalently, a set A of a bitopological space (X, τ1 , τ2 ) is called τ1 τ2 - semi closed if there exists a τ1 - closed set F such that τ2 − int(F ) ⊆ A ⊆ F , (c) τ1 τ2 - generalized closed (τ1 τ2 − g closed ) if τ2 − cl(A) ⊆ U whenever A ⊆ U and U is τ1 - open in X, (d) τ1 τ2 - generalized open (τ1 τ2 − g open ) if X − A is τ1 τ2 − g closed, (e) τ1 τ2 - generalized semi open (τ1 τ2 −gs open) if F ⊆ τ2 −sint(A) whenever F ⊆ A and F is τ1 - closed in X, (f) τ1 τ2 - generalized semi closed (τ1 τ2 − gs closed) if X − A is τ1 τ2 − gs open, (g) τ1 τ2 - semi star generalized closed (τ1 τ2 − s∗ g closed ) if τ2 − cl(A) ⊆ U whenever A ⊆ U and U is τ1 - semi open in X, (h) τ1 τ2 - semi star generalized open (τ1 τ2 − s∗ g open ) if X − A is τ1 τ2 − s∗ g closed in X, (i) τ1 τ2 - generalized w - closed (τ1 τ2 − gw closed ) if τ2 − clw (A) ⊆ U whenever A ⊆ U and U is τ1 - open in X, (j) τ1 τ2 - generalized w - open (τ1 τ2 − gw open ) if X − A is τ1 τ2 − gw closed. (k) τ1 τ2 - regular generalized w - closed (τ1 τ2 −rgw closed ) if τ2 −clw (A) ⊆ U whenever A ⊆ U and U is τ1 τ2 - regular open in X, (l) τ1 τ2 - regular generalized w - open (τ1 τ2 −rgw open ) if X −A is τ1 τ2 −rgw closed.

2

Semi Star Generalized w - Closed Sets

Recall that a set A of a topological space (X, τ ) is called semi star generalized w - closed ( s∗ gw - closed )[6] if clw (A) ⊆ U wheneverA ⊆ U and U is semi open in X.

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Semi star generalized w - closed sets

Definition 2.1 A set A of a bitopological space (X, τ1 , τ2 ) is called τ1 τ2 - semi star generalized w closed ( τ1 τ2 -s∗ gw closed ) if τ2 − clw (A) ⊆ U wheneverA ⊆ U and U is τ1 -semi open in X. Example 2.2 Let X be the set of all real numbers , τ1 = {φ, , − Q}, τ2 = {φ, , Q},where Q is the set of all rational numbers. Then − Q is τ1 τ2 − s∗ gw closed. Theorem 2.3 Let (X, τ1 , τ2 ) be a bitopological space and A ⊆ X. Then the following are true. (a) If A is τ2 − w closed , then A is τ1 τ2 − s∗ gw closed. (b) If A isτ1 - semi open and τ2 − s∗ gw closed, then A is w - closed . (b) If A is τ2 − s∗ gw - closed, then A is gw - closed. Proof. ( a ) Suppose that A is τ2 − w closed . Let A ⊆ U and U is τ1 - semi open in X. Then τ2 − clw (A) = A ⊆ U. Consequently A is τ1 τ2 − s∗ gw closed. ( b )Suppose that A is τ1 - semi open and τ1 τ2 − s∗ gwclosed. Let A ⊆ A and A is τ1 - semi open. Then τ2 − clw (A) ⊆ A .Therefore, τ2 − clw (A) = A. Consequently A is τ2 − w closed. ( c ) Suppose that A is τ1 τ2 − s∗ gw closed. Let A ⊆ U and U is τ1 - open in X. Since U is τ1 - semi open in X, we have τ2 − clw (A) ⊆ U. Consequently A is τ1 τ2 − gw closed. Since, τ2 − clw (A) ⊆ τ2 − cl(A), we have the following theorem Theorem 2.4 Every τ1 τ2 − s∗ g closed set is τ1 τ2 − s∗ gw closed and every τ2 - closed set is τ1 τ2 − s∗ gw closed Remark 2.5 From the Theorem 2.3 , Theorem 2.4 and above definitions, we have the following relations. τ1 τ2 − s∗ gw closed⇒ ⇑ τ2 − w closed ⇑ τ2 -closed ⇒ ⇓ τ2 - semi closed ⇒

τ1 τ2 − gw closed ⇒ ⇑ τ1 τ2 − g closed ⇒ ⇑ τ1 τ2 − s∗ g closed ⇒ ⇓ τ1 τ2 − sg closed

τ1 τ2 − rgw closed ⇑ τ1 τ2 − rg closed τ1 τ2 − gs closed ⇑

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Theorem 2.6 If A is τ1 τ2 − s∗ gw closed in X and A ⊆ B ⊆ τ2 − clw (A), then B is τ1 τ2 − s∗ gw closed. Proof. Suppose that A is τ1 τ2 − s∗ gw closed in X and A ⊆ B ⊆ τ2 − clw (A). Let B ⊆ U and U is τ1 - semi open in X. Then A ⊆ U. Since A is τ1 τ2 − s∗ gw closed, we have τ2 − clw (A) ⊆ U. Since B ⊆ τ2 − clw (A), τ2 − clw (B) ⊆ τ2 − clw (A) ⊆ U. Hence B is τ1 τ2 − s∗ gw closed. Theorem 2.7 If A and B are τ1 τ2 − s∗ gwclosed sets then so is A ∪ B . Proof. Suppose that A and B are τ1 τ2 − s∗ gw closed sets. Let U be τ1 - semi open in X and A ∪ B ⊆ U. Then A ⊆ U and B ⊆ U . Since A and B are τ1 τ2 − s∗ gw closed sets, we have τ2 − clw (A) ⊆ U and τ2 − clw (B) ⊆ U. Consequently τ2 − clw (A ∪ B) ⊆ U. Therefore, A ∪ B is τ1 τ2 − s∗ gw closed. Theorem 2.8 Let B ⊆ A ⊆ X where A is τ1 - open and τ1 τ2 − s∗ gw closed in X. Then B is τ1 τ2 − s∗ gw closed relative to A if and only if B is τ1 τ2 − s∗ gw closed relative to X. Proof. Suppose that B ⊆ A ⊆ X where A is τ1 - open and τ1 τ2 − s∗ gw closed. Suppose that B is τ1 τ2 − s∗ gw closed relative to A. Let B ⊆ U and U is τ1 - semi open in X. Since A ⊆ X , A is τ1 - open, we have A ∩ U is τ1 - semi open in X. Consequently A ∩ U is τ1 - semi open in A. Since B ⊆ A , B ⊆ U, we have B ⊆ A ∩ U. Since B is τ1 τ2 − s∗ gw closed relative to A , we have τ2 − clw (BA ) ⊆ A ∩ U . Therefore, τ2 − clw (BA ) ⊆ U . Since A is τ1 - open, we have τ2 − clw (BA ) = τ2 − clw (B) ∩ A = τ2 − clw (B) ⊆ U.Hence B is τ1 τ2 − s∗ gw closed relative to X. Conversely, suppose that B is τ1 τ2 − s∗ gw closed relative to X. Let B ⊆ U and U is τ1 - semi open in A. Since A ⊆ X , we have U is τ1 - semi open in X. Since B is τ1 τ2 − s∗ gw closed relative to X , we have τ2 − clw (B) ⊆ U . Now, τ2 − clw (BA ) = τ2 − clw (B) ∩ A = τ2 − clw (B) ⊆ U . Therefore, B is τ1 τ2 − s∗ gw closed relative to A. Corollary 2.9 If A is τ1 τ2 − s∗ gw closed, τ1 - semi open in X and F is τ2 − w closed in X, then A ∩ F is τ2 − w closed in X. Proof. Since A is τ1 τ2 − s∗ gw closed, τ1 - semi open in X, we have A is τ2 − w closed in X. { By Theorem 2.3 ( b ) }. Since F is τ2 − w - closed in X, A ∩ F is τ2 − w - closed in X. Theorem 2.10 If a set A is τ1 τ2 − s∗ gw closed in X then τ2 − clw (A) − A contains no nonempty τ1 - semi closed set.


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Proof. Suppose that A is τ1 τ2 − s∗ gw closed in X . Let F be τ1 - semi closed and F ⊆ τ2 − clw (A) − A. Since F be τ1 - semi closed, we have F C is τ1 - semi open. Since F ⊆ τ2 − clw (A) − A, we have F ⊆ τ2 − clw (A) and A ⊆ F C . Since A is τ1 τ2 − s∗ gw closed in X, we have τ2 − clw (A) ⊆ F C . Consequently, F = φ. Hence τ2 − clw (A) − A contains no nonempty τ1 - semi closed set. Corollary 2.11 Let A be τ1 τ2 − s∗ gwclosed. Then A is τ2 − w - closed if and only if τ2 − clw (A) − A is τ1 - semi closed. Proof. Suppose that A is τ1 τ2 − s∗ gw closed. Since A is τ2 − w - closed , we have τ2 −clw (A) = A. Then τ2 −clw (A)−A = φ is τ1 - semi closed. Conversely, suppose that A is τ1 τ2 − s∗ gw closed and τ2 − clw (A) − A is τ1 - semi closed. Since A is τ1 τ2 − s∗ gw closed, we have τ2 − clw (A) − A contains no nonempty τ1 - semi closed set { by Theorem 2.10 }. Since τ2 − clw (A) − A is itself τ1 semi closed, we have τ2 − clw (A) − A = φ. Then τ2 − clw (A) = A. Hence A is τ2 − w - closed. Theorem 2.12 If A is τ1 τ2 − s∗ gwclosed and A ⊆ B ⊆ τ2 − clw (A) then τ2 − clw (B) − B contains no nonempty τ1 - semi closed set. Proof. Let A be τ1 τ2 − s∗ g closed and A ⊆ B ⊆ τ2 − clw (A). Then B is τ1 τ2 − s∗ gw closed. { By Theorem 2.6 }. Hence τ2 − clw (B) − B contains no nonempty τ1 - semi closed set.{ By Theorem 2.10 }

3

Semi Star Generalized w - Open Sets

Definition 3.1 A set A of a topological space (X, τ1 , τ2 ) is called τ1 τ2 - semi star generalized w open ( τ1 τ2 − s∗ gw open ) if and only if AC is τ1 τ2 − s∗ gw closed. Example 3.2 In Example 2.2, Q is τ1 τ2 − s∗ gw open Theorem 3.3 A set A is τ1 τ2 − s∗ gw open if and only if F ⊆ τ2 − intw (A) whenever F is τ1 -semi closed and F ⊆ A. Proof. Suppose that A is τ1 τ2 −s∗ gw open. Suppose that F is τ1 - semi closed and F ⊆ A. Then F C is τ1 - semi open and AC ⊆ F C . Since AC is τ1 τ2 − s∗ gw closed , we have τ2 − clw (AC ) ⊆ F C . Since τ2 − clw (AC ) = [τ2 − intw (A)]C , we have F ⊆ τ2 − intw (A). Conversely, suppose that F ⊆ τ2 − intw (A) whenever F is τ1 - semi closed and F ⊆ A. Then AC ⊆ F C and F C is τ1 - semi open. Since F ⊆ τ2 − intw (A) , and τ2 − clw (AC ) = [τ2 − intw (A)]C , we have τ2 − clw (AC ) ⊆ U. Then AC is τ1 τ2 − s∗ gw closed. Consequently A is τ1 τ2 − s∗ gw open.

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Theorem 3.4 If A and B are separated τ1 τ2 − s∗ gw open sets then A ∪ B is τ1 τ2 − s∗ gw open set. Proof. Suppose A and B are separated τ1 τ2 − s∗ gwopen sets. Let F be τ1 - semi closed and F ⊆ A ∪ B. Since A and B are separated, we have τ1 − cl(A) ∩ B = A ∩ τ1 − cl(B) = φ and τ2 − cl(A) ∩ B = A ∩ τ2 − cl(B) = φ. Then, F ∩ τ2 − cl(A) ⊆ (A ∪ B) ∩ τ2 − cl(A) = A.Similarly, we can prove F ∩ τ2 − cl(B) ⊆ B. Since F is τ1 - semi closed, we have F ∩ τ1 − cl(A) and F ∩ τ1 − cl(B) are τ1 - semi closed. Since A and B are τ1 τ2 − s∗ gw open, we have F ∩ τ2 − cl(A) ⊆ τ2 − intw (A) and F ∩ τ2 − cl(B) ⊆ τ2 − intw (B) . Now, F = F ∩ (A ∪ B) ⊆ {F ∩ τ2 − cl(A)} ∪ {F ∩ tau2 − cl(B)} ⊆ τ2 − intw (A ∪ B). Therefore , A ∪ B is τ1 τ2 − s∗ gwopen. Theorem 3.5 If A and B are τ1 τ2 − s∗ gw open sets then so is A ∩ B . Proof. Suppose that A and B are τ1 τ2 − s∗ gw open sets. Let F be τ1 - semi closed and F ⊆ A ∩ B. Then, F ⊆ A and F ⊆ B. Since A and B are τ1 τ2 − s∗ gw open , we have F ⊆ τ2 − intw (A) and F ⊆ τ2 − intw (B). Hence F ⊆ τ2 − intw (A ∩ B). Consequently A ∩ B is τ1 τ2 − s∗ gw open set. Theorem 3.6 If A is τ1 τ2 − s∗ gw open in X and τ2 − intw (A) ⊆ B ⊆ A, then B is τ1 τ2 − s∗ gw open. Proof. Suppose that A is τ1 τ2 − s∗ gw open in X and τ2 − intw (A) ⊆ B ⊆ A. Let F be τ1 - semi closed and F ⊆ B. Since F ⊆ B, B ⊆ A, we have F ⊆ A. Since A is τ1 τ2 −s∗ gw open, we have F ⊆ τ2 −intw (A). Since τ2 −intw (A) ⊆ B, we have τ2 − intw (A) ⊆ τ2 − intw (B). Then F ⊆ τ2 − intw (B). Therefore, B is τ1 τ2 − s∗ gw open set. Theorem 3.7 If a set A is τ1 τ2 − s∗ gw closed in X then τ2 − clw (A) − A is τ1 τ2 − s∗ gw open. Proof. Suppose that A is τ1 τ2 − s∗ gw closed in X. Let F be τ1 - semi closed and F ⊆ τ2 − clw (A) − A. Since A is τ1 τ2 − s∗ gw closed in X, τ2 − clw (A) − A contains no nonempty τ1 - semi closed set. Since F ⊆ τ2 − clw (A) − A , F = φ ⊆ τ2 − intw [τ2 − clw (A) − A]. Therefore, τ2 − clw (A) − A is τ1 τ2 − s∗ gwopen. Theorem 3.8 If A set A is τ1 τ2 −s∗ gw open in a bitopological space (X, τ1 , τ2 ), then G = X whenever G is τ1 - semi open and τ2 − intw (A) ∪ AC ⊆ G. Proof. Suppose that A is τ1 τ2 − s∗ gw open in a bitopological space (X, τ1 , τ2 ) and G is τ1 - semi open and τ2 − intw (A) ∪ AC ⊆ G. Then GC ⊆ {τ2 − intw (A)] ∪ AC }C = τ2 − clw (AC ) − AC . Since GC is τ1 - semi closed and AC is τ1 τ2 − s∗ gw closed , we have τ2 − clw (AC ) − AC contains no nonempty τ1 semi closed set in X { By Theorem 2.10 }. Therefore , GC = φ . Hence G = X.


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Theorem 3.9 The intersection of a τ1 τ2 − s∗ gw open set and τ1 − w open set is always τ1 τ2 − s∗ gw open. Proof. Suppose that A is τ1 τ2 − s∗ gw open and B is τ1 − w open. Then B C is τ2 − w closed. Therefore , B C is τ1 τ2 − s ∗ gw closed. { By Theorem 2.3 ( a ) }. Hence B is τ1 τ2 − s∗ gw open. Consequently , A ∩ B is τ1 τ2 − s∗ gw open. { By Theorem 4.5 } Theorem 3.10 If A × B is τ1 τ2 − s∗ gwopen subset of (X × Y, τ1 × σ1 , τ2 × σ2 ) , then A is τ1 τ2 − s∗ gw open subset in (X, τ1 , τ2 ) and B is σ1 σ2 − s∗ gw open subset in (Y, , σ1 , σ2 ). Proof. Let F be a τ1 - semi closed subset of (X, τ1 , τ2 ) and let G be a σ1 semi closed subset of (Y, σ1 , σ2 ) such that F ⊆ A and G ⊆ B. Then F × G is τ1 × σ1 - semi closed in (X × Y, τ1 × σ1 , τ2 × σ2 ) such that F × G ⊆ A × B. By assumption A × B is τ1 τ2 − s∗ gw open in (X × Y, τ1 × σ1 , τ2 × σ2 ) and so F × G ⊆ τ2 − intw (A × B) ⊆ τ2 − intw (A) × τ2 − intw (B) Therefore F ⊆ τ2 − intw (A) and G ⊆ τ2 − intw (B). Hence A and B are τ1 τ2 − s∗ gw open.

4

Pairwise Semi Star Generalized w−T1/2 Spaces

Recall that a space (X, τ ) is called T1/2 [10] if every g - closed set is closed. A space (X, τ ) is a regular generalized w − T1/2 (simply, rgw − T1/2 ) [1] if every rgw - closed set is w - closed. A space (X, τ1 , τ2 ) is called pairwise T1/2 if every τ1 − g closed set is τ2 - closed and or every τ2 − g closed set is τ1 - closed. A space (X, τ1 , τ2 ) is a pairwise regular generalized w − T1/2 (simply, pairwise rgw − T1/2 ) if every τ1 τ2 − rgw closed set in (X, τ1 , τ2 ) is τ2 − w closed and τ2 τ1 − rgw closed set in (X, τ1 , τ2 ) is τ1 − w closed. A space (X, τ ) is a semi star generalized w − T1/2 (simply, s∗ gw − T1/2 ) [6] if every s∗ gw - closed set is w - closed. We introduce the following relatively new definition. Definition 4.1 A space (X, τ1 , τ2 ) is a pairwise semi star generalized w − T1/2 (simply, pairwise s∗ gw − T1/2 ) if every τ1 τ2 − s∗ gw closed set in (X, τ1 , τ2 ) is τ2 − w closed and τ2 τ1 − s∗ gw closed set in (X, τ1 , τ2 ) is τ1 − w closed . Theorem 4.2 For a space (X, τ1 , τ2 ) , the following are equivalent. (a) X is pairwise s∗ gw − T1/2 , (b) Every singleton is either τi - semi closed or τj − w open, i = j.

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Proof. (1) ⇒ (2): Suppose {x} is not a τ1 - semi closed subset for some x ∈ X. Then X − {x} is not τ1 - semi open and hence X is the only τ1 - semi open set containing X − {x}. Therefore X − {x} is τ1 τ2 − s∗ gw closed . Since (X, τ1 , τ2 ) is τ1 τ2 − s∗ gw − T1/2 space, X − {x} is τ2 − w closed and thus {x} is τ2 − w open. (2) ⇒ (1) Let A be an τ1 τ2 − s∗ gw closed subset of (X, τ1 , τ2 ) and x ∈ τ2 − clw (A).We show that x ∈ A. Case: 1 If {x} is τ1 - semi closed and x does not belong to A, then x ∈ τ2 − clw (A) − A. Thus τ2 − clw (A) − A contains a nonempty τ1 - semi closed set {x}, which is a contradiction to the fact that A is τ1 τ2 − s∗ gw closed. So x ∈ A. Case:2 If {x} is τ2 − w open, since x ∈ τ2 − clw (A), then for every τ2 − w open set U containing x, we have U ∩ A = φ . But {x} is τ2 − w open, then {x} ∩ A = φ. Hence, x ∈ A. So in both cases we have x ∈ A. Therefore A is τ2 − w closed. Similarly we can prove every τ2 τ1 − s∗ gw closed set in (X, τ1 , τ2 ) is τ1 − w closed. Hence X is pairwise s∗ gw − T1/2 . Since, every τ1 τ2 − s∗ gwclosed set is τ1 τ2 − rgw closed, we have the following Theorem 4.3 Every pairwise rgw − T1/2 space is pairwise s∗ gw − T1/2 space.

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