sg - Locally Closed Sets in Bitopological Spaces 1 Introduction - m-hikari

Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 12, 597 - 607

s∗g - Locally Closed Sets in Bitopological Spaces K. Chandrasekhara Rao and K. Kannan Department of Mathematics, Srinivasa Ramanujan Centre SASTRA University, Kumbakonam - 612 001, India k.chandrasekhara@rediffmail.com anbukkannan@rediffmail.com Abstract The aim of this paper is to introduce the concepts of semi star generalized locally closed sets, s∗ g submaximal spaces and study their basic properties in bitopological spaces.

Mathematics Subject Classification: 54E55 Keywords: τ1 τ2 - semi star generalized locally closed sets, τ1 τ2 - semi star generalized closed sets, τ1 τ2 - submaximal spaces, τ1 τ2 −s∗ g submaximal spaces

1

Introduction

The study of generalization of closed sets has been found to ensure some new separation axioms which have been very useful in the study of certain objects of digital topology. In recent years many generalizations of closed sets have been developed by various authors. K. Chandrasekhara Rao and K. Joseph [3] introduced the concepts of semi star generalized open sets and semi star generalized closed sets in unital topological spaces. On the other hand K. Chandrasekhara Rao and K. Kannan [4] introduced the concepts of semi star generalized open sets and semi star generalized closed sets in bitopological spaces. Ganster and Reilly [12] introduced locally closed sets in topological spaces and Stone called locally closed sets as F G sets. H. Maki, P. Sundaram and K. Balachandran [21] introduced the concept of generalized locally closed sets and obtained seven different notions of generalized continuities. Ganster, Arockiarani and Balachandran [11] introduced regular generalized locally closed sets and RGLC continuous functions and discussed some of their properties.K. Chandrasekhara Rao and K. Kannan [6] introduced the concepts

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of semi star generalized locally closed sets and s∗ g - submaximal spaces in unital topological spaces. In this paper, we introduce the concept of semi star generalized locally closed sets, s∗ g - submaximal spaces and study their basic properties in bitopological spaces.

2

Preliminaries

Let (X, τ1 , τ2 ) or simply X denote a bitopological space. By τi −S ∗ GO(X, τ1 , τ2 ) {resp. τi − S ∗ GC(X, τ1 , τ2 )}, we shall mean the collection of all τi − s∗ g open sets (resp. τi − s∗ g closed sets) in (X, τ1 , τ2 ). 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 respectively. AC denotes the complement of A in X unless explicitly stated. We shall require the following known definitions. Definition 2.1 A subset of a bitopological space (X, τ1 , τ2 ) is called (a) τ1 τ2 - semi open if there exists a 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 subset 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 - 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. (f) τ1 τ2 - semi star generalized open (τ1 τ2 − s∗ g open) if X − A is τ1 τ2 − s∗ g closed in X.

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Semi Star Generalized Locally Closed Sets

Definition 3.1 A subset A of a bitopological space (X, τ1 , τ2 ) is said to be (a) τ1 τ2 − s∗ g locally closed set if A = G ∩ F where G is τ1 − s∗ g open set and F is τ2 − s∗ g closed set in X.

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(b) τ1 τ2 − s∗ g locally closed∗ if A = G ∩ F where G is τ1 − s∗ g open set and F is τ2 - closed in X. (c) τ1 τ2 − s∗ g locally closed∗∗ if A = G ∩ F where G is τ1 - open and F is τ2 − s∗ g closed in X. Remark 3.2 (a) The class of all τ1 τ2 − s∗ g locally closed sets in (X, τ1 , τ2 ) is denoted by τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ). (b) The class of all τ1 τ2 − s∗ g locally closed∗ sets in (X, τ1 , τ2 ) is denoted by τ1 τ2 − S ∗ GLC ∗ (X, τ1 , τ2 ). (c) The class of all τ1 τ2 − s∗ g locally closed∗∗ sets in (X, τ1 , τ2 ) is denoted by τ1 τ2 − S ∗ GLC ∗∗ (X, τ1 , τ2 ). Example 3.3 Let X = {a, b, c}, τ1 = {φ, X, {b, c}}, τ2 = {φ, X, {a}}. Then τ1 − s∗ g open sets in (X, τ1 , τ2 ) are φ, X, {b}, {c}, {b, c} and τ2 − s∗ g closed sets in (X, τ1 , τ2 ) are X, φ, {b, c}. Then (a) τ1 τ2 − s∗ g locally closed sets in (X, τ1 , τ2 ) are φ, X, {b}, {c}, {b, c}. (b) τ1 τ2 − s∗ g locally closed∗ sets in (X, τ1 , τ2 ) are φ, X, {b}, {c}, {b, c}. (c) τ1 τ2 − s∗ g locally closed∗∗ sets in (X, τ1 , τ2 ) are φ, X, {b, c}. Remark 3.4 Every τ1 τ2 −s∗ g locally closed set in (X, τ1 , τ2 ) are not τ2 - closed in general as can be seen from the following example. Example 3.5 In Example 3.3, {b} is τ1 τ2 −s∗ g locally closed set in (X, τ1 , τ2 ), but {b} is not τ2 - closed in (X, τ1 , τ2 ). Remark 3.6 Every τ1 τ2 − s∗ g locally closed set in (X, τ1 , τ2 ) are not τ1 - open in general as can be seen from the following example. Example 3.7 In Example 3.3, {c} is τ1 τ2 −s∗ g locally closed set in (X, τ1 , τ2 ), but {c} is not τ1 - open in (X, τ1 , τ2 ). Theorem 3.8 In any bitopological space (X, τ1 , τ2 ), (i) A ∈ τ1 τ2 − S ∗ GLC ∗ (X, τ1 , τ2 ) ⇒ A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ). (ii) A ∈ τ1 τ2 − S ∗ GLC ∗∗ (X, τ1 , τ2 ) ⇒ A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ). (iii) A ∈ τ2 − S ∗ GC(X, τ1 , τ2 ) ⇒ A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ). (iv) A ∈ τ1 − S ∗ GO(X, τ1 , τ2 ) ⇒ A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ).

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Proof. (i) Since A is τ1 τ2 − s∗ g locally closed∗ subset in (X, τ1 , τ2 ), we have A = G ∩ F where G is τ1 − s∗ g open set and F is τ2 - closed in X. Since every τ2 closed sets are τ2 − s∗ g closed in (X, τ1 , τ2 ), A = G ∩ F where G is τ1 − s∗ g open and F is τ2 − s∗ g closed in (X, τ1 , τ2 ). Therefore A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ). (ii) Since A is τ1 τ2 − s∗ g locally closed∗∗ subset in (X, τ1 , τ2 ), we have A = G ∩ F where G is τ1 - open and F is τ2 − s∗ g closed in (X, τ1 , τ2 ). Since every τ1 - open sets are τ1 − s∗ g open in (X, τ1 , τ2 ), A = G ∩ F where G is τ1 − s∗ g open and F is τ2 − s∗ g closed in (X, τ1 , τ2 ). Therefore A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ). (iii) Since A = A ∩ X and A is τ2 − s∗ g closed and X is τ1 − s∗ g open in (X, τ1 , τ2 ), we have A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ). (iv) Since A = A ∩ X and A is τ1 − s∗ g open and X is τ2 − s∗ g closed in (X, τ1 , τ2 ), we have A ∈ τ1 τ2 − S ∗ GLC(X, τ1 , τ2 ).  Remark 3.9 The converses of (i), (ii), (iii) and (iv) of the above theorem are not true in general as can be seen from the following examples. Example 3.10 Let X = {a, b, c, d}, τ1 = {φ, X, {a}, {a, b}}, τ2 = {φ, X, {a}, {c, d}}. Then {a, c} is τ1 τ2 − s∗ g locally closed in (X, τ1 , τ2 ), but not τ1 τ2 − s∗ g locally closed∗ in (X, τ1 , τ2 ). Example 3.11 In Example 3.3, {b} is τ1 τ2 − s∗ g locally closed in (X, τ1 , τ2 ), but not τ1 τ2 − s∗ g locally closed∗∗ in (X, τ1 , τ2 ). Example 3.12 In Example 3.10, {a, c} is τ1 τ2 −s∗ g locally closed in (X, τ1 , τ2 ), but not τ1 − s∗ g open in (X, τ1 , τ2 ) and {a} is τ1 τ2 − s∗ g locally closed in (X, τ1 , τ2 ), but not τ2 − s∗ g closed in (X, τ1 , τ2 ). Theorem 3.13 If (X, τ1 , τ2 ) is pairwise door space, then every subset of X is both τ1 τ2 − s∗ g locally closed and τ2 τ1 − s∗ g locally closed. Proof. Since (X, τ1 , τ2 ) is pairwise door space, every subset of (X, τ1 , τ2 ) is either τ1 - open or τ2 - closed and τ2 - open or τ1 - closed. Since every τ1 - open (resp. τ2 - closed) subset of (X, τ1 , τ2 ) is τ1 − s∗ g open (resp. τ2 − s∗ g closed), we have every subset of (X, τ1 , τ2 ) is either τ1 − s∗ g open or τ2 − s∗ g closed. Since every τ1 − s∗ g open and τ2 − s∗ g closed subset of (X, τ1 , τ2 ) is τ1 τ2 − s∗ g locally closed, we have every subset of X is τ1 τ2 − s∗ g locally closed. Similarly  we can prove that every subset of X is τ2 τ1 − s∗ g locally closed. Theorem 3.14 For a subset A of a bitopological space (X, τ1 , τ2 ), the following are equivalent. (a) A ∈ τ1 τ2 − S ∗ GLC ∗ (X, τ1 , τ2 ). (b) A = G ∩ [τ2 - cl (A)] for some τ1 − s∗ g open set G.

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(c) A ∪ {X − [τ2 - cl (A)]} is τ1 − s∗ g open. (d) [τ2 - cl (A)] − A is τ1 − s∗ g closed. Proof. (a) ⇒ (b) : Since A is τ1 τ2 − s∗ g locally closed∗ set in (X, τ1 , τ2 ), we have A = G ∩ F where G is τ1 − s∗ g open set and F is τ2 - closed in X. Since A ⊆ τ2 - cl (A) and A ⊆ G, we have A ⊆ G ∩ [τ2 - cl (A)] ..........(1) Since A ⊆ F and F is τ2 - closed in X, we have τ2 - cl (A) ⊆ F. Therefore G ∩ [τ2 - cl (A)] ⊆ G ∩ F = A. Hence G ∩ [τ2 - cl (A)] ⊆ A ........(2) From(1) and(2), we have A = G ∩ [τ2 - cl (A)] for some τ1 − s∗ g open set G in (X, τ1 , τ2 ). (b) ⇒ (a) : Suppose that A = G ∩ [τ2 - cl (A)] for some τ1 − s∗ g open set G in (X, τ1 , τ2 ). Since τ2 - cl (A) is τ2 - closed in (X, τ1 , τ2 ) and G is τ1 −s∗ g closed in (X, τ1 , τ2 ), we have A ∈ τ1 τ2 − S ∗ GLC ∗ (X, τ1 , τ2 ) (b) ⇒ (c) : Since A = G ∩ [τ2 - cl (A)] for some τ1 − s∗ g open set G in (X, τ1 , τ2 ), we have A ∪ {X − [τ2 - cl (A)]} = {G ∩ [τ2 - cl (A)]} ∪ {X − [τ2 - cl (A)]} = G. Therefore A ∪ {X − [τ2 - cl (A)]} is τ1 − s∗ g open. (c) ⇒ (b) : Suppose that A ∪ {X − [τ2 - cl (A)]} is τ1 − s∗ g open in (X, τ1 , τ2 ). Let G = A ∪ {X − [τ2 - cl (A)]}. Then G is τ1 − s∗ g open in (X, τ1 , τ2 ). Now, G ∩ [τ2 − cl(A)] = [A ∪ {X − [τ2 − cl(A)]}] ∩ [τ2 − cl(A)] = {[A ∪ [τ2 − cl(A)]C } ∩ [τ2 − cl(A)] = {A ∩ [τ2 − cl(A)]} ∪ {[τ2 − cl(A)]C ∩ [τ2 − cl(A)]} =A∪φ = A. Therefore A = G ∩ [τ2 - cl (A)] for some τ1 − s∗ g open set G in (X, τ1 , τ2 ). (c) ⇒ (d) : Suppose that A ∪ {X − [τ2 - cl (A)]} is τ1 − s∗ g open in (X, τ1 , τ2 ). Let G = A ∪ {X − [τ2 - cl (A)]}. Since G is τ1 − s∗ g open in (X, τ1 , τ2 ), we have X − G is τ1 − s∗ g closed in (X, τ1 , τ2 ). Now, X − G = X − [A ∪ {X − [τ2 − cl(A)]}] = (X − A) ∩ {X − [τ2 − cl(A)]} = (X − A) ∩ [τ2 − cl(A)] = τ2 − cl(A) − A.

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Therefore, τ2 - cl (A) − A is τ1 − s∗ g closed in (X, τ1 , τ2 ). (d) ⇒ (c) : Suppose that τ2 - cl (A) − A is τ1 − s∗ g closed in (X, τ1 , τ2 ). Let F = τ2 - cl (A) − A. Then F is τ1 − s∗ g closed in (X, τ1 , τ2 ) implies that X − F is τ1 − s∗ g open in (X, τ1 , τ2 ). Now, X − F = X − {[τ2 − cl(A)] − A} = X ∩ {[τ2 − cl(A)] − A}C = X ∩ {[τ2 − cl(A)] ∩ AC }C = X ∩ {[τ2 − cl(A)]C ∪ (AC )C } = X ∩ {[τ2 − cl(A)]C ∪ A} = {X ∩ [τ2 − cl(A)]C } ∪ {X ∩ A} = [τ2 − cl(A)]C ∪ A = {X − [τ2 − cl(A)]} ∪ A. Hence A ∪ {X − [τ2 - cl (A)]} is τ1 − s∗ g open in (X, τ1 , τ2 ).



Theorem 3.15 In a bitopological space (X, τ1 , τ2 ), the following are equivalent. (a) A − [τ1 - int (A)] is τ2 − s∗ gopen in (X, τ1 , τ2 ). (b) [τ1 - int (A)] ∪ [X − A] is τ2 − s∗ g closed in (X, τ1 , τ2 ). (c) G ∪ [τ1 - int (A)] = A for some τ2 − s∗ g open set G in (X, τ1 , τ2 ). Proof. (a) ⇒ (b) : Now, X − {A − [τ1 − int(A)]} = X ∩ {A − [τ1 − int(A)]}C = X ∩ [A ∩ {τ1 − int(A)}C ]C = X ∩ {AC ∪ [{τ1 − int(A)}C ]C } = X ∩ {AC ∪ [τ1 − int(A)]} = {AC ∪ [τ1 − int(A)]} = [τ1 − int(A)] ∪ [X − A]. Since A − [τ1 - int (A)] is τ2 − s∗ g open, we have X − {A − [τ1 - int (A)]} = [τ1 − int(A)] ∪ [X − A] is τ2 − s∗ g closed in (X, τ1 , τ2 ). (b) ⇒ (a) : Suppose that [τ1 - int (A)] ∪ [X − A] is τ2 − s∗ g closed in (X, τ1 , τ2 ). Since [τ1

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- int (A)] ∪ [X − A] is τ2 − s∗ g closed, we have X − {[τ1 - int (A)] ∪ [X − A]} is τ2 − s∗ g open. Now, X − {[τ1 − int(A)] ∪ [X − A]} = X ∩ {[τ1 − int(A)] ∪ [X − A]}C = X ∩ {[τ1 − int(A)] ∪ AC }C = X ∩ {[τ1 − int(A)]C ∩ (AC )C } = X ∩ {[τ1 − int(A)]C ∩ A} = A ∩ [τ1 − int(A)]C = A − [τ1 − int(A)]. Therefore A − [τ1 - int (A)] is τ2 − s∗ g open in (X, τ1 , τ2 ). (b) ⇒ (c) : Suppose that [τ1 - int (A)] ∪ [X − A] is τ2 − s∗ g closed. Let U = [τ1 - int (A)] ∪ [X − A]. Then U is τ2 − s∗ g closed. Then U C is τ2 − s∗ g open. Now, U C ∪ [τ1 − int(A)] = {[τ1 − int(A)] ∪ [X − A]}C ∪ [τ1 − int(A)] = {[τ1 − int(A)]C ∩ (AC )C } ∪ [τ1 − int(A)] = {[τ1 − int(A)]C ∩ A} ∪ [τ1 − int(A)] = {[τ1 − int(A)]C ∪ [τ1 − int(A)]} ∩ {A ∪ [τ1 − int(A)]} =X ∩A = A. Take G = U C . Then A = G ∪ [τ1 - int (A)] = A for some τ2 − s∗ g open set in (X, τ1 , τ2 ). (c) ⇒ (b) : Suppose that A = G ∪ [τ1 - int (A)] = A for some τ2 − s∗ g open set G in (X, τ1 , τ2 ). Now, [τ1 − int(A)] ∪ [X − A] = τ1 − int(A) ∪ AC = [τ1 − int(A)] ∪ {G ∪ [τ1 − int(A)]}C = [τ1 − int(A)] ∪ {GC ∩ [τ1 − int(A)]C } = {[τ1 − int(A)] ∪ GC } ∩ {[τ1 − int(A)] ∪ [τ1 − int(A)]C } = {[τ1 − int(A)] ∪ GC } ∩ X} = {[τ1 − int(A)] ∪ GC } = X − G. Since G is τ2 − s∗ g open in (X, τ1 , τ2 ), we have X − G is τ2 − s∗ g closed in (X, τ1 , τ2 ). Therefore [τ1 - int (A)] ∪ [X − A] is τ2 − s∗ g closed in (X, τ1 , τ2 ). 

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Remark 3.16 The union of two τ1 τ2 − s∗ g locally closed sets in (X, τ1 , τ2 ) is not τ1 τ2 − s∗ g locally closed in general as can be seen from the following example. Example 3.17 Let X = {a, b, c, d}, τ1 = {φ, X, {a}, {a, b}}, τ2 = {φ, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}}. Then A = {a, d}, B = {b, d} are τ1 τ2 − s∗ g locally closed sets in (X, τ1 , τ2 ), but A ∪ B = {a, b, d} is not τ1 τ2 − s∗ g locally closed set in (X, τ1 , τ2 ). Remark 3.18 Even A and B are not τ1 τ2 −s∗ g locally closed sets in (X, τ1 , τ2 ), A∪ B is τ1 τ2 − s∗ g locally closed in general as can be seen from the following example. Example 3.19 Let X = {a, b, c, d}, τ1 = {φ, X, {a}, {a, b}}, τ2 = {φ, X, {c, d}, {b, c, d}}. Then A = {b}, B = {a, d} are not τ1 τ2 − s∗ g locally closed sets in (X, τ1 , τ2 ), but A ∪ B = {a, b, d} is τ1 τ2 − s∗ g locally closed set in (X, τ1 , τ2 ).

4

s∗g - Submaximal Spaces

Definition 4.1 A bitopological space (X, τ1 , τ2 ) is (i) τ1 τ2 - submaximal space if every τ1 - dense subset of X is τ2 - open in X. (ii) τ2 τ1 - submaximal space if every τ2 - dense subset of X is τ1 - open in X. (iii) τ1 τ2 − s∗ g submaximal space if every τ1 - dense subset of X is τ2 − s∗ g open in X. (iv) τ2 τ1 − s∗ g submaximal space if every τ2 - dense subset of X is τ1 − s∗ g open in X. Example 4.2 In Example 3.17, (i) τ1 - dense subsets of (X, τ1 , τ2 ) are X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}. (ii) τ2 − s∗ g open sets of (X, τ1 , τ2 ) are φ, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}. (iii) τ2 - open sets of (X, τ1 , τ2 ) are φ, X, {a}, {a, b}{a, c}{a, d}, {a, b, c}, {a, b, d}, {a, c, d}. Therefore (X, τ1 , τ2 ) is both τ1 τ2 − s∗ g submaximal space and τ1 τ2 - submaximal space. Theorem 4.3 If (X, τ1 , τ2 ) is τ1 τ2 - submaximal space then X is τ1 τ2 − s∗ g submaximal space.

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Proof. Since (X, τ1 , τ2 ) is τ1 τ2 - submaximal space, we have every τ1 - dense subset of X is τ2 - open in X. Since every τ2 - open set in X is τ2 − s∗ g open in X, we have every τ1 - dense subset of X is τ2 − s∗ g - open in X. Therefore (X, τ1 , τ2 ) is τ1 τ2 − s∗ g submaximal space.  Remark 4.4 The converse of the above theorem is not true in general as can be seen from the following example. Example 4.5 Let X = {a, b, c, d}, τ1 = {φ, X, {a}, {a, b}}, τ2 = {φ, X, {a}, {b, c, d}}. Then τ1 - dense subsets of (X, τ1 , τ2 ) are X, {a}, {a, b}{a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}. Therefore (X, τ1 , τ2 ) is τ1 τ2 −s∗ g submaximal space but not τ1 τ2 - submaximal space. Theorem 4.6 A bitopological space (X, τ1 , τ2 ) is τ1 τ2 −s∗ g submaximal space if and only if τ2 τ1 − S ∗ GLC ∗ (X, τ1 , τ2 ) = P (X). Proof. Suppose that (X, τ1 , τ2 ) is τ1 τ2 − s∗ g submaximal space. Obviously τ2 τ1 − S ∗ GLC ∗ (X, τ1 , τ2 ) ⊆ P (X). Let A ∈ P (X) and U = A ∪ {X − [τ1 - cl (A)]}. Since τ1 - cl (U) = X, we have U is τ1 - dense subset of X. Since (X, τ1 , τ2 ) is τ1 τ2 − s∗ g submaximal space, we have U is τ2 − s∗ g open in X. Since every τ2 − s∗ g open set in X is τ2 τ1 − s∗ g locally closed∗ set in (X, τ1 , τ2 ), we have U ∈ τ2 τ1 − S ∗ GLC ∗ (X, τ1 , τ2 ). Therefore P (X) ⊆ τ2 τ1 − S ∗ GLC ∗ (X, τ1 , τ2 ). Hence τ2 τ1 − S ∗ GLC ∗ (X, τ1 , τ2 ) = P (X). Conversely, suppose that τ2 τ1 − S ∗ GLC ∗ (X, τ1 , τ2 ) = P (X). Let A be the τ1 dense subset of (X, τ1 , τ2 ). Then A ∪ {X − [τ1 - cl (A)]} = A ∪ [τ1 - cl (A)]C = A. Therefore A ∈ τ2 τ1 − S ∗ GLC ∗ (X, τ1 , τ2 ) implies that A is τ2 − s∗ g open in X {By Theorem 3.14}. Hence X is τ1 τ2 − s∗ g submaximal space. 

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