Generalized Alpha Star Star Closed Sets in Bitopological Spaces

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Generalized Alpha Star Star Closed Sets in Bitopological Spaces

Qays Hatem Imran

Al-Muthanna University, College of Education Department of Mathematics, Al-Muthanna, Iraq

E-mail: [email protected]

(Received: 12-2-14 / Accepted: 20-3-14)

Abstract

The aim of this paper is to introduce the concepts of generalized alpha star star closed sets, generalized alpha star star open sets and studies their basic properties in bitopological spaces.

Keywords: Bitopological space,τ₁τ₂-generalized alpha star closed sets,

2 -

1τ

τ generalized alpha star star closed sets,τ₁τ₂ -generalized alpha star star open sets.

1 Introduction

Levine, [6] initiated the study of generalized closed sets in topological spaces in 1970. In 1963, J.C. Kelly, [2] defined: a set equipped with two topologies is called a bitopological space, denoted by (^X,

τ

₁,

τ

₂)^where⁽^X^,τ1⁾^and⁽^X^,τ2⁾ are two topological spaces. In 1986, T. Fukutake, [7] generalized this notion to bitopological spaces and he defined a set A of a bitopological space X to be an ij- generalized closed set (briefly ij-g-closed) if j-cl(A)⊆U whenever A⊆U and U is

τ

_i-open^{in X,} i^, j =¹^,² and i≠ j. Semi generalized closed sets and

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generalized semi closed sets are extended to bitopological settings by F. H. Khedr and H. S. Al-saadi, [1]. K. Chandrasekhara Rao and K. Kannan, [4, 5] introduced the concepts of semi star generalized closed sets in bitopological spaces.

The aim of this communication is to introduce the concepts of

τ

₁

τ

₂-generalized alpha star star closed sets,

τ

₁

τ

₂-generalized alpha star star open sets and study their basic properties in bitopological spaces.

2 Preliminaries

Throughout this paper, spaces always mean a bitopological spaces, for a subset A of X

τ

_i -cl(A)^(resp.

τ

i ^-^int⁽^A⁾^,

τ

i ^-

α

^cl⁽^A⁾) denote the closure (resp. interior,

α

^- closure) of A with respect to the topology

τ

_i^{, for}i =1,2 .

Definition 2.1: A subset A of a bitopological space (^X,

τ

₁,

τ

₂)^{is called}

(i)

τ

₁

τ

₂ -

α

-open [3] if A⊆

τ

₁-int(

τ

₂ -cl(

τ

₁-int(A)))^.

(ii)

τ

₁

τ

₂ -

α

-closed [3] if X −A is

τ

₁

τ

₂ -

α

-open. Equivalently, a subset A of a bitopological space (^X,

τ

₁,

τ

₂) is called

τ

₁

τ

₂ -

α

-closed if

A A

cl

cl ( ₂ )))⊆

2 - (

τ

₁-int

τ

- (

τ

^.

(iii)

τ

₁

τ

₂-generalized closed (briefly

τ

₁

τ

₂ -g-closed) [7] if

τ

₂ -cl(A)⊆U whenever A⊆U and U is

τ

₁-open in X.

(iv)

τ

₁

τ

₂-generalized open (briefly

τ

₁

τ

₂ -g-^open) ^{[7] if} ^X −^Ais -

2 -

1

τ

g

τ

^closed.

(v)

τ

₁

τ

₂ -alpha generalized closed (briefly

τ

₁

τ

₂ -

α

g-closed) [3] if U

A cl( ) ⊆

2 α

τ ^- ^whenever^A⊆^U and U is

τ

₁-open in X .

(vi)

τ

₁

τ

₂ -alpha generalized open (briefly

τ

₁

τ

₂ -

α

g-open) [3] if X −Ais -

2 -

1

τ α

g

τ

^closed.

(vii)

τ

₁

τ

₂ -generalized alpha closed (briefly

τ

₁

τ

₂ -^g

α

-closed) [3] if U

A cl( ) ⊆

2 α

τ ^- ^whenever^A⊆^U and U is

τ

₁-

α

-open in X.

(viii)

τ

₁

τ

₂ -generalized alpha open (briefly

τ

₁

τ

₂-^g

α

-open) [3] if X −Ais -

2 -

1

τ α

τ

^g ^closed.

τ

₁,

τ

₂) is called

2 -

1

τ

generalized alpha star closed (briefly τ₁τ₂ -^gα^*-closed) if

τ

₂ -cl(A)⊆U whenever A⊆Uand U is

τ

₁-

α

-open in X. The family of all τ₁τ₂ -^gα^*-^closed sets of X is denoted by

τ

₁

τ

₂ -g

α

^*C(X)^.

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Example 2.3: LetX ={a,b,c},

τ

₁={

φ

,X,{a},{b,c}}^,

τ

₂ ={

φ

,X,{a},{b},{a,b}}^. Then {a,b} is

τ

₁

τ

₂ -^g

α

^* -closed and {a} is not

τ

₁

τ

₂ -^g

α

^*-^closed.

τ

₁,

τ

₂) is called

2-

1

τ

generalized alpha star open (briefly

τ

₁

τ

₂ -^g

α

^* -open) if and only if A

X − is

τ

₁

τ

₂ -^g

α

^*-closed. The family of all

τ

₁

τ

₂ -^g

α

^*-open sets of X is denoted by

τ

₁

τ

₂ -g

α

^*O(X)^.

3 Generalized Alpha Star Star Closed Sets

In this section we define and study the concept of

τ

₁

τ

₂ -generalized alpha star star closed sets in bitopological spaces.

τ

₁,

τ

₂) is called

2 -

1

τ

generalized alpha star star closed (briefly

τ

₁

τ

₂-^g

α

^*^*-closed) if U

A cl( )⊆

2 -

τ

whenever A⊆U and U is

τ

₁-^g

α

^*-open in X. The family of all -

- ^*^*

2

1

τ α

τ

^g closed sets of X is denoted by

τ

₁

τ

₂-g

α

^*^*C(X)^.

Example 3.2: LetX ={a,b,c,d},

τ

₁={

φ

,X,{a,b}}^,

τ

₂ ={

φ

,X,{b},{c},{b,c}}^. Then ϕ, X,{c},{d}, {a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},{a,c,d},{a,b,d},{b,c,d}

are

τ

₁

τ

₂ -^g

α

^*^*-closed sets.

Now, the characterization of

τ

₁

τ

₂-^g

α

^*^*-closed sets by using different types of generalization of closed sets and

τ

₁-^g

α

^*-open sets are established in the following theorem.

Theorem 3.3: Let (^X,

τ

₁,

τ

₂) be a bitopological space andA⊆ X . Then the following are true.

(i) If A is

τ

₂-closed, then A is

τ

₁

τ

₂-^g

α

^*^*-^closed.

(ii) If A is

τ

₁-^g

α

^*-^{open and}

τ

1

τ

2^-^g

α

^*^*^-closed, then A is

τ

₂ -^closed.

(iii) If A is

τ

₁

τ

₂ -^g

α

^*^*-closed, then A is

τ

₁

τ

₂ -g-^closed.

(iv) If A is

τ

₁

τ

₂ -^g

α

^*^*-closed, then A is

τ

₁

τ

₂ -

α

g-^closed.

Proof:

(i) It is obvious that every

τ

₂ -closed set is

τ

₁

τ

₂ -^g

α

^*^*-^closed.

(ii) Suppose that A is

τ

₁-^g

α

^*-^{open and}

τ

1

τ

2^-^g

α

^*^*^-closed. Then A⊆ A implies that

τ

-cl(A)⊆ A. Obviously, A⊆

τ

-cl(A). Therefore, A is

τ

-^closed.

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(iii) Suppose that A is

τ

₁

τ

₂-^g

α

^*^*-closed. Let A⊆Uand U is

τ

₁-open in X. Since every

τ

₁-open set is

τ

₁-^g

α

^*-open in X, we have U is

τ

₁-^g

α

^*-open in X. Then,

U A cl( )⊆

2 -

τ

^{since A is}

τ

1

τ

2^-^g

α

^*^*^-closed. Consequently, A is

τ

₁

τ

₂ -g-^closed.

(iv) Suppose that A is

τ

₁

τ

₂-^g

α

^*^*-closed. Let A⊆Uand U is

τ

₁-open in X. Since every

τ

₁-open set is

τ

₁-^g

α

^*-open in X, we have U is

τ

₁-^g

α

^*-open in X.

Then,

τ

₂ -cl(A)⊆U since A is

τ

₁

τ

₂-^g

α

^*^*-closed. Since

τ

₂ -

α

cl(A)⊆

τ

₂-cl(A)^, we have

τ

₂ -

α

cl(A)⊆U. Consequently, A is

τ

₁

τ

₂ -

α

g-^closed.

In the following examples it is proved that the converses of the assertions of the above theorem are not true in general.

Example 3.4: In example (3.2), {c} is

τ

₁

τ

₂-^g

α

^*^*-closed but not

τ

₂ -closed. Also {a, d} is

τ

₂ -^closed,

τ

1

τ

2^-^g

α

^*^*^-closed but not

τ

₁-^g

α

^*-^open.

τ

₁ ={

φ

,X,{b,c}}^,

τ

₂ ={

φ

,X,{a}}. Then {b} is -

2 -

1

τ

g

τ

closed but not

τ

₁

τ

₂-^g

α

^*^*-closed in X.

Example 3.6: In example (3.2), {a} is

τ

₁

τ

₂ -

α

g-closed but not

τ

₁

τ

₂-^g

α

^*^*-^closed in X.

Remark 3.7:

τ

₁

τ

₂-^g

α

-closed sets and

τ

₁

τ

₂-^g

α

^*^*-closed sets are independent in general. The following example supports our claim. In Example (3.2), {a} is

-

2-

1

τ α

τ

^g closed but not

τ

₁

τ

₂-^g

α

^*^*-closed in X. Also {a,b,c} is

τ

₁

τ

₂-^g

α

^*^*- closed but not

τ

₁

τ

₂-^g

α

-closed in X.

Theorem 3.8: If A is

τ

₁

τ

₂-^g

α

^*^*-^closed,

τ

1^-^g

α

^*^-open in X and F is

τ

₂ -^closed in X then A∩Fis

τ

₂-closed in X.

Proof: Since A is

τ

₁

τ

₂-^g

α

^*^*-^closed,

τ

1^-^g

α

^*^-open in X, we have A is

τ

₂ -^closed in X [by theorem (3.3) (ii)]. Since F is

τ

₂ -closed in X, A∩Fis

τ

₂-closed in X.

Remark 3.9:

(i)

τ

₁

τ

₂-^g

α

^*-closed and

τ

₁

τ

₂ -

α

-closed sets are independent in general.

(ii)

τ

₁

τ

₂ -^g

α

^*^*-closed and

τ

₁

τ

₂ -^g

α

^*-closed sets are independent in general.

Example 3.10: In example (3.2), {c} is

τ

₁

τ

₂-^g

α

^*^*-closed but not

τ

₁

τ

₂ -^g

α

^* - closed in X.

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τ

₁

τ

₂-^g

α

^*^*-closed in X andA⊆B⊆

τ

₂ -cl(A), then B is -

- ^*^*

2

1

τ α

τ

^g ^closed.

Proof: Suppose that A is

τ

₁

τ

₂ -^g

α

^*^*-closed in X andA⊆B⊆

τ

₂-cl(A)^{. Let} U

B⊆ and U is

τ

₁-^g

α

^*-open in X. Since A⊆BandB⊆U , we have A⊆U . Hence

τ

₂-cl(A)⊆U (Since A is

τ

₁

τ

₂-^g

α

^*^*-closed). Since B⊆

τ

₂ -cl(A)^{, we} have

τ

₂-cl(B)⊆

τ

₂-cl(A)⊆U . Therefore, B is

τ

₁

τ

₂-^g

α

^*^*-^closed.

Theorem 3.12: If A and B are

τ

₁

τ

₂-^g

α

^*^*-closed sets then so isA∪B.

Proof: Suppose that A and B are

τ

₁

τ

₂-^g

α

^*^*-closed sets. Let U be

τ

₁-^g

α

^*-^open in X andA∪B⊆U. SinceA∪B⊆U , we have A⊆U and B⊆U . Since U is

-

- ^*

1

α

τ

^g open in X and A and B are

τ

₁

τ

₂-^g

α

^*^*-closed sets, we have U

A cl( )⊆

2 -

τ

^and

τ

₂-^cl(^B)⊆^U. Therefore,

τ

₂ -cl(A∪B)⊆

τ

₂ -cl(A)∪ U

B cl( )⊆

2 -

τ

^{. Hence}^A^∪^B^is

τ

1

τ

2^-^g

α

^*^*^-^closed.

Remark 3.13: The following diagram shows the relations among the different types of weakly closed sets that were studied in this section:

Theorem 3.14: The arbitrary union of

τ

₁

τ

₂-^g

α

^*^*-closed sets A_i,i∈Iin a bitopological space (^X,

τ

₁,

τ

₂)^is

τ

1

τ

2^-^g

α

^*^*^-closed if the family {A_i,i∈I}is locally finite in (^X,

τ

₁)^.

closed

2 -

τ

₁

τ

₂ -^g

α

^*^* -closed

closed -

2 -

1

τ α

g

τ

closed -

2 -

1

τ

g

τ

closed

-

2 -

1

τ α

τ

^g

+

open -

- ^*

1

α

τ

^g closed -

2 -

1

τ α

τ

closed -

- ^*

2

1

τ α

τ

^g

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Proof: Let {A_i,i∈I}be locally finite in X and A is _i

τ

₁

τ

₂-^g

α

^*^*-closed in X for each i∈I. Let ∪A_i ⊆U and U is

τ

₁-^g

α

^*-open in X. Then A_i ⊆U and U is

-

- ^*

1

α

τ

^g open in X for each i. Since A_i is

τ

₁

τ

₂-^g

α

^*^*-closed in X for each i∈I, we have

τ

₂ -cl(A_i)⊆U . Consequently, ^∪[

τ

₂ -cl(A_i)]⊆U . Since the family

} ,

{A_i i∈I is locally finite in X,

τ

₂ -cl[^∪(A_i)]=^∪[

τ

₂ -cl(A_i)]⊆U . Therefore, Ai

∪ is

τ

₁

τ

₂-^g

α

^*^*-closed in X.

Remark 3.15: The intersection of any two

τ

₁

τ

₂-^g

α

^*^*-closed sets is not necessary

τ

₁

τ

₂ -^g

α

^*^*-closed set as in the following example.

Example 3.16: In example (3.2),A={ ca, },B={ da, } are

τ

₁

τ

₂ -^g

α

^*^*-closed but }

{a B

A∩ = is not

τ

₁

τ

₂-^g

α

^*^*-closed in X.

Theorem 3.17: If a set A is

τ

₁

τ

₂ -^g

α

^*^*-closed in X , then

τ

₂ -cl(A)−A^contains no nonempty

τ

₁-^g

α

^*-closed set.

τ

₁

τ

₂-^g

α

^*^*-closed in X. Let F be

τ

₁-^g

α

^*-closed and A

A cl

F ⊆

τ

₂ - ( )− . Since F is

τ

₁-^g

α

^*-closed, we have F is ^c

τ

₁-^g

α

^*-^open.

SinceF ⊆

τ

₂ -cl(A)−A^{, we have}F ⊆

τ

2 ^-cl⁽A⁾^andF ⊆ A^c^{. Hence}A⊆F^c^. Consequently

τ

₂ -cl(A)⊆F^c {Since A is

τ

₁

τ

₂ -^g

α

^*^*-closed in X}. Therefore,

A c

cl

F ⊆[

τ

₂ - ( )] . Hence F ⊆[

τ

₂ -cl(A)]^c∩

τ

₂-cl(A)=

φ

. Hence

τ

₂ -cl(A)−A contains no nonempty

τ

₁-^g

α

^*-closed set.

Corollary 3.18: Let A be

τ

₁

τ

₂-^g

α

^*^*-closed. Then A is

τ

₂ -closed if and only if A

A cl( )−

2 -

τ

^is

τ

1^-^g

α

^*^-^closed.

τ

₁

τ

₂-^g

α

^*^*-closed and

τ

₂ -closed. Since A is

τ

₂-^closed, we have

τ

₂ -cl(A)= A. Therefore,

τ

₂ -cl(A)−A=

φ

which is

τ

₁-^g

α

^*-^closed.

Conversely, suppose that A is

τ

₁

τ

₂-^g

α

^*^*-closed and

τ

₂ -cl(A)−A^is

τ

1^-^g

α

^*^- closed. Since A is

τ

₁

τ

₂-^g

α

^*^*-closed, we have

τ

₂-cl(A)−A contains no nonempty

τ

₁-^g

α

^*-closed set {by Theorem 3.17}. Since

τ

₂ -cl(A)−A is itself

-

- ^*

1

α

τ

^g closed, we have

τ

₂ -cl(A)−A=

φ

. Therefore,

τ

₂ -cl(A)= A^implies that A is

τ

₂-^closed.

τ

₁

τ

₂-^g

α

^*^*-closed and A⊆B⊆

τ

₂ -cl(A)^then B

B cl( )−

2 -

τ

contains no nonempty

τ

₁-^g

α

^*-closed set.

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Proof: Let A be

τ

₁

τ

₂-^g

α

^*^*-closed andA⊆B⊆

τ

₂ -cl(A). Then B is -

- ^*^*

2

1

τ α

τ

^g closed {by theorem (3.11)}. Therefore,

τ

₂ -cl(B)−Bcontains no nonempty

τ

₁-^g

α

^*-closed set {by theorem (3.17)}.

Theorem 3.20: For each x∈X, the singleton {x} is either

τ

₁-^g

α

^*-closed or its complement{x}^cis

τ

₁

τ

₂-^g

α

^*^*-^{closed in}⁽^X^,

τ

1^,

τ

2⁾^.

Proof: Let x∈X. Suppose that {x} is not

τ

₁-^g

α

^*-closed. Then{x}^cis not -

- ^*

1

α

τ

^g open. Consequently, X itself is the only

τ

₁-^g

α

^*-^{open set} containingX −{x}. Therefore,

τ

₂ -cl(X −{x})⊆ X which implies that X −{x} is

τ

₁

τ

₂-^g

α

^*^*-^{closed in}⁽^X^,

τ

1^,

τ

2⁾^.

4 Generalized Alpha Star Star Open Sets

We begin this section with a relatively new definition.

τ

₁,

τ

₂) is called

2-

1

τ

generalized alpha star star open (briefly

τ

₁

τ

₂ -^g

α

^*^*-open) if and only if A

X − is

τ

₁

τ

₂-^g

α

^*^*-closed. The family of all

τ

₁

τ

₂ -^g

α

^*^*-open sets of X is denoted by

τ

₁

τ

₂ -g

α

^*^*O(X)^.

Example 4.2: In example (3.2), ϕ, X, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {a, b, c}, {a, b, d} are

τ

₁

τ

₂-^g

α

^*^*-open sets in X.

The following theorem will give an equivalent definition of

τ

₁

τ

₂-^g

α

^*^*-open sets.

Theorem 4.3: A set A is

τ

₁

τ

₂ -^g

α

^*^*-open if and only if F ⊆

τ

₂ -int(A)^whenever F is

τ

₁-^g

α

^*-closed and F ⊆ A.

τ

₁

τ

₂-^g

α

^*^*-^{open. Then}^{A is}^c

τ

1

τ

2^-^g

α

^*^*^-closed. Suppose that F is

τ

₁-^g

α

^*-closed and F ⊆ A. Then F is ^c

τ

₁-^g

α

^*-^{open and}^A^c ⊆ ^F^c^.

Therefore,

τ

₂ -cl(A^c)⊆ F^c^(since ^{A is}^c

τ

1

τ

2^-^g

α

^*^*^-closed). Since

τ

₂ -cl(A^c)= A)]c

( int -

[

τ

₂ ^{, we have}[

τ

₂ -int(^A)]^c ⊆^F^c^{. Hence}F ⊆

τ

2 ^-^int⁽A⁾^.

Conversely, suppose that F ⊆

τ

₂ -int(A) whenever F is

τ

₁-^g

α

^*-closed and A

F ⊆ . Then A^c ⊆F^c^and^{F is}^c

τ

1^-^g

α

^*^-open. Take U =F^c.

Since F ⊆

τ

₂ -int(A), we have [

τ

₂ -int(A)]^c ⊆F^c =U^{. Since}

τ

₂ -cl(A^c)= A)]c

( int -

[

τ

^{, we have}

τ

-^cl(^A^c)⊆^U. Therefore, A is ^c

τ τ

-^g

α

^*^*-^closed.

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Thus, A is

τ

₁

τ

₂-^g

α

^*^*-^open.

Remark 4.4: Every

τ

₁-open set is

τ

₁

τ

₂ -^g

α

^*^*-open but the converse is not true in general as can be seen from the following example.

Example 4.5: In example (3.2), {a, c} is

τ

₁

τ

₂ -^g

α

^*^*-open in X but not

τ

₁-^open in X.

Remark 4.6:

τ

₁

τ

₂ -^g

α

^*^* -^{open and}

τ

1

τ

2 ^-^g

α

^* ^-open sets are in general, independent as can be seen from the following two examples.

τ

₁ ={

φ

,X,{a},{b,c}}^,

τ

₂ ={

φ

,X,{b},{a,c}}^. Then {c} is

τ

₁

τ

₂ -^g

α

^*-open in X but not

τ

₁

τ

₂ -^g

α

^*^* -open in X.

Example 4.8: In example (3.2), {d} is

τ

₁

τ

₂ -^g

α

^*^* -open in X but not -

- ^*

2

1

τ α

τ

^g open in X.

Remark 4.9: The union of any two

τ

₁

τ

₂-^g

α

^*^*-open sets is not necessary -

- ^*^*

2

1

τ α

τ

^g open set as in the following example.

Example 4.10: In example (3.2),A={ cb, }, B={ db, } are

τ

₁

τ

₂ -^g

α

^*^*-^{open sets} but A∪B={b,c,d} is not

τ

₁

τ

₂ -^g

α

^*^*-open in X.

Theorem 4.11: If A and B are

τ

₁

τ

₂-^g

α

^*^*-open sets then so is A∩B.

Proof: Suppose that A and B are

τ

₁

τ

₂-^g

α

^*^*-open sets. Let F be

τ

₁-^g

α

^*-^closed in X and F ⊆ A∩B. Since F ⊆ A∩B, we have F ⊆ Aand F ⊆B. Since A and B are

τ

₁

τ

₂-^g

α

^*^*- open sets. Then F ⊆

τ

₂ -int(A)^andF ⊆

τ

₂ -int(B). Therefore,

) int(

- ) ( int

- ₂

2 A B

F ⊆

τ

^∩

τ

⊆

τ

₂ -int(A∩B)^{. Hence}^A^∩^B^is

τ

1

τ

2^-^g

α

^*^*^-^open.

Theorem 4.12: The arbitrary intersection of

τ

₁

τ

₂-^g

α

^*^*-open sets A_i,i∈Iin a bitopological space (^X,

τ

₁,

τ

₂)^is

τ

1

τ

2 ^-^g

α

^*^*^-open if the family {A_i^c,i∈I}^is locally finite in (^X,

τ

₁)^.

Proof: Let {A_i^c,i∈I}be locally finite in(^X,

τ

₁)^and A is i

τ

₁

τ

₂-^g

α

^*^*-open in X for each i∈I. Then A_i^c is

τ

₁

τ

₂-^g

α

^*^*-closed in X for each i∈I. Then by theorem (3.14), we have ∪(A_i^c) is

τ

₁

τ

₂-^g

α

^*^*-closed in X. Consequently, let

c

A )i

(∩ is

τ

₁

τ

₂-^g

α

^*^*-closed in X. Therefore, ∩A_i is

τ

₁

τ

₂-^g

α

^*^*-open in X.

(9)

τ

₁

τ

₂-^g

α

^*^*-open in X and

τ

₂ -int(A)⊆ B⊆ A, then B is -

- ^*^*

2

1

τ α

τ

^g ^open.

τ

₁

τ

₂-^g

α

^*^*-open in X and

τ

₂ -int(A)⊆B⊆ A. Let F is -

- ^*

1

α

τ

^g closed and F ⊆B. Since F ⊆ Band B⊆ A, we have F ⊆ A. Since A is

τ

₁

τ

₂-^g

α

^*^*-open, we have F ⊆

τ

₂-int(A)^{. Since}

τ

₂ -int(A)⊆ B, we have

) ( int

2 - A

F ⊆

τ

⊆

τ

₂ -int(B). Hence B is

τ

₁

τ

₂-^g

α

^*^*-open in X.

τ

₁

τ

₂-^g

α

^*^*-closed in X, then

τ

₂ -cl(A)−A^is -

- ^*^*

2

1

τ α

τ

^g ^{open set.}

τ

₁

τ

₂-^g

α

^*^*-closed in X. Let F be

τ

₁-^g

α

^*-closed and A

A cl

F ⊆

τ

₂ - ( )− . Since A is

τ

₁

τ

₂-^g

α

^*^*-closed in X, we have

τ

₂ -cl(A)−A contains no nonempty

τ

₁-^g

α

^*-closed set. Since F ⊆

τ

₂ -cl(A)−A, we have

] ) ( - int[

- ₂

2 cl A A

F =

φ

⊆

τ τ

− . Therefore,

τ

₂ -cl(A)−A^is

τ

1

τ

2^-^g

α

^*^*^-^open.

τ

₁

τ

₂-^g

α

^*^*-open in a bitopological space (^X,

τ

₁,

τ

₂)^, then G = X whenever G is

τ

₁-^g

α

^*-^{open and}

τ

₂ -int(^A)∪^A^c ⊆^G^.

τ

₁

τ

₂-^g

α

^*^*-open in a bitopological space (^X,

τ

₁,

τ

₂)^and G is

τ

₁-^g

α

^*-^{open and}

τ

₂ -int(^A)∪^A^c ⊆^G^{. Then}G^c ⊆[

τ

₂ -int(A)∪A^c]^c =

c

c A

A cl( )−

2 -

τ

. Since G is

τ

₁-^g

α

^*-open, we have G^c is

τ

₁-^g

α

^*-^closed.

Since A is

τ

₁

τ

₂-^g

α

^*^*-open, we have A^c is

τ

₁

τ

₂-^g

α

^*^*-closed. Therefore,

c

c A

A cl( )−

2 -

τ

contains no nonempty

τ

₁-^g

α

^*-closed set in X {by theorem (3.17)}. Consequently, G^c =

φ

. Hence G = X.

Remark 4.16: The converse of the above theorem is not true in general as can seen from the following example.

Example 4.17: In example (3.2), if we take A = {c,d}, then

τ

₂ -int(A)∪A^c ⊆ X^, X is

τ

₁-^g

α

^*-open, but A is not

τ

₁

τ

₂-^g

α

^*^*-^open.

Lemma 4.18: The intersection of

τ

₁

τ

₂-^g

α

^*^*-open set and

τ

₂ -open set is always -

- ^*^*

2

1

τ α

τ

^g ^open.

τ

₁

τ

₂-^g

α

^*^*-open and B is

τ

₂ -open. Since B is

2 -

τ

open, we have B^c is

τ

₂ -closed. Then B^c is

τ

₁

τ

₂-^g

α

^*^*-closed {by theorem

(10)

(3.3) (i)}. Hence, B is

τ

₁

τ

₂-^g

α

^*^*-open. Hence A∩Bis

τ

₁

τ

₂-^g

α

^*^*-^{open {by} theorem (4.11)}.

Remark 4.19: The following diagram shows the relations among the different types of weakly open sets that were studied in this section:

References

[1] F.H. Khedr and H.S. Al-Saadi, On pair wise semi-generalized closed sets, JKAU Sci., 21(2) (2009), 269-295.

[2] J.C. Kelly, Bitopological spaces, Proc. London Math. Soc., 3(13) (1963), 71-89.

[3] K. Kannan,

τ

₁

τ

₂ -Semi star star generalized closed sets, International Journal of Pure and Applied Mathematics, 76(2) (2012), 277-294.

[4] K.C. Rao and K. Kannan, Semi star generalized closed sets and semi star generalized open sets in bitopological spaces, Varahimir Journal of Mathematics, 5(2) (2005), 473-485.

[5] K.C. Rao, K. Kannan and D. Narasimhan, Characterizations of

τ

₁

τ

₂ -s^*g closed sets, Acta Ciencia Indica, XXXIII M (3) (2007), 807-810.

[6] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19(2) (1970), 89-96.

[7] T. Fukutake, On generalized closed sets in bitopological spaces, Bull.

Fukuoka Univ. Ed., Part III, 35(1986), 19-28.

open

2 -

τ

₁

τ

₂ -^g

α

^*^*-open

open -

2 -

1

τ

g

τ

open

-

2 -

1

τ α

τ

^g

+

closed -

- ^*

1

α

τ

^g open -

2 -

1

τ α

τ

open -

- ^*

2

1

τ α

τ

^g

open -

2 -

1

τ α

g