Operation on $lak$-closed sets

https://doi.org/10.26637/MJM0S01/0003

Operation on

Ω

ˆ

-closed sets

S.M. Meenarani

*, K. Poorani

* and M. Anbuchelvi

Abstract

This work is based on operation in a topological space. An operation has been extended to the class ofΩ-openˆ sets. The new class ofγΩˆ-open sets has been introduced and two kinds of closures such as,γΩˆCl and(ΩˆCl)γ

are studied. Necessary basic properties have been derived. Moreover,Ω-regular operation onˆ ΩˆO(X,τ)has been introduced in which intersection of any twoγΩˆ-closed sets isγΩˆ-closed. Also three types of separation

axioms are defined and few results on them have been derived.

Keywords

γ_Ωˆ-open set,γ_ΩˆCl,(ΩˆCl)γ,Ω-regular operation,ˆ Ω-open operation,ˆ γ_Ωˆ-Tispaces(i=0,1,2).

AMS Subject Classification

54A05, 54A10, 54D10.

1,3_{Department of Mathematics, V.V.Vanniaperumal College for Women, Virudhunagar-626001, Tamil Nadu, India.}

*Corresponding author: ,2_{[email protected]}

Article History: Received21December2018; Accepted11February2019 c2019 MJM.

Contents

1 Introduction. . . 7

2 Preliminaries. . . 7

3 Operation on_ΩOˆ _{(X,τ). . . 8}

4 Basic properties of Closures. . . 9

5 Separation axioms. . . 11

6 Conclusion. . . 11

References. . . 11

1. Introduction

Generalized open sets play a vital role in research area of General Topology. Levine [7] introduced the concept of semi-open sets in topology. In 1987, Bhattacharyya and Lahiri [3] used semi-open sets to define the notion of semi-generalized closed sets. Kasahara [8] introduced the notion of anα

op-eration approaches on a classτof sets and studied the con-cept of α-continuous functions with α-closed graphs and

α-compact spaces. Jankovic [5] introduced the concept ofα -closure of a set inX viaα-operation and investigated further characterizations of a function withα-closed graph. Later Ogata [10] defined and studied the concept ofγ-open sets and applied it to investigate functions and operation-separation.Recently several researchers developed many con-cepts of operationγ in a spaceX. Krishnan, Gangster and

Balachandran [9] introduced and studied the concept of the operationγ on the class of all semiopen sets of(X,τ)and

defined the notion of semiγ-open sets and investigated some of their properties. An, Cuong and Maki [1] defined and inves-tigated an operationγon the class of all preopen sets of(X,τ)

and introduced the notion of pre-γ-open sets and developed some of their properties. Asaad [2] defined the notion of an operationγ on the class of generalized open sets in(X,τ)

and studied some of its applications. Recently, the concept of ˆΩ-closed set was introduced and investigated by Lellis Thivagar et al. [6]. In this paper, the concept of an operation

γhas been extended to the class of ˆΩ-open sets and it leads

to the introduction of the notion ofγ_Ωˆ-open sets on a

topo-logical spaces(X,τ). Furthermore, some basic properties of

γΩˆ-Closures have been derived. In last Section,γΩˆ-Tispaces

wherei∈ {0,1,2}are introduced and investigated using the operationγonτΩˆ.

2. Preliminaries

In this section, some definitions and results that are used in this work have been dealt. Throughout this paper,(X,τ)

orXrepresents a topological space on which no separation axioms are assumed, unless otherwise mentioned.

Definition 2.1. [7] A subset A of a topological space(X,τ)is called asemi-openset if A⊆cl(int(A)). SO(X)denotes the set of all semi-open sets in(X,τ).It’s complement is known as asemi-closedset on X.

whereδcl(A) ={x∈X:int(cl(U))∩A6=/0,U∈O(X,x)}. The complement of aδ-closed set in(X,τ)is called aδ-open set in(X,τ). The set of allδ-closed sets in X is denoted by δC(X). From[4], lemma 3,δcl(A) =T{F∈δC(X):A⊆F} and from corollary 4,δcl(A)is aδ-closed set for a subset A in a topological space(X,τ)

Definition 2.3. ([6], Definition 3.1) Let(X,τ)be a topolog-ical space. A is said to beΩˆ-closedset ifδcl(A)⊆U when A⊆U , where U is a semi-open subset of X . The complement ofΩˆ-closed set isΩˆ-open set.

Definition 2.4. ([6], Definition 5.1) Let A be a subset of a topological space(X,τ). ThenΩˆ-closureof A is defined to be the intersection of allΩˆ-closed sets containing A and it is denoted byΩˆcl(A). That isΩˆcl(A) =T

{F:A⊆F and F∈ΩˆC(X)}. Always A⊆Ωˆcl(A).

Remark 2.5. ([6], Remark 5.2) From the definition and The-orem 4.16, arbitrary intersection ofΩˆ-closed sets in a topo-logical space(X,τ)isΩˆ-closed set in(X,τ),Ωˆcl(A)is the smallestΩˆ-closed set containing A.

Theorem 2.6. ([6], Theorem 5.3) Let A be any subset of a topological space(X,τ). Then, A is aΩˆ-closed set in(X,τ) if and only if A=Ωˆcl(A).

Theorem 2.7. ([6], Theorem 5.11) In a topological space

(X,τ), for x∈X , x∈_Ωˆ_{cl(A)if and only if U∩A6=/0 for} everyΩˆ-open set U containing x.

Definition 2.8. [8] Let(X,τ)be a topological space. An operationγ on the topologyτis a mapping fromτ→P(X) such that V⊆Vγ_{for each V∈τ,where V}γ_{denotes the value} ofγ at V.It is denoted byγ:τ→P(X).

Notation 2.9.

i) U∈ΩˆO(X,x) denotes the set of all Ωˆ-open sets in

(X,τ)containing x.

ii) ΩˆO(X,τ)orΩˆO(X)orτ_Ωˆ denotes the set of allΩˆ-open sets in a topological space(X,τ).

iii) The closure (res.interior,complement )of A is denoted by cl(A)(res.int(A), Ac).

iv) SO(X)denotes the set of all semi-open sets in a topo-logical space(X,τ).

3. Operation on

ΩO(X

ˆ

,

τ

)

Definition 3.1. A functionγ : ΩˆO(X,τ)→ P(X) is called anoperation onΩˆO(X,τ), if U⊆γ(U)for every set U∈

ˆ

ΩO(X,τ).

Remark 3.2. For any operationγ: ˆΩO(X,τ)→P(X),γ(X) = X , andγ(/0) =/0.

Definition 3.3. A non-empty subset A of X is calledγ_Ωˆ-open

setif for each x∈A,there exists anΩˆ-open set U such that x∈U andγ(U)⊆A. The complement ofγ_Ωˆ-open set isγ_Ωˆ -closed set. Assume that the empty set /0is alwaysγ_Ωˆ-open for any operationγ onΩˆO(X,τ). τγ_Ωˆ denotes the set of all

γΩˆ-open sets on(X,τ).τγ_Ωˆ ={/0} S

{A/for each x∈A there exists anΩˆ-open set U3x∈U andγ(U)⊆A.}

Example 3.4.X={a,b,c,d},τ={/0,{a},{b},{a,b},{a,b,c}, {a,b,d},X},ΩˆO(X) ={/0,{a},{b},{a,b},X}. γis defined byγ(/0) =/0,γ({a}) ={a},γ({b}) ={a,b},γ({a,b}) ={a,b,d}, γ(X) =X . Here,γis an operation onτ_Ωˆ;{/0,{a},{a,b},X} areγ_Ωˆ-open sets.

Theorem 3.5. Arbitrary union ofγΩˆ-open sets is aγΩˆ-open

set in a topological space X .

Proof. Let{Aα}α∈Jbe any family ofγ_Ωˆ-open sets in a space (X,τ). LetA= S

α∈J

Aαandx∈Abe arbitrary. Thenx∈Aα

for someα ∈J. By the definition ofγ_Ωˆ-open, there exist U∈_Ωˆ_{O(X,x)such thatγ(U)⊆A}

α⊆

S

α∈J

Aα=A. Therefore,

Aisγ_Ωˆ-open.

Remark 3.6. Arbitrary intersection of γΩˆ-closed sets is a γ_Ωˆ-closed set in a topological space X .

Example 3.7. The intersection of any twoγΩˆ-open sets is not necessarily anγΩˆ-open set in(X,τ). Let X ={a,b,c}and P(X) =ΩˆO(X,τ). Define an operationγ: ˆΩO(X,τ)→P(X) as follows. For every U∈ΩˆO(X,τ)

γ(U) =

(

U for U6={a} {a,b} for U={a}

Here{a,b}and{a,c}areγ_Ωˆ-open sets but{a}is not aγ_Ωˆ -open set.

Proposition 3.8. EveryγΩˆ-open set isΩˆ-open in a space X.

Proof. LetAbe anyγΩˆ-open subset ofX.Letx∈Abe

arbi-trary. Then there exists ˆΩ-open setUxcontainingxsuch that Ux⊆γ(Ux)⊆A.ThenS{Ux/x∈A}=A.By ([6], Theorem

4.16),Ais ˆΩ-open subset ofX.

Remark 3.9. From Example 3.7, everyΩˆ-open is not neces-sarilyγΩˆ-open as{a} ∈ΩˆO(X)and{a}∈/τγ_Ωˆ. It turns out

to find a space in whichΩˆO(X) =τγ_ˆ Ω.

Definition 3.10. A space(X,τ)with an operationγonΩˆO(X,τ) is calledγ_Ωˆ-regular if for each x∈X and for each U ∈

ˆ

ΩO(X,x), there exists anΩˆ-open set V such that x∈V and

γ(V)⊆U.

Theorem 3.12. Let (X,τ)be a topological space and γ :

ˆ

ΩO(X,τ)→P(X)be an operation onΩˆO(X,τ).Then the following conditions are equivalent:

i) EveryΩˆ-open set isγ_Ωˆ-open set.

ii) X is anγ_Ωˆ-regular space.

iii) For every x∈X and for every U∈ΩˆO(X,x), there exists anγΩˆ-open set V of(X,τ)containing x such that V⊆U.

Proof. i)⇒ii)Letx∈Xbe arbitrary andU∈_Ωˆ_O(X,x).By hypothesis, there existsV∈_Ωˆ_{O(X,x)such thatγ(V)⊆U.} ii)⇒iii)Let xbe any point ofX andU∈_Ωˆ_{O(X,x). By} hypothesis, there exists ˆΩ-open setV such thatx∈V and

γ(V)⊆U.Again apply hypothesis to the setV.Then, there exists ˆΩ-open setV1∈ΩˆO(X,x)such thatγ(V1)⊆V.Then, V isγΩˆ-open set containingxsuch thatV⊆U.

iii)⇒i)LetUbe any ˆΩ-open set inXandx∈Ube arbitrary. By hypothesis, there existsγΩˆ-open setVxcontaining x such

thatVx⊆U. By Theorem 3.5,U= S x∈U

VxisγΩˆ-open.

Definition 3.13. Let (X,τ)be any topological space. an operationγonΩˆO(X,τ)is calledΩˆ-openif for each x∈X and for every U ∈ΩˆO(X,x),there exists anγΩˆ-open set V containing x such that V ⊆γ(U).

Example 3.14. X={a,b,c,d},τ={/0,{a},{b},{a,b},{a,b,c}, {a,b,d},X},_Ωˆ_{O(X) ={/0,{a},{b},{a,b},X}. γ is defined} byγ(/0) =/0,γ({a}) ={a},γ({b}) ={a,b},γ({a,b}) ={a,b,d}, γ(X) =X . Here,the operationγonτ_Ωˆ isΩˆ-open.

Definition 3.15. Let (X,τ)be any topological space. An operationγonΩˆO(X,τ)is calledΩˆ-regularif for each x∈X and for every pair of sets U1, U2∈ΩˆO(X,x), there exists a set V∈ΩˆO(X,x)such thatγ(V)⊆γ(U1)∩γ(U2).

Example 3.16. X={a,b,c,d},τ={/0,{a,b},X},ΩˆO(X) = {/0,{a},{b},{a,b},X}. γ is defined byγ(/0) = /0,γ({a}) = {a,b},γ({b}) ={a,b},γ({a,b}) ={a,b},γ(X) =X.Here, an operationγonτ_Ωˆ isΩˆ-regular.

Proposition 3.17. Intersection of any twoγ_Ωˆ-open sets is a

γΩˆ-open in aΩˆ-regular operation onΩˆO(X,τ).

Proof. LetUandVbe any twoγΩˆ-open sets inX. Letx∈U∩ Vbe any point. Then,x∈Uandx∈V. By the definition, there existsU1∈ΩˆO(X,x)such thatγ(U1)⊆U.Similarly for the

setV,there existsU2∈ΩˆO(X,x)such thatγ(U2)⊆V.Now γ(U1)∩γ(U2)⊆U∩V. By hypothesis, there exists ˆΩ-open

setW containingxsuch thatγ(W)⊆γ(U1)∩γ(U2)⊆U∩V.

HenceU∩Visγ_Ωˆ-open subset ofX.

Remark 3.18. By Proposition 3.17, the family of allγ_Ωˆ-open sets satisfy the axioms topology provided an operationγis a

ˆ

Ω-regular.

4. Basic properties of Closures

Definition 4.1. Letγ be an operation onΩˆO(X,τ).then for any subset A of X ,γΩˆ-closureis denoted byγΩˆCl(A), defined

asγΩˆCl(A) =

T

{F/A⊆F,X\F∈τγ_Ωˆ}.It follows that A⊆

γΩˆCl(A),γΩˆCl(/0) =/0andγΩˆCl(X) =X.Moreover,γΩˆCl(A)

isγΩˆ-closed as any intersection ofγΩˆ-closed sets isγΩˆ-closed.

Proposition 4.2. A isγΩˆ-closed if and only ifγΩˆCl(A) =A

for any subset A of X.

Proof. It follows straight forward from the definition.

Theorem 4.3. Let A be any subset of a topological space

(X,τ)andγbe an operation onΩˆO(X,τ). Then, x∈γ_ΩˆCl(A) iff everyγ_Ωˆ-open set containing x meets A.

Proof. Necessary: Let Abe any subset of(X,τ). Assume that there existsγ_Ωˆ-open setU containingxsuch thatU∩ A=/0. ThenUc is anγ_Ωˆ-closed set such thatA⊆Uc. By

the definition ofγ_ΩˆCl(A),A⊆γ_ΩˆCl(A)⊆Uc. Nowx∈/Uc

impliesx∈/γ_ΩˆCl(A).

Sufficiency: Assume thatx∈/γ_ΩˆCl(A). Then, there exists a γ_Ωˆ-closed setF such thatA⊆F andx∈/F. Now,Fcis an γΩˆ-open set containingxsuch thatA∩F

c_=/0.

Proposition 4.4. If A and B are any two subsets of the space X,then the following statements hold.

i) A⊆B,thenγΩˆCl(A)⊆γΩˆCl(B)

ii) γ_ΩˆCl(A∩B)⊆γ_ΩˆCl(A)∩γ_ΩˆCl(B)

iii) γΩˆCl(A)∪γΩˆCl(B)⊆γΩˆCl(A∪B)

iv) γ_ΩˆCl(γ_ΩˆCl(A)) =γ_ΩˆCl(A).

Proof. i) IfU is anyγ_Ωˆ-open subset ofX containingx,

then by Theorem 4.3 and hypothesisB∩U6=/0.Again by Theorem 4.3,x∈γ_Ωˆ Cl(B)

ii) Suppose thatx∈/(γΩˆCl(A))∩(γΩˆCl(B)). Then, there

are two possibilities such as, eitherx∈/γΩˆCl(A)orx∈/ γΩˆCl(B). Ifx∈/γΩˆCl(A), then by Theorem 4.3,there

existsγ_Ωˆ-open setUofX containingxsuch thatU∩ A=/0. It follows thatUdoes not meetA∩B.Again by Theorem 4.3,x∈/γ_ΩˆCl(A∩B). Similarly for the case x∈/γ_ΩˆCl(B).

iii) Proof is analogous to that ofii).

iv) If follows from proposition 4.2.

Definition 4.5. For a subset A of a topological space X,Ωˆ -closure with respect to an operationγis denoted by(ΩˆCl)γ(A) and defined as(ΩˆCl)γ(A) ={x∈X/γ(U)∩A6=/0for each U∈_Ωˆ_{O(X,x)}.Always,(Ω}ˆ_Cl)

γ(/0) =/0and(ΩˆCl)γ(X) = X.

Proof. LetAbe any subset ofXandF= (ΩˆCl)γ(A).Always, F ⊆(ΩˆCl)γ(F).If x∈/ F, then there existsU∈ΩˆO(X,x)

such thatγ(U)∩F=/0.Then,U∩F=/0.By Theorem 2.7, x∈/_Ωˆ_{Cl(F).Therefore,F=Ω}ˆ_{Cl(F).By Theorem 2.6,Fis}

ˆ

Ω-closed.

Lemma 4.7. In a topological space(X,τ)with an operation γ onΩˆO(X,τ), the following statements hold for any two subsets A and B of X.

i) A⊆(ΩˆCl)γ(A)⊆γ_ΩˆCl(A).

ii) A isγ_Ωˆ-closed if and only if(ΩˆCl)γ(A) = A.

iii) If A⊆B,(ΩˆCl)γ(A)⊆(ΩˆCl)γ(B).

iv) (ΩˆCl)γ(A∩B)⊆(ΩˆCl)γ(A)∩(ΩˆCl)γ(B).

v) (ΩˆCl)γ(A)∪(ΩˆCl)γ(B)⊆(ΩˆCl)γ(A∪B).

Proof. i) Letx∈Abe arbitrary. IfU∈ΩˆO(X,x), then by [Theorem 5.11]U∩A6=/0. Thusγ(U)∩A6=/0. By the definition,x∈(ΩˆCl)γ(A). For another part, assume that x∈/γΩˆCl(A).Then there existsγΩˆ-closed setFsuch that

A⊆Fandx∈/F.Since everyγ_Ωˆ-closed set is ˆΩ-closed, F is ˆΩ-closed subset ofX.Then,Fc∈_Ωˆ_O(X,x)such thatFc_{∩A=/0.Thus,x∈/(Ω}ˆ_Cl)

γ(A).

ii) Assume thatAisγ_Ωˆ-closed. Ifx∈/A,then x∈Ac= U(say). Now U is γ_Ωˆ-open subset of X containing x.By the definition, there existsV ∈_Ωˆ_O(X,x)such thatγ(V)⊆U=Ac.Thusγ(V)∩A=/0 saysx∈/(ΩˆCl)γ(A)

iii) Letx∈(ΩˆCl)γ(A)and LetUbe any ˆΩ-open set

con-taining x. By hypothesis, γ(U)∩A6=/0 and hence γ(U)∩B=/0. Thusx∈(ΩˆCl)γ(B).

Therefore,(ΩˆCl)γ(A)⊆(ΩˆCl)γ(B).

iv) Suppose thatx∈/ (ΩˆCl)γ(A)

∩ (ΩˆCl)γ(B)

. Then, there are two possibilities such as, eitherx∈/(ΩˆCl)_γ(A)

orx∈/(ΩˆCl)γ(B). Ifx∈/(ΩˆCl)γ(A), then there exists

ˆ

Ω-open setUofXcontainingxsuch thatγ(U)∩A=

/0. It follows thatγ(U)does not meetA∩B.By the

definition, x∈/(ΩˆCl)γ(A∩B). Similarly for the case x∈/(ΩˆCl)γ(B).

v) Proof is analogous to that ofiv).

Theorem 4.8. Ifγ is anΩˆ-regular operation onΩˆO(X,τ), then for any two subsets A,B of X the following results hold.

i) γ_ΩˆCl(A)∪γ_ΩˆCl(B) =γ_ΩˆCl(A∪B).

ii) (ΩˆCl)γ(A)∪(ΩˆCl)γ(B)= (ΩˆCl)γ(A∪B)

Proof. i) Alwaysγ_ΩˆCl(A)∪γ_ΩˆCl(B)⊆γ_ΩˆCl(A∪B).For

other inclusion, ifx∈/γ_ΩˆCl(A)∪γ_ΩˆCl(B), then there

exist γ_Ωˆ-open sets U andV containing x such that A∩U=/0 andB∩V=/0.By proposition 3.17,U∩V isγ_Ωˆ-open set in X such that(U∩V)∩(A∪B) = /0.

Therefore,x∈/γ_ΩˆCl(A∪B).

ii) Always(ΩˆCl)γ(A)∪(ΩˆCl)γ(B)⊆(ΩˆCl)γ(A∪B). For

other inclusion, ifx∈/ (ΩˆCl)γ(A) ∪(ΩˆCl)γ(B), then x∈/ (ΩˆCl)γ(A) andx∈/ (ΩˆCl)γ(B). By the definition, γ(U1)∩A=/0=γ(U2)∩Bfor someU1,U2∈ΩˆO(X,x).

By the definition of ˆΩ-regular operation, there exists V ∈_Ωˆ_{O(X,x)such thatγ(V)⊆γ(U1)∩γ(U2)⊆(A∪} B). Then,(A∪B)∩γ(V) =/0 impliesx∈/(ΩˆCl)γ(A∪ B).

Theorem 4.9. Let A be any subset of a topological space

(X,τ). Ifγ is anΩˆ-open operation onΩˆO(X,τ), then the following statements are true.

i) (ΩˆCl)γ(A)=γ_ΩˆCl(A)

ii) (ΩˆCl)γ (ΩˆCl)γ(A)

= (ΩˆCl)γ(A)

iii) (ΩˆCl)γ(A)isγ_Ωˆ-closed set in X .

Proof. i) Always(ΩˆCl)γ(A)⊆γ_ΩˆCl(A).Letx∈/(ΩˆCl)γ(A).

Then there exists anU∈_Ωˆ_{O(X,x)such thatγ(U)∩A=} /0.By the choice ofγ,there exists anγ_Ωˆ-open setV

con-taining xsuch thatV ⊆γ(U).NowV∩A⊆γ(U)∩ A=/0.By Theorem 4.3,x∈/γΩˆCl(A).So,γΩˆCl(A)⊆

(ΩˆCl)γ(A). Hence(ΩˆCl)γ(A)=γΩˆCl(A).

ii) Byi)and Proposition 4.4.(iv),(ΩˆCl)γ((ΩˆCl)γ(A))= (ΩˆCl)γ(A).

iii) It follows from i)and by the result thatγ_ΩˆCl(A) is γ_Ωˆ-closed.

Theorem 4.10. Let A be any subset of a topological space (X,τ)andγbe an operation onΩˆO(X,τ). Then the following statements are equivalent.

i) A isγ_Ωˆ-open set.

ii) (ΩˆCl)γ(X\A) =X\A.

iii) γΩˆCl(X\A) =X\A.

iv) X\A isγ_Ωˆ-closed set.

Proof. It follows from the definition.

Proof. Assume thatx∈γ_ΩˆCl(A)∩U for everyγ_Ωˆ-open set Uand every subsetAofX. LetV be anyγ_Ωˆ-open subset of X containingx. By Proposition 3.17,U∩V isγ_Ωˆ-open set

containingx. Since,x∈γ_ΩˆCl(A),A∩(U∩V)6=/0.That is, (A∩U)∩V6=/0.By Theorem 4.3,x∈γ_ΩˆCl(A∩U).

5. Separation axioms

Definition 5.1. A topological space(X,τ)with an operation γonΩˆO(X,τ)is calledγΩˆ-T0if for any two points x,y in X such that x6=y there exists an U∈ΩˆO(X,τ), such that x∈U and y∈/γ(U)or y∈U and x∈/γ(U).

Definition 5.2. A topological space(X,τ)with an operation γonΩˆO(X,τ)is calledγ_Ωˆ-T1if for any two points x,y in X such that x6=y,there exist twoΩˆ-open sets U and V contain-ing x and y respectively such that y∈/γ(U)and x∈/γ(V).

Definition 5.3. A topological space(X,τ)with an operation γonΩˆO(X,τ)is calledγ_Ωˆ-T2if for any two points x,y in X such that x6=y there exist twoΩˆ-open sets U and V containing x and y respectively such thatγ(U)∩γ(V) =/0.

Theorem 5.4. Letγ be anΩˆ-open operation onΩˆO(X,τ). Then(X,τ)is anγΩˆ-T0space iff(ΩˆCl)γ({x})6= (ΩˆCl)γ({y}) for every pair x,y of X with x6=y.

Proof. Letx,ybe any two distinct points of anγ_Ωˆ-T0space (X,τ).Then,there exists aγΩˆ-open setUsuch thatx∈Uand y∈/γ(U).Sinceγis an ˆΩ-open, there exists aγΩˆ-open setV

such thatx∈V andV ⊆γ(U). Therefore,y∈X\γ(U)⊆ X\V. Now X\V is an γΩˆ-closed set in (X,τ) such that (ΩˆCl)γ({y})⊆X\V. Thus(ΩˆCl)γ({x})6= (ΩˆCl)γ({y}).

Conversely,ifx,yare any two distinct points ofXthen,

(ΩˆCl)γ({x})6= (ΩˆCl)γ({y}). Choosez∈X such that z∈ (ΩˆCl)γ({x}), andz∈/(ΩˆCl)γ({y}). Ifx∈(ΩˆCl)γ({y}), then (ΩˆCl)γ({x})⊆(ΩˆCl)γ({y}). That is,z∈(ΩˆCl)γ({y}), which

is a contradiction. So, x∈/ (ΩˆCl)γ({y}). Then, there

ex-ists an ˆΩ-open setUcontainingxsuch thatγ(U)∩ {y}=/0.

Now,x∈Uandy∈/γ(U)that satisfy the condition ofγ_Ωˆ-T0

space.

Theorem 5.5. The space(X,τ)isγΩˆ-T1if and only if for every point x∈X,{x}is anγΩˆ-closed set.

Proof. Let(X,τ)be aγ_Ωˆ-T1space andxbe any point ofX.

Then for any pointy∈X such thatx6=y, there exists an ˆ Ω-open setVysuch thaty∈Vybutx∈/γ(Vy). Thus,y∈γ(Vy)⊆ X\ {x}.This implies thatX\ {x}=S

{γ(Vy):y∈X\ {x}}.

NowX\ {x}is γΩˆ-open set in(X,τ)and hence{x} isγΩˆ

-closed set in(X,τ).

Conversely, letx,y∈Xsuch thatx6=y.By hypothesis, we get X\ {y}andX\ {x}areγ_Ωˆ-open sets such thatx∈X\ {y}=U

(say) andy∈X\ {x}=V (say). Therefore, there exist ˆ Ω-open setsUandV such thatx∈U,y∈V,γ(U)⊆X\ {y}and γ(V)⊆X\ {x}.So,y∈/γ(U)andx∈/γ(V).This implies that

(X,τ)isγ_Ωˆ-T1.

Theorem 5.6. For any topological space(X,τ)and any op-erationγonτ_Ωˆ, the following properties hold.

i) Everyγ_Ωˆ-T2space isγ_Ωˆ-T1.

ii) Everyγ_Ωˆ-T1space isγ_Ωˆ-T0.

Proof. It follows from definitions.

6. Conclusion

In this paper, an attempt has been made to define operation on the class of ˆΩ-open sets. with the help of this operation,the new class ofγ_Ωˆ-open sets has been introduced and two kinds

of closures such as,γΩˆCl and(ΩˆCl)γ studied. Their basic

properties have been derived. Moreover, it is shown by an example that intersection of any twoγΩˆ-closed sets is not

necessarily aγΩˆ-closed but that holds in a ˆΩ-regular operation

on ˆΩO(X,τ)has been derived.

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