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Volume 2007, Article ID 16292,11pages doi:10.1155/2007/16292

Research Article

Regular Generalized

ω

-Closed Sets

Ahmad Al-Omari and Mohd Salmi Md Noorani

Received 12 September 2006; Revised 1 December 2006; Accepted 16 January 2007

Recommended by Lokenath Debnath

In 1982 and 1970, Hdeib and Levine introduced the notions ofω-closed set and gener-alized closed set, respectively. The aim of this paper is to provide a relatively new notion of generalized closed set, namely, regular generalizedω-closed, regular generalizedω -continuous,a-ω-continuous, and regular generalizedω-irresolute maps and to study its fundamental properties.

Copyright © 2007 A. Al-Omari and M. S. M. Noorani. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-erly cited.

1. Introduction

All through this paper (X,τ) and (Y,σ) stand for topological spaces with no separation axioms assumed, unless otherwise stated. LetA⊆X, the closure ofAand the interior ofA

will be denoted by Cl(A) and Int(A), respectively.Ais regular open ifA=Int(Cl(A)) and

Ais regular closed if its complement is regular open; equivalentlyAis regular closed ifA= Cl(Int(A)), see [1]. Let (X,τ) be a space and letAbe a subset ofX. A pointx∈Xis called a condensation point ofAif for eachU∈τ withx∈U, the setU∩Ais uncountable.

Ais calledω-closed [2] if it contains all its condensation points. The complement of an

ω-closed set is called ω-open. It is well known that a subsetW of a space (X,τ) isω -open if and only if for eachx∈W, there existsU∈τ such that x∈U andU−W is countable. The family of allω-open subsets of a space (X,τ), denoted byτωorωO(X), forms a topology onX finer thanτ. Theω-closure andω-interior, that can be defined in a manner similar to Cl(A) and Int(A), respectively, will be denoted by Clω(A) and

Intω(A), respectively. Several characterizations ofω-closed subsets were provided in [3,

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set (brieflyg-closed) if Cl(A)⊆U wheneverU∈τandA⊆U. Generalized semiclosed [6] (resp.,α-generalized closed [7],θ-generalized closed [8], generalized semi-preclosed [9],δ-generalized closed [10],ω-generalized closed [3,11]) sets are defined by replacing the closure operator in Levine’s original definition by the semiclosure (resp.,α-closure,

θ-closure, semi-preclosure,δ-closure,ω-closure) operator.

2. Regular generalizedω-closed sets

A subsetAof (X,τ) is called regular generalized closed (simply,rg-closed) (see [12]) if Cl(A)⊂UwheneverA⊂UandUis regular open. Analogously, we begin this section by introducing the class of regular generalizedω-closed sets.

Definition 2.1. A subsetAof (X,τ) is called regular generalizedω-closed (simply,rgω -closed) if Clω(A)⊂UwheneverA⊂UandUis regular open. A subsetBof (X,τ) is called

regular generalizedω-open (simply,rgω-open) if the complement ofBisrgω-closed sets.

We have the following relation forrgω-closed with the other known sets:

ω-c-closed

closed g-closed rg-closed

ω-closed -closed rgω-closed

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Example 2.2. LetRbe the set of all real numbers, letQbe the set of all rational numbers, with the topologyτ= {R,φ,RQ}. ThenA=RQis not-closed, sinceAis open, thusω-open andA⊆A, Clω(A)A(becauseAis notω-closed). Also the only regular

open set containingAisX. ThusAisrgω-closed.

Example 2.3. LetX= {a,b,c,d}, with the topologyτ= {φ,X,{a},{b},{a,b},{a,b,c}}. Then the set{a}is notrg-closed, see [13]. But{a}isrgω-closed set, sinceXis finite and

τωis discrete topology.

It is clear that if (X,τ) is a countable space, thenrgω(X,τ)=ᏼ(X), wherergω(X,τ) is the set of allrgω-closed subsets ofXandᏼ(X) is the power set ofX.

Since every closed set isω-closed we have the following.

Lemma 2.4. For every subsetAof (X,τ), Clω(A)Cl(A).

The proof of the following result follows from the fact that every regular open set is an open set together withLemma 2.4.

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Theorem 2.6. LetAbe anrgω-closed subset of (X,τ). Then Clω(A)−Adoes not contain

any nonempty regular closed set.

Proof. LetFbe a regular closed subset of (X,τ) such thatF⊆Clω(A)−A. ThenF⊆X− Aand henceA⊆X−F. SinceAisrgω-closed set andX−Fis a regular open subset of (X,τ), Clω(A)⊆X−Fand soF⊆X−Clω(A). ThereforeF⊆Clω(A)(X−Clω(A))=

φ.

Theorem 2.7. A subsetAof (X,τ) isrgω-open if and only ifF⊆Intω(A) wheneverFis a

regular closed subset such thatF⊆A.

Proof. LetAbe anrgω-open subset ofX and letFbe a regular closed subset ofXsuch thatF⊆A. ThenX−Ais anrgω-closed set andX−A⊆X−F. SinceX−A isrgω -closed,X−Intω(A)=Clω(X−A)⊆X−F. ThusF⊆Intω(A). Conversely, ifF⊆Intω(A)

whereFis a regular closed subset of (X,τ) such thatF⊆A, then for any regular open subsetUsuch thatX−A⊆U, we haveX−U⊆Aand thusX−U⊆Intω(A). That is, X−Intω(A)=Clω(X−A)⊆U. ThereforeX-Aisrgω-closed.

Lemma 2.8 [14]. For every openUin a topological spaceXand everyA⊆X, Cl(U∩A)= Cl(U∩Cl(A)).

Recall that two nonempty setsAandBofXare said to be separated if Cl(A)∩B=φ= A∩Cl(B).

Theorem 2.9. IfAandBare open,rgω-open, and separated sets, thenA∪Bisrgω-open.

Proof. LetFbe a regular closed subset ofA∪B. ThenF∩Cl(A)⊆A, sinceAis open and byLemma 2.8we haveF∩Cl(A) is regular closed hence by Theorem 2.7F∩Cl(A) Intω(A). Similarly,F∩Cl(B)Intω(B). Then we haveF⊆Intω(A∪B) and henceA∪B

isrgω-open.

The following example shows that the union ofrgω-open sets need not bergω-open.

Example 2.10. LetXbe an uncountable set and letA,B,C,Dbe subsets ofX, such that each of them is uncountable set and the family{A,B,C,D}is a partition ofX. We defined the topologyτ= {φ,X,{A},{B},{A,B},{A,B,C}}. Choosex,y /∈Aandx=y. ThenH= A∪ {x}andG=A∪ {y}arergω-closed, since only regular open set containingH,Gis

X. ButH∩G= {A}and{A}is regular open inXand Clω(A)A, since{A}is notω

-closed. ThusH∩Gis notrgω-closed. Therefore the union ofrgω-open sets need not be

rgω-open.

The proof of the following result is straightforward sinceτω is a topology onX and

thus omitted.

Theorem 2.11. IfAandBarergω-closed sets, thenA∪Bisrgω-closed.

Theorem 2.12. LetAbe argω-closed subset of (X,τ). IfB⊆Xsuch thatA⊆B⊆Clω(A),

thenBis alsorgω-closed. LetBbe a subset of (X,τ) and letAbe anrgω-open subset such that Intω(A)⊆B⊆A. ThenBis alsorgω-open.

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Theorem 2.13. IfAbe anrgω-closed subset of (X,τ), then Clω(A)−Aisrgω-open set.

Proof. LetAbe anrgω-closed subset of (X,τ) and letF be a regular closed subset such thatF⊆Clω(A)−A. ByTheorem 2.6,F=φand thusF⊆Intω(Clω(A)−A). ByTheorem

2.7, Clω(A)−Aisrgω-open set.

We first recall the following lemmas to obtain further results forrgω-closed sets.

Lemma 2.14 [3]. If Y is an open subspace of a space X and A is a subset of Y, then Clω|Y(A)=Clω(A)(Y).

Lemma 2.15. IfAis a regular open andrgω-closed subset of a spaceX, thenAisω-closed inX.

The proof is obvious.

Theorem 2.16. LetY be an open subspace of a spaceXandA⊆Y. IfAisrgω-closed inX, thenAisrgω-closed inY.

Proof. LetUbe a regular open set ofY such thatA⊆U. ThenU=V∩Yfor some regu-lar open setVofX. SinceAisrgω-closed inX, we have Clω(A)⊆Uand byLemma 2.14,

Clω|Y(A)=Clω(A)(Y)⊆V∩Y=U. HenceAisrgω-closed inX.

Corollary 2.17. IfAis anrgω-closed regular open set andBis anω-closed set of a space

X, thenA∩Bisrgω-closed.

Theorem 2.18. LetAbe anrgω-closed set. ThenA=Clω(Intω(A)) if and only if Clω(Intω

(A))−Ais regular closed.

Proof. If A=Clω(Intω(A)), then Clω(Intω(A))−A=φand hence Clω(Intω(A))−A is

regular closed. Conversely, let Clω(Intω(A))−Abe regular closed, since Clω(A)−A

con-tains the regular closed set Clω(Intω(A))−A. ByTheorem 2.6Clω(Intω(A))−A=φand

henceA=Clω(Intω(A)).

Lemma 2.19 [3]. Let (A,τA) be an antilocally countable subspace of a space (X,τ). Then Cl(A)=Clω(A).

We call (X,τ) an antilocally countable space if each nonempty open set is an uncount-able set.

Corollary 2.20. In an antilocally countable subspace of a space (X,τ), the concepts of

rgω-closed set andrg-closed set coincide.

Lemma 2.21 [3]. Let (X,τ) and (Y,σ) be two topological spaces. Then (τ×σ)ω⊆τω×σω.

Theorem 2.22. IfA×Bisrgω-open subset of (X×Y,τ×σ), thenAisrgω-open subset in (X,τ) andBisrgω-open subset in (Y,σ).

Proof. LetFA be a regular closed subset of (X,τ) and letFB be a regular closed subset

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FA×FB⊆Intω(A×B)Intω(A)×Intω(B) byLemma 2.21. ThereforeFA⊆Intω,FB⊆

Intω(B). HenceA,Barergω-open.

The converse of the above need not be true in general.

Example 2.23. LetX=Y =Rwith the usual topologyτ. LetA= {{RQ} ∪[2, 5]} andB=(1, 7). ThenAandBarergω-open (ω-open) subsets of (R,τ), whileA×B is notrgω-open in (R×R,τ×τ), since the setF=[2, 3]×[3, 5] is regular closed set con-tained inA×BandFIntω(A×B). The point (

2, 4)∈F and (2, 4)∈/ Intω(A×B),

because if (2, 4)Intω(A×B), then there exist open setU containing

2 and open setV containing 4 such that (U×V)(A×B) is countable but (U×V)(A×B) is uncountable for any open setUcontaining2 and open setV containing 4.

3. Regular generalizedω-T1/2space

Recall that a space (X,τ) is calledT1/2 [5] if everyg-closed set is closed or equivalently

if every singleton is open or closed, Dunham [15]. We introduce the following relatively new definition.

Definition 3.1. A space (X,τ) is a regular generalizedω-T1/2(simply,rgω-T1/2) if every rgω-closed set in (X,τ) isω-closed.

Theorem 3.2. For a space (X,τ), the following are equivalent. (1)Xis argω-T1/2.

(2) Every singleton is either regular closed orω-open.

Proof. (1)⇒(2) Suppose{x}is not a regular closed subset for somex∈X. ThenX− {x} is not regular open and henceXis the only regular open set containingX− {x}. Therefore

X− {x}isrgω-closed. Since (X,τ) isrgω-T1/2space,X− {x}isω-closed and thus{x}is ω-open.

(2)(1) LetAbe anrgω-closed subset of (X,τ) andx∈Clω(A). We show thatx∈A.

If{x}is regular closed andx /∈A, thenx∈(Clω(A)−A). Thus Clω(A)−Acontains a

nonempty regular closed set{x}, a contradiction toTheorem 2.6. Sox∈A. If{x}isω -open, sincex∈Clω(A), then for everyω-open setUcontainingx, we haveU∩A=φ. But {x}isω-open then{x} ∩A=φ. Hencex∈A. So in both cases we havex∈A. Therefore

Aisω-closed.

Theorem 3.3. Let (X,τ) be an antilocally countable space. Then (X,τ) is aT1-space if every rgω-closed set isω-closed.

Proof. Letx∈X, and suppose that {x}is not closed. ThenA=X− {x}is not open, and thusAisrgω-closed (the only regular open set containingA isX). Therefore, by assumption,Aisω-closed, and thus{x}isω-open. So there existsU∈τsuch thatx∈U

andU− {x}is countable. It follows thatUis a nonempty countable open subset ofx∈X,

a contradiction.

Definition 3.4. A map f :X→Y is said to be

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(ii) approximately continuous [16] (a-continuous) provided that Cl(A) f−1(V)

wheneverVis an open subset ofY,Ais ag-closed subset ofX, andA⊆f−1(V).

Definition 3.5. A mapf :X→Yis said to be approximatelyω-closed (simply,a-ω-closed) provided thatf(F)Intω(A) wheneverFis a regular closed subset ofX,Ais anrgω-open

ofY, and f(F)⊆A.

Definition 3.6. A map f :X→Y is said to be approximatelyω-continuous (simply,a-ω -continuous) provided that Clω(A)⊆f−1(V) wheneverVis a regular open subset ofY,A

is anrgω-closed subset ofX, andA⊆f−1(V).

The notions ofa-closed (resp.;a-continuous) anda-ω-closed (resp.;a-ω-continuous) are independent.

Example 3.7. LetX= {a,b,c,d}with the topologyτ= {φ,X,{a},{b},{a,b},{a,b,c}}. Let

f : (X,τ)(X,τ) be a function defined byf(a)=a, f(b)=d, f(c)=b, f(d)=c. Then

f isa-ω-closed, sinceXis finite and thusτωis a discrete topology, and f is nota-closed

function. Because the set A= {b,c} isg-open and F= {c,d}is closed, f(F)⊆A, but

f(F)Int(A).

Example 3.8. LetX=Rwith the topologyτ= {φ,X,RQ}. Let f : (X,τ)(X,τ) be a function defined by f(x)=0, for allx∈X. Thenf isa-closed, since for any closed setF

ofX, the onlyg-open set containingf(F) isX. Andf is nota-ω-closed function. Because the setA=Qisrgω-open andF=Ris regular closed, f(F)⊆A, but f(F)Intω(A).

Theorem 3.9. A spaceXisrgω-T1/2-space if and only if every spaceY and every function f :X→Yarea-ω-continuous.

Proof. LetV be a regular open subset of Y and A is anrgω-closed subset of X such thatA⊆ f−1(V), sinceX isrgω-T

1/2-space thenAisω-closed thusA=Clω(A), hence

Clω(A) f−1(V) and f isa-ω-continuous. LetAbe a nonemptyrgω-closed subset ofX

and letYbe the setXwith the topology{Y,A,φ}. Let f :X→Ybe the identity mapping. By assumption f isa-ω-continuous. SinceAisrgω-closed subset inX and open inY

such thatA⊆f−1(A), it follows that Cl

ω(A)⊆f−1(A)=A. HenceAisω-closed inXand

thereforeXisrgω-T1/2-space.

Lemma 3.10. If the regular open and regular closed sets ofXcoincide, then all subsets ofX

arergω-closed (and hence all arergω-open).

Proof. LetAbe any subset ofXsuch thatA⊆U andUis regular open, then Clω(A)

Clω(U)Cl(U)=U. ThereforeAisrgω-closed.

Theorem 3.11. If the regular open and regular closed sets ofY coincide, then a function

f :X→Yisa-ω-closed if and only if f(F) isω-open for every regular closed subsetFofX.

Proof. Assume f isa-ω-closed byLemma 3.10all subsets of Y arergω-closed. So for any regular closed subsetFofX, f(F) isrgω-closed inY. Since f isa-ω-closed, f(F) Intω(f(F)), therefore f(F)=Intω(f(F)) thus f(F) is ω-open. Conversely if f(F)⊆A

whereFis regular closed andAisrgω-open, then f(F)=Intω(f(F))Intω(A) hence f

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The proof of the following result for a-ω-continuous function is analogous and is omitted.

Theorem 3.12. If the regular open and regular closed sets ofX coincide, then a function

f :X→Y isa-ω-continuous if and only if f−1(V) isω-closed for every regular open subset VofY.

4.rgω-continuity

In this section, we will introduce some new classes of maps and study some of their char-acterizations. In [11,3] a map f :X→Y is calledω-irresolute (resp.,R-map [17]) if the inverse image of everyω-closed (resp., regular closed) subset ofY isω-closed (resp., reg-ular closed) inX. In [3], a mapf :X→Y is called-closed if the image of every closed subset ofXis-closed inY. Relatively new definitions are given next.

Definition 4.1. A mapf :X→Y is calledrgω-closed (resp., ro-preserving, pre-ω-closed) if f(V) isrgω-closed (resp., regular open,ω-closed) inY for every closed (resp., regular open,ω-closed) subsetVofX.

Example 4.2. LetX= {a,b,c,d}with the topologyτ= {φ,X,{a},{b},{a,b},{a,b,c}}. Let

f : (X,τ)(X,τ) be a function defined byf(a)=a, f(b)=b, f(c)=d, f(d)=c. Then

f is ro-preserving, since the family of all regular open sets of Xis,X,{a},{b}}. But if we definedg: (X,τ)(X,τ) asg(a)=c,g(b)=d,g(c)=a,g(d)=b, theng is not ro-preserving function.

Definition 4.3. A map f :X→Y is calledrgω-continuous (resp.,rgω-irresolute) if the inverse image of everyω-closed (resp.,rgω-closed) subsetVofY isrgω-closed subset of

X.

From the definition stated above we obtain the following diagram of implications:

continuous

ω-continuous -continuous rgω-continuous

ω-irresolute -irresolute rgω-irresolute

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Theorem 4.4. Let f :X→Y be a surjective,rgω-irresolute, and pre-ω-closed map ifXis

rgω-T1/2-space, thenY is also anrgω-T1/2-space.

Proof. LetAbergω-closed subset ofY. Since f is anrgω-irresolute map, then f−1(A) is

anrgω-closed subset ofX. SinceXisrgω-T1/2-space, then f−1(A) is anω-closed subset

of X. Since f is a pre-ω-closed map, then f(f−1(A))=Ais anω-closed subset ofY.

ThereforeY is alsorgω-T1/2-space.

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Theorem 4.5. A map f :X→Yisgω-closed if and only if for eachA⊆Yand each open set

Ucontaining f−1(A), there exists a-open subsetVofY such thatAV and f1(V) U.

Proof. LetF be a-closed map,A⊆Y, and letU be an open set containing f−1(A).

Then V =Y f(X−U) is -open subset of Y containing A and f−1(V)U.

Conversely let F be closed subset of X and let H be an open subset of Y such that

f(F)⊆H. Then f−1(Y f(F))XF and XF is open by hypothesis, there

ex-ists a -open subset V of Y such thatY− f(F)⊆V and f−1(V)XF.

There-fore,F⊆X−f−1(V) and hence f(F)YV. SinceYHYf(F), f1(YH) f−1(Y f(F)) f1(V)XF, by taking complement, we get FXf1(V) X− f−1(Yf(F))Xf1(YH). Therefore f(F)YV H. Since YV is -closed set and Clω(f(F))Clω(Y−V)⊆H, hence f(F) is-closed. Thus f is a

-closed map.

Since everyω-closed set isrgω-closed, we have the following.

Theorem 4.6. Everyrgω-irresolute map isrgω-continuous map.

Definition 4.7. A subsetA⊆Xis said to beω-c-closed provided that there is a proper subset Bfor which A=Clω(B). A map f :X→Y is said to be-c-closed if f(A) is -closed inYfor everyω-c-closed subsetA⊆X.

Since closed sets are obviouslyω-c-closed,-closed maps are-c-closed. In a sim-ilar manner, we say a map f :X→Y isrgω-c-closed if f(A) isrgω-closed inY for every

ω-c-closed subsetA⊆X.

Theorem 4.8. Let f :X→Y be anR-map andrgω-c-closed. Then f(A) isrgω-closed in

Y for everyrgω-closed subsetAofX.

Proof. Let A be an rgω-closed subset of X and let U be a regular open subset of Y

such that f(A)⊆U. Since f is an R-map, f−1(U) is a regular open subset ofX and A⊆ f−1(U). AsAis anrgω-closed subset, Cl

ω(A) f−1(U). Hence f(Clω(A))(U).

Because Clω(A) isω-c-closed andFisrgω-c-closed map,f(Clω(A)) isrgω-closed.

There-fore, Clω(f(A))Clω(f(Clω(A))) f(Clω(A))⊆U. Hence f(A) is anrgω-closed subset

ofY.

Theorem 4.9. Let f :X→Y be ro-preserving andω-irresolute function, ifBisrgω-closed inY, thenf−1(B) isrgω-closed inX.

Proof. LetGbe a regular open subset ofX such that f−1(B)G. ThenB f(G) and f(G) is regular open. SinceBisrgω-closed, then Clω(A) f(G) and f−1(Clω(B))⊆G.

Since f isω-irresolute then f−1(Cl

ω(B)) isω-closed and Clω(f−1(Clω(B)))= f−1(Clω

(B)), therefore Clω(f−1((B)))Clω(f−1(Clω(B)))⊆Gthus f−1(B) isrgω-closed inX.

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Proof. Assume thatAis anrgω-closed inY and f−1(A)U, whereUis a regular open

subset ofX. Taking complements we obtainX−U⊆X− f−1(A) f1(YA) and f(X−U)⊆Y−A. Since f is a-ω-closed, f(X−U)Intω(Y−A)=Y−Clω(A). It

follows thatX−U⊆X− f−1(Cl

ω(A)) and f−1(Clω(A))⊆U, since f is ω-irresolute, f−1(Cl

ω(A)) is ω-closed thus we have f−1(A) f−1(Clω(A))⊆U and Clω(f−1(A))

Clω(f−1(Clω(A)))= f−1(Clω(A))⊆U. Therefore Clω(f−1(A))⊆Uand f−1(A) isrgω

-closed inX.

Theorem 4.11. If f :X→Y isR-map andrgω-closed andAisg-closed subset ofX, then

f(A) isrgω-closed.

Proof. Letf(A)⊆U, whereUis regular open subset ofXthenf−1(U) is regular open set

containingA. SinceAisg-closed, we have then Cl(A) f−1(U) and f(Cl(A))U. Since f isrgω-closed,f(Cl(A)) isrgω-closed. Therefore Clω(f(Cl(A)))⊆Uwhich implies that

Clω(f(A))⊆U, hence f(A) isrgω-closed.

The proof ofTheorem 4.8can be easily modified to obtain the following result.

Theorem 4.12. Let f :X→Y bea-ω-map andrgω-c-closed. Then f(A) is rgω-closed subset ofYfor everyrgω-closed subsetAofX.

Theorem 4.13. Let f :X→Y beR-map and pre-ω-closed. Then f(A) isrgω-closed inY

for everyrgω-closed subsetAofX.

Proof. LetAbe anyrgω-closed subset ofXand letU be any regular open subset ofY

such that f(A)⊆U. Since f isR-map, f−1(U) is regular open andAf1(U). AsAis rgω-closed, Clω(A)⊆f−1(U). Hencef(Clω(A))⊆U. Therefore Clω(f(A))Clω(f(Clω

(A)))=f(Clω(A))⊆U. Hence f(A) isrgω-closed inY.

Definition 4.14. A map f :X→Y is said to beω-contra-R-map if for every regular open subsetVofY, f−1(V) isω-closed.

Example 4.15. LetX=Rwith the usual topologyτand letY= {a,b,c,d}, with the topol-ogyσ= {φ,Y,{a},{b},{a,b},{a,b,c}}. Then the function f : (X,τ)(Y,σ) defined by

f(x)=

⎧ ⎨ ⎩

a, ifx∈Q,

c, ifx /∈Q, (4.2)

isω-contra-R-map, sinceQisω-closed. But the function f(x) defined by

f(x)=

⎧ ⎨ ⎩

a, ifx∈Q,

b, ifx /∈Q, (4.3)

is notω-contra-R-map, since the family of all regular open set in (Y,σ) is,Y,{a},{b}} and f−1({b}) is notω-closed.

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Proof. LetAbe any subset ofXand letUbe any regular open subset ofYsuch thatf(A) U. ThenA⊆ f−1(U). Since f isω-contra-R-map, f1(U) isω-closed and so Cl

ω(A)

Clω(f(U))= f(U). Hence f(Clω(A))⊆U. As Clω(A) isω-c-closed subset ofX and f

isrgω-c-closed map, f(Clω(A)) isrgω-closed. Therefore Clω(f(A))Clω(f(Cl(A)))

f(Clω(A))⊆U. Thus f(A) isrgω-closed inY.

Theorem 4.17. If map f :X→Y isrgω-continuous (resp.,rgω-irresolute) andXisrgω

-T1/2, then f isω-continuous (resp.,rgω-irresolute).

Proof. LetAbe any closed (resp.,ω-closed) subset ofY. Since f is anrgω-continuous (resp.,rgω-irresolute) map, f−1(A) is anrgω-closed subset ofX. As (X,τ) isrgω-T

1/2

space, f−1(A) is anω-closed subset ofX. Therefore, f is anω-continuous (resp.,rgω

-irresolute).

Theorem 4.18. Let f :X→Ybe a bijective, ro-preserving, andrgω-continuous map. Then

f isrgω-irresolute map.

Proof. LetVbe anyrgω-closed subset ofXand letUbe any regular open subset ofYsuch that f−1(V)U. ClearlyVf(U). Sincef is a ro-preserving map, f(U) is regular open

and, by assumption,V isrgω-closed set. Hence Clω(V) f(U) and f−1(Clω(V))⊆U.

Since f isrgω-continuous and Clω(V) isω-closed inY, thenf−1(Clω(V)) is argω-closed

subset ofUand so Clω(f−1(Clω(V)))⊆U. Since Clω(f−1(V))Clω(f−1(Clω(V)))⊆U,

Clω(f−1(V))⊆U. Thereforef−1(V) is anrgω-closed subset. Hencef isf rgω-irresolute

map.

Theorem 4.19. A map f :X→Y is f rgω-closed if and only if for each subsetBofY and for each open setUcontaining f−1(B), there is anrgω-open setVofYsuch thatBVand

f−1(V)U.

Proof. Suppose f isrgω-closed, letBbe a subset ofY, andUis an open set ofX such that f−1(B)U. Then f(XU) isrgω-closed inY. LetV=Yf(XU), thenV is rgω-open set and f−1(V)= f1(Yf(XU))=X(XU)U thereforeV is an rgω-open set containingBsuch that f−1(V)U. Conversely suppose thatFis a closed

set ofXthen f−1(Yf(F))XF, andXFis open. By hypothesis, there is anrgω

-open setV ofYsuch thatY−f(F)⊆V and f−1(V)XFthereforeFXf1(V).

HenceY−V⊆ f(F) f(X− f−1(V))YV implies that f(F)=YV, thus f is

rgω-closed.

Acknowledgment

The authors would like to thank the referees for useful comments and suggestions.

References

[1] S. Willard, General Topology, Addison-Wesley, Reading, Mass, USA, 1970.

[2] H. Z. Hdeib, “ω-closed mappings,” Revista Colombiana de Matem´aticas, vol. 16, no. 1-2, pp. 65–78, 1982.

[3] K. Y. Al-Zoubi, “On generalizedω-closed sets,” International Journal of Mathematics and

(11)

[4] H. Z. Hdeib, “ω-continuous functions,” Dirasat Journal, vol. 16, no. 2, pp. 136–153, 1989. [5] N. Levine, “Generalized closed sets in topology,” Rendiconti del Circolo Matematico di Palermo.

Serie II, vol. 19, pp. 89–96, 1970.

[6] S. P. Arya and T. M. Nour, “Characterizations ofs-normal spaces,” Indian Journal of Pure and

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[8] J. Dontchev and H. Maki, “Onθ-generalized closed sets,” International Journal of Mathematics

and Mathematical Sciences, vol. 22, no. 2, pp. 239–249, 1999.

[9] J. Dontchev, “On generalizing semi-preopen sets,” Memoirs of the Faculty of Science Kochi

Uni-versity. Series A. Mathematics, vol. 16, pp. 35–48, 1995.

[10] J. Dontchev and M. Ganster, “Onδ-generalized closed sets andT3/4-spaces,” Memoirs of the

Faculty of Science Kochi University. Series A. Mathematics, vol. 17, pp. 15–31, 1996.

[11] T. A. Al-Hawary, “ω-generalized closed sets,” International Journal of Applied Mathematics, vol. 16, no. 3, pp. 341–353, 2004.

[12] N. Palaniappan and K. C. Rao, “Regular generalized closed sets,” Kyungpook Mathematical

Jour-nal, vol. 33, no. 2, pp. 211–219, 1993.

[13] I. A. Rani and K. Balachandran, “On regular generalised continuous maps in topological spaces,”

Kyungpook Mathematical Journal, vol. 37, no. 2, pp. 305–314, 1997.

[14] R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics, Heldermann, Berlin, Germany, 2nd edition, 1989.

[15] W. Dunham, “T1/2-spaces,” Kyungpook Mathematical Journal, vol. 17, no. 2, pp. 161–169, 1977.

[16] C. W. Baker, “On preservingg-closed sets,” Kyungpook Mathematical Journal, vol. 36, no. 1, pp. 195–199, 1996.

[17] F. H. Khedr and T. Noiri, “Onθ-irresolute functions,” Indian Journal of Mathematics, vol. 28, no. 3, pp. 211–217, 1986.

Ahmad Al-Omari: School of Mathematical Sciences, Faculty of Science and Technology, National University of Malaysia (UKM), Selangor 43600, Malaysia

Email address:omarimutah1@yahoo.com

Mohd Salmi Md Noorani: School of Mathematical Sciences, Faculty of Science and Technology, National University of Malaysia (UKM), Selangor 43600, Malaysia

References

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