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Generalized star semi regular closed sets in topological spaces

D. Sreejaa,∗ and S. Sasikalab

aDepartment of Mathematics,CMS College of Science and Commerce, Coimbatore, Tamil Nadu, India.

bDepartment of Mathematics, Pioneer College of Arts and Science, Coimbatore, Tamil Nadu, India.

Abstract

In this paper a new class of sets called generalized star semi regular closed sets is introduced and

its various properties are discussed. g∗srcontinuous function is defined and its results are studied.

Also comparative study has been done with the existing sets.

Keywords: Topological spaces, generalized closed sets, g-closed, gs-closed, g∗sr-closed sets, g∗sr

-closed map,g∗sr-continuous functions.

2010 MSC:54D10. c2012 MJM. All rights reserved.

1

Introduction

The notion of closed set is fundamental in the study of topological spaces. In 1970 Levine [12]

introduced the class of generalized closed sets in the topological space by comparing the closure of

subset with its open supersets. The investigation on generalized of closed set has lead to significant

contribution to the separation axiom, covering properties and generalization of continuity. T.K.

Kong, R. Kopperman and P. Meyer [12] shown some of the properties of generalized closed set have

been found to be useful in computer science and digital topology . Maki et al. [15] definedαg-closed

sets in 1994. S.P. Arya and N. Tour [2] defined gs-closed sets in 1990. Dontchev [6], Gnanambal and

Palaniappan and Rao [18] introduced gsp-closed sets, gpr closed sets and rg-closed sets respectively.

M.K.R.S. Veerakumar [11] introducedg∗- closed sets in 1991. J. Dontchev [6] introduced gsp-closed

sets in 1995. In 1993, N.Palaniappan and Chandra Sekran Rao [18] introduced rg-closed sets.

Corresponding author.

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2

Preliminaries

Throughout this paper(X, τ)and(Y, σ)represent nonempty topological spaces are mentioned.

For a subset A of a space (X, τ) cl A and int A denote the closure of A and the interior of A

respectively. (X, τ) will be replaced by X if there is no chance of confusion. Let us recall the

following definitions which we shall require later.

Definition 2.1. A subset A of a space(X, τ)is called

(1) The closure of A is defined as the intersection of all closed sets containing A.

(2) a pre-open set [13] ifA⊆int(cl(A))and a pre-closed set ifcl(int(A))⊆A.

(3) a semi-open set [11] ifA⊆cl(int(A))and semiclosed set ifint(cl(A))⊆A.

(4) aα-open set [16] ifA⊆int(cl(int(A)))andα-closed set ifcl(int(cl(A)))⊆A.

(5) a semi pre-open set [1] ifA⊆cl(int(cl(A))and a semi pre-closed set ifint(cl(int(A)))⊆A.

(6) a regular-open set ifA=int(cl(A))and regular closed setA=cl(int(A)).

The family of all pre-open (resp.preclosed, semi preopen, semi preclosed, regularopen, regularcosed) subsets

of space (X, τ) wil be denoted by po(τ)(resp, pc(τ), spo(X), spc(X), ro(τ), rc(τ)). The intersection of all

semi-closed subsets of x containing A is called semiclosure of A is denoted by scl(A).

Definition 2.2. A subset A of a space(X, τ)is called a

(1) generalized closed (briefly g-closed)[12] ifcl(A)⊆U. wheneverA⊆U andU is open inX.

(2) semi-generalized closed set (briefly sg-closed) [3] ifscl(A)⊆U, wheneverA ⊆U andU is semi open

inX.

(3) generalized semi-closed set (briefly gs-closed) [2] ifscl(A)⊆U, wheneverA⊆U andU is open inX.

(4) α - generalized -closed set (brieflyαg-closed) [15] ifαcl(A) ⊆ U, wheneverA ⊆U andU is open in

X.

(5) a generalizedα-closed set (briefly gα-closed) [14]. ifαcl(A) ⊆U, wheneverA ⊆U andU is open in

X.

(6) generalized semi-pre closed set (briefly gsp-closed) [6] ifspcl(A)⊆U, wheneverA⊆U andU is open

inX.

(7) regular generalized closed set (briefly rg-closed) [18] ifcl(A) ⊆U, wheneverA ⊆U andU is regular

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(8) generalized pre-closed set (briefly gp-closed) [4] ifpcl(A)⊆U, wheneverA⊆U andU is open inX.

(9) generalized pre-regular closed set (briefly gpr-closed) ifpcl(A)⊆U, wheneverA⊆U andU is regular

open inX.

(10) sg∗rclosed ifrcl(A)⊆U wheneverA⊆U andU is sg open.

Definition 2.3. A mapf : (x, τ)→(y, σ)is called

(1) generalized closed (briefly g-closed ) if[12]f(U)is g-open in(y, σ)for every open set ofU of(x, τ)

(2) regular closed (briefly r-closed) iff(U)is r-open in(y, σ)for every open set ofU of(x, τ)

(3) αgeneralized closed (brieflyαg-closed) [15] iff(U)isαg-open in(y, σ)for every open set ofU of(x, τ)

(4) generalized pre-closed (briefly gp-closed)[4] iff(U)is rg-open in(y, σ)for every open set ofU of(x, τ)

(5) regular generalized closed (briefly rg-closed)[18] iff(U)is rg-open in(y, σ)for every open set ofU of

(x, τ)

(6) generalized semi-pre closed (briefly gsp-closed) [6] iff(U)is rg-open in(y, σ)for every open set ofU of

(x, τ).

Definition 2.4. A functionf : (x, τ)→(y, σ)is called

(1) r continuous iff−1(V)is r closed set of(X, τ)→(Y, σ)for every closed set v of(Y, σ).

(2) αg continuous iff−1(V)isαg closed set of(X, τ)→(Y, σ)for every closed set v of(Y, σ).

(3) gpr continuous iff−1(V)is gpr closed set of(X, τ)→(Y, σ)for every closed set v of(Y, σ).

(4) gp continuous iff−1(V)is gp closed set of(X, τ)→(Y, σ)for every closed set v of(Y, σ)

(5) rg continuous iff−1(V)is r closed set of(X, τ)→(Y, σ)for every closed set v of(Y, σ).

3

Basic properties ofg∗srclosed sets

Definition 3.1. A subset A of a topological space(X, τ)is said to beg∗srclosed set ifrcl(A)⊆U whenever

A⊆U andU is gs open inX.

Example 3.1. LetX = {a, b, c, d} with topologyτ ={ϕ,{a},{a, b},{c, d},{a, c, d}, X}. The closed sets

are

τc ={ϕ,{b, c, d},{c, d},{a, b},{b}, X}.

gs open sets are {ϕ,{a},{c},{d},{a, c},{a, b},{a, d},{c, d},{a, b, d},{a, c, d},{a, b, c}, X} g∗sr closed

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Theorem 3.1. Every regular closed set isg∗srclosed

Proof. Let A be a regular closed set inX. ThenA = rcl(A). Let us prove that A isg∗sr closed in

X. LetA ⊆ U andU is gs open. Thenrcl(A) = A ⊆U wheneverA ⊆ U andU is gs open. Hence

rcl(A)⊆U. Hence A isg∗srclosed set inX.

Remark 3.1. The converse of the above theorem need not be true in general, as shown in the following example.

Example 3.2. Let X = {a, b, c, d} with topology τ = {ϕ,{a},{b},{a, b}, X}, Regular closed sets are

{ϕ,{a, c, d},{b, c, d}, X}. g∗srclosed sets are{ϕ,{c, d},{b, c, d},{a, c, d}, X}. HenceA= {c, d}isg∗sr

closed but not regular closed.

Theorem 3.2. Everyg∗srclosed set is g closed

Proof. Let A be g∗sr closed set in X. Then rcl(A) ⊆ U wheneverA ⊆ U andU is gs open. Let

U be open set such that A ⊆ U. Since every open set is g open and A is g∗sr closed . We have

rcl(A)⊆cl(A)⊆U. Hencecl(A)⊆U. Therefore A is g closed set inX.

Remark 3.2. The converse of the above theorem need not be true in general, as shown in the following example.

Example 3.3. Let X = {a, b, c, d} with topology τ = {ϕ,{a},{b},{a, b}, X}. g closed sets are

{ϕ,{a, c, d},{b, c, d}, X}.

g∗srclosed sets are{ϕ,{c, d},{b, c, d},{a, c, d}, X}. HenceA={c, d}isg∗srclosed but not g closed.

Theorem 3.3. Everyg∗srclosed set is gs closed.

Proof. Let A be ag∗srclosed set inX. (i.e)rcl(A)⊆U wheneverA⊆U andU is gs open. To prove

that A is gs closed. LetU be open set such thatA⊆U. Since every open set is gs-open and A isg∗sr

closed, We havescl(A)⊆rcl(A)⊆U. Therefore A is gs- closed set inX.

Remark 3.3. The converse of the above theorem need not be true in general, as shown in the following example.

Example 3.4. Let X = {a, b, c, d} with topology

τ = {ϕ,{b},{d},{b, d},{a, b},{b, c},{a, b, c},{b, c, d},{a, b, d}, X} gs closed sets are

{ϕ,{a},{c},{d},{c, d},{a, c},{a, d},{a, c, d},{a, b, c}, X} g∗sr closed sets are {ϕ,{d},{a, c, d}, X}

HenceA={a}is gs closed but notg∗srclosed.

Theorem 3.4. Everyg∗srclosed set is rg closed.

Proof. Let A beg∗srclosed set inX. Thenrcl(A) ⊆ U andU is gs open. To prove A is rg closed in

X. LetU be r open set such thatA ⊆U. Since every r open set is gs open and A isg∗srclosed. We

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Remark 3.4. The converse of the above theorem need not be true in general,as shown in the following example.

Example 3.5. Let X = {a, b, c, d} with topology τ = {ϕ,{a},{b},{a, b}, X} rg closed are sets

{ϕ,{c},{d},{a, b},{a, c},{a, d},{b, d},{c, d},{a, c, d},{a, b, d},{a, b, c},{b, c, d}, X} g∗sr closed sets

are{ϕ,{a, c, d},{b, c, d},{c, d}, X}HenceA={c}rg closed but notg∗srclosed.

Theorem 3.5. Everyg∗srclosed set isαg closed

Proof. Let A beg∗sr closed set inX. Let us prove that A isαg closed. LetU be open set such that

A⊆U. ThenA⊆UandUis gs open andrcl(A)⊆Asince A isg∗srclosed. Butαcl(A)⊆rcl(A)⊆U

andU is open. Therefore A isαgclosed.

Remark 3.5. The converse of the above theorem need not be true in general, as shown in the following example.

Example 3.6. LetX={a, b, c, d}with topologyτ ={ϕ,{a},{a, b},{c, d},{a, c, d}, X}αgclosed sets are

{ϕ,{b},{c},{d},{a, b},{b, c},{b, d},{c, d},{a, b, c},{b, c, d},{a, b, d},{a, c, d}, X} g∗sr closed sets are

{ϕ,{b},{a, b},{c, d},{b, c, d}}

HenceA={c}isαg closed but notg∗srclosed.

Theorem 3.6. Everyg∗srclosed set is gp closed.

Proof. Let A beg∗srclosed set inX. Thenrcl(A) ⊆U wheneverA⊆ U andU is gs open. To prove

that A is gp closed. LetU be open set such thatA⊆U. Since every open set is gs open and A isg∗sr

closed. We haveP cl(A)⊆U. Hencepcl(A)⊆U. Therefore A is gp closed set inX.

Remark 3.6. The converse of the above theorem need not be true in general, as shown in the following example.

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

gp closed sets are {ϕ,{b},{a, b},{b, c},{b, d},{a, b, c},{a, b, d},{b, c, d}, X} g∗sr closed sets are

{ϕ,{b},{a, b},{b, d},{b, c, d}, X}HenceA={a, b, c}is gp closed but notg∗srclosed.

Theorem 3.7. Everyg∗srclosed set is gsp closed

Proof. Let A be g∗sr closed set in X. Then rcl(A)U andU is gs open. To prove A is gsp closed

in X. Let U be open set such thatA ⊆ U. Since every open set is gs open and A is g∗sr closed.

Spcl(A)⊆scl(A)⊆rcl(A)⊆U. Hencespcl(A)U. Therefore A is gsp closed.

Remark 3.7. The converse of the above theorem need not be true in general, as shown in the following example.

Example 3.8. LetX ={a, b, c, d}wih topologyτ ={ϕ,{a},{c},{d},{a, c},{ad},{c, d},{a, c, d}, X}gsp

closed sets are{ϕ,{a},{b},{c},{d},{a, b},{a, c},{a, d},{b, c},{b, d},{c, d},{a, b, c},{a, b, d}, X}g∗sr

closed sets are{ϕ,{b},{a, b},{b, c, d},{b, d}, X}

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Theorem 3.8. Everyg∗srclosed set is gpr closed

Proof. Let A beg∗srclosed set inX. ThenA ⊆U andU is gs open. To prove A is gpr closed. Let

U be open set such thatA ⊆ U. Since every r-open set is gs open and A is g∗sr closed. We have

P cl(A)⊆rcl(A)⊆U. Therefore A is gpr closed set inX.

Remark 3.8. The converse of the above theorem need not be true in general, as shown in the following example.

Example 3.9. LetX ={a, b, c, d}with topology

τ ={ϕ,{a},{a, b},{c, d},{a, c, d}, X}g∗srclosed sets are{ϕ,{b},{a, b},{c, d},{b, c, d}}gprclosed sets

are

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

Hence

A = {{a},{c},{d},{b, c},{a, c},{a, d},{b, d},{a, b, c},{a, c, d},{a, b, d}} are gpr closed but not g∗sr

closed.

Theorem 3.9. The finite union of theg∗srclosed set isg∗srclosed.

Proof. Let A and B beg∗sr closed sets inX. LetU be a gs open inX such that AU B ⊆ U. Then

A ⊆ U and B ⊆ U . Since A and B are g∗sr closed set rcl(A) ⊆ U and rcl(B) ⊆ U. Hence

rcl(AU B) =rcl(A)U rcl(B)⊆U. ThereforeAU Bisg∗srclosed.

Theorem 3.10. The finite intersection of twog∗srclosed sets is closed.

Proof. Let A and B beg∗srclosed sets in X. Let U be a gs open inX such thatA∩B ⊆ U. Then

A ⊆ U and B ⊆ U . Since A and B are g∗sr closed set rcl(A) ⊆ U and rcl(B) ⊆ U. Hence

rcl(A∩B) =rcl(A)∩rcl(B)⊆U. ThereforeA∩Bisg∗srclosed.

Theorem 3.11. Let A and B be subsets such thatA⊆B ⊆rcl(A). If A isg∗srclosed then B isg∗srclosed.

Proof. Let A and B be subsets such thatA ⊆ B ⊆ rcl(A). Suppose that A isg∗srclosed. To show

that B isg∗srclosed setB ⊆ U andU is gs open inX. SinceA ⊆ B andB ⊆ U, we haveA ⊆ U.

HenceA⊆U andU is gs open inX. Since A isg∗srclosed we havercl(A)⊆U . SinceB ⊆rcl(A),

we havercl(B)⊆rcl(rcl(A)) =rcl(A)⊆U. Hencercl(B)⊆U. Hence B isg∗srclosed.

Theorem 3.12. Ifx∈rcl(A)if f U∩A6=φfor every regular open setU containing x.

Proof. Suppose thatx ∈ rcl(A). To show thatU ∩A 6=ϕfor every gs open setU containing x such

thatU ∩A=φ. ThenA⊆U andU is gs closed set. SinceA⊆U, rcl(A)⊆rcl(U)c. Sincex∈rcl(A)

We havex∈rcl(U)c. SinceU is gs closed set. We havexUc. Hencex /U which is contradiction

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Conversely , suppose thatU ∩A 6= φ. for every open set containing x. To show that x ∈ rcl(A).

Suppose thatx /∈rcl(A). Then there exists a gs open setU containing x such thatU∩A=φ. This is

contradiction toU ∩A6=φ. Hencex∈rcl(A).

Theorem 3.13. A subset A of X isg∗sr-closed set inXiffscl(A)−Acontains no non empty gs-closed set

inX.

Proof. Suppose that F is a non empty gs-closed subset of scl(A)-A. Now F ⊆ scl(A)−A. Then

F ⊆ scl(A)∩Ac. ThereforeF ⊆scl(A)andF ⊆Ac. SinceFc is gs-open set and A isg∗sr-closed ,

scl(A) ⊆Fc. That isF ⊆scl(A)c. SinceFc is gs-open set and A isg∗sr-closed ,scl(A) ⊆Fc . That

isF ⊆scl(A)c . HenceF ⊆scl(A)∩[scl(A)]c. That isF =Thusscl(A)−Acontains no non empty

gs-closed set.

Conversely, assume that scl(A) −A contains no non empty gs-closed set. Let A ⊆ U , U is

gs-open. Suppose thatscl(A)is not contained inU. Thenscl(A)∩Uc is a non empty gs closed set and

contained inscl(A)−AWhich is a contradiction. Thereforescl(A) ⊆U and hence A isg∗sr-closed

set.

Definition 3.2. An topological space(X, τ)is called ag∗srcompact space if everyg∗srcovering has a finite

subcover.

Definition 3.3. Let(X, τ)be ag∗srcompact space. If A isg∗srclosed subset ofX, then A isg∗srcompact.

Theorem 3.14. Every closed subset of ag∗srcompact. space isg∗srcompact.

Proof. Let Y be a closed set in a g∗s-I compact space(X, τ). Therefore, Y isg∗s-I compact, since

every closed set isg∗sr closed. Given, a covering A of Y by g∗sr open sets inX, we can form an

open covering B ofX by adjoining to A the singleg∗sropen setX−Y. ie,B =A∪(X−Y). Since

X is g∗sr compact, some finite subcollection of B covers X. If this subcollection contains the set

X−Y, discardX−Y, otherwise leave the subcollection alone. The resulting collection is a finite

subcollection A that coversY. HenceY isg∗srcompact.

Theorem 3.15. Let(X, τ)be a compact topological space. If A isg∗sr-closed subset ofX, then A iscompact.

Proof. Let{Ui}be a open cover of A. Since every open set is gs-open and A isg∗sr-closed. We get

scl(A) ⊆UiSince a closed subset of a compact space is compact,scl(A)is compact. Therefore there

exist a finite subover say{U1∪U2...∪Un}of{Ui}for Ascl(A). So,A⊆scl(A) ⊆U1∪U2∪...∪Un.

Therefore A is compact

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Proof. If{x}is not gs-closed, then the only gs-open set containing{x}c inX. Thus semi closure of

{x}c is contained inXand hence{x}cisgs-closed inX.

4

g∗srclosed maps

In this section,we introduce the concepts ofg∗sr- closed maps in topological Spaces and study

their properties.

Definition 4.1. A mapf : (X, τ)→(Y, σ)is calledg∗sr- closed map if the image of each closed set inXis

g∗sr- closed inY.

Theorem 4.1. Every regular closed map isg∗srclosed.

Proof. Let F be a closed set in(X, τ) andf : (X, τ) → (Y, σ)is regular closed map. Hencef(F)is

regular closed inY. As every regular closed set isg∗srclosed set inY. Thereforef(F)isg∗srclosed

inY. Hence f isg∗srclosed map.

Remark 4.1. The converse of the above theorem need not be true in general .

Example 4.1. Everyg∗srclosed map is not regular closed. LetX={a, b, c, d} Y ={a, b, c, d}

τ ={ϕ,{a},{b},{a, b}, X}

τc={X,{b, c, d},{a, c, d},{c, d}, ϕ}

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σc = {X,{b, c, d},{a, c, d},{c, d}, ϕ} Letf : (X, τ) → (Y, σ)be the identity mapping. g∗sr-closed sets

of (Y, σ) are {ϕ, X,{c, d},{a, c, d},{b, c, d}}. Regular closed sets of (Y, σ) are {ϕ, X,{a, c, d},{b, c, d}}

consider the closed setsA ={ϕ, X,{c, d},{a, c, d},{b, c, d}}. f(A) =f({c, d}) ={c, d}isg∗srclosed in

(Y, σ)f(A) =f({a, c, d}) ={a, c, d}isg∗srclosed in(Y, σ)f(A) = f({c, d}) ={b, c, d}isg∗srclosed in

(Y, σ)therefore f isg∗srclosed map.f(A) =f({c, d}) ={c, d}isg∗srclosed, but not regular closed.

Theorem 4.2. Everyg∗srclosed map isαgclosed.

Proof. Let F be a closed set in(X, τ)andf : (X, τ)→(Y, σ)beg∗srclosed map in(Y, σ). Hencef(F)

isg∗srclosed inY. As everyg∗srclosed set isαgclosed set inY. Thereforef(F)isαgclosed inY.

Hence f isαgclosed map.

Remark 4.2. The converse of the above theorem need not be true in general.

Example 4.2. Everyαgclosed map is notg∗srclosed. LetX={a, b, c} Y ={a, b, c}

τ ={ϕ,{a},{a, b}, X}

τc ={X,{b, c},{c}, ϕ}

σ={ϕ,{a},{a, b}, X}

σc = {X,{b, c},{c}, ϕ} Let f : (X, τ) → (Y, σ) be the identity mapping. g∗sr -closed sets of (Y, σ)

are{ϕ, X,{b, c}}. αg closed sets of(Y, σ) are{ϕ, X,{b},{c},{a, c},{b, c}} consider the closed setsA =

{ϕ, X,{c},{b, c}}. f(A) =f({b, c}) = {b, c}isg∗srclosed in(Y, σ)Butf(A) =f{a, c}isαgclosed but

notg∗srclosed set.

Theorem 4.3. Everyg∗srclosed map is gp closed.

Proof. Let F be a closed set in(X, τ)andf : (X, τ)→(Y, σ)beg∗srclosed map in(Y, σ). Hence f(F)

isg∗srclosed inY. As everyg∗sr closed set is gp closed set inY. Therefore f(F) is gp closed inY.

Hence f is gp closed map.

Remark 4.3. The converse of the above theorem need not be true in general.

Example 4.3. Every gp closed map is notg∗srclosed. LetX ={a, b, c, d} Y ={a, b, c, d}

τ ={ϕ,{a},{c},{d},{a, d},{a, c},{c, d},{a, c, d}X}

τc ={X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ}

σ={ϕ,{a},{c},{d},{a, d},{a, c},{c, d},{a, c, d}X}

σc = {X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ} Letf : (X, τ) → (Y, σ) be the identity

mapping. g∗sr - closed sets of (Y, σ) are {ϕ, X,{b},{a, b},{b, d},{b, c, d}}. gp closed sets of (Y, σ) are

{ϕ, X,{b, c, d},{a, b, d},{b, d},{b, c},{a, b},{b}, ϕ}consider the closed sets

A = {X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ}f(A) = f{b, c}is gp closed but notg∗sr

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Theorem 4.4. Everyg∗srclosed map is rg closed.

Proof. Let F be a closed set in(X, τ)andf : (X, τ)→(Y, σ)beg∗srclosed map in(Y, σ). Hencef(F)

isg∗srclosed in Y. As everyg∗srclosed set is rg closed set in Y. Therefore f(F) is rg closed inY.

Hence f is rg closed map

Remark 4.4. The converse of the above theorem need not be true in general.

Example 4.4. Every rg closed map is notg∗srclosed. LetX={a, b, c}Y ={a, b, c}

τ ={ϕ,{a},{a, b}, X}

τc={X,{b, c},{c}, ϕ}

σ ={ϕ,{a},{a, b}, X}

σc = {X,{b, c},{c}, ϕ} Letf : (X, τ) → (Y, σ)be the identity mapping. g∗sr- closed sets of (Y, σ)are

{ϕ, X,{b, c}}. rg closed sets of(Y, σ)are{ϕ, X,{a},{b},{c},{a, b},{a, c},{b, c}. Consider the closed sets

A ={ϕ, X,{c},{b, c}}. f(A) =f({b, c}) = {b, c}isg∗srclosed in(Y, σ)f(A) =f{a, b}is rg closed but

notg∗srclosed set.

Theorem 4.5. Everyg∗srclosed map is gsp closed.

Proof. Let F be a closed set in(X, τ)andf : (X, τ)→(Y, σ)beg∗srclosed map in(Y, σ). Hencef(F)

isg∗srclosed inY. As everyg∗srclosed set is gsp closed set inY. Thereforef(F)is gsp closed inY.

Hence f is gsp closed map.

Remark 4.5. The converse of the above theorem need not be true in general.

Example 4.5. Every gsp closed map is notg∗srclosed. LetX ={a, b, c, d} Y ={a, b, c, d}

τ ={ϕ,{a},{c},{d},{a, d},{a, c},{c, d},{a, c, d}X}

τc={X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c}, a, b},{b}, ϕ}

σ ={ϕ,{a},{c},{d},{a, d},{a, c},{c, d},{a, c, d}X}

σc ={X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ}

Let f : (X, τ) → (Y, σ) be the identity mapping. g∗sr - closed sets of (Y, σ) are

{ϕ, X,{b},{a, b},{b, d},{b, c, d}}. gsp closed sets of (Y, σ) are

{ϕ, X,{a},{b},{z},{d},{a, c},{a, d},{c, d},{b, c, d},{a, b, d},{b, d},{b, c},{a, b},{a, b, c}} consider

the closed sets A = {X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ} f(A) = f{a, b, c} is gsp

closed but notg∗srclosed set.

Theorem 4.6. A mapf :x →yisg∗sr-closed if and only if for each subset S of y and for each open setU

Containingf−1(s)there is ag∗s-open set V of y such thatS ⊆V and⊆(V)

Proof. Suppose f isg∗s-closed. Let S be a sub set of Y and is an open set of X such thatf−1(V)⊆(U)

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By hypothesis, there is ag∗s-open set V of Y such thatf−1(F) ⊆U V andf−1(v)⊆X−F is open.

Thereforey−v⊆f(F)f(x− ⊆y−v)which impliesf(F) =y−v. Sincey−visg∗s-closed, f(F) is

g∗s-closed and thus f isg∗s-closed map.

Theorem 4.7. Iff :X→Y is closed andh:Y →Zisg∗srclosed thenh◦f :X →Zisg∗srclosed

Proof. Letf :X→Y is closed map andh:Y →Zis ag∗srclosed map. Let f be any closed set inX.

Sincef :X →Y is a closed f(F) is closed inY. Sinceh :Y → Zis ag∗srclosed map. Thenh(f(F))

isg∗srclosed set inZ. Hencehf :X→Zis ag∗srclosed map.

5

g

sr

ontinuous functions

In this section,we introduce the concept ofg∗sr- continuous functions in topological Spaces and

study their properties.

Definition 5.1. A mapf : X → Y from a topological spaceX into a topological space Y is calledg∗sr

continuous if the inverse image of every closed set inY isg∗srclosed inX.

Theorem 5.1. If a mapf :X →Y is regular continuous then it isg∗srcontinuous.

Proof. Let f : X → Y be regular continuous. Let F be any closed set in Y. The inverse image of

f−1(F)is regular closed inX. Since every regular closed set is g∗srclosed . Hence f−1(F)isg∗sr

closed inX. Hence f isg∗srcontinuous.

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Example 5.1. If a map f : X → Y isg∗srcontinuous but not regular continuous. LetX = {a, b, c, d}

Y ={a, b, c, d}

τ ={ϕ,{a},{b},{a, b}, X}

τc={X,{b, c, d},{a, c, d},{c, d}, ϕ}

σ ={ϕ,{a},{b},{a, b}, X}

σc = {X,{b, c, d},{a, c, d},{c, d}, ϕ} Letf : (X, τ) → (Y, σ) be the inverse mapping. g∗sr- closed sets

of (X, τ) are{ϕ, X,{c, d},{a, c, d},{b, c, d}}. Regular closed sets of(X, τ) are{ϕ, X,{a, c, d},{b, c, d}}.

Let F be a closed set in (Y, σ). Let F = {c, d}, f−1(F) = f−1(c, d) = {c, d}isg∗srclosed set in(X, τ).

Therefore f is g∗sr continuous then f−1(F) = f−1(c, d) is not regular continuous. Hence here f is g∗sr

continuous but not regular continuous.

Theorem 5.2. If a mapf :X→Y isg∗srcontinuous then it is gpr continuous.

Proof. Letf :X →Y beg∗srcontinuous. Let F be any closed set inY. The inverse image off−1(F)

is g∗sr closed in X. Since everyg∗sr closed set is gpr closed . Hencef−1(F) is gpr closed inX.

Hence f is gpr continuous.

Remark 5.2. The converse of the above theorem need not be true in general as shown in the following example.

Example 5.2. If a mapf :X →Y is gpr continuous but notg∗srcontinuous. LetX={a, b, c}

Y ={a, b, c}

τ ={ϕ,{a},{a, b}, X}

τc={X,{b, c},{c}, ϕ}

σ ={ϕ,{a},{a, b}, X}

σc = {X,{b, c},{c}, ϕ} Let f : (X, τ) → (Y, σ) be the inverse mapping. g∗sr- closed sets of(X, τ)are

{ϕ, X,{b, c}}gpr closed sets of(X, τ)are{ϕ, X,{a},{b},{c},{a, b},{a, c},{b, c}}Let F be a closed set in

(Y, σ). LetF ={a, b}, f−1(F) =f−1(a, b) ={a, b}is gpr closed set in(X, τ). Therefore f is gpr continuous

thenf−1(F) =f−1(a, b)is notg∗srcontinuous. Hence here f is gpr continuous but notg∗srcontinuous.

Theorem 5.3. If a mapf :X→Y isg∗srcontinuous then it isαgcontinuous.

Proof. Letf :X →Y beg∗srcontinuous. Let F be any closed set inY. The inverse image off−1(F)

isg∗srclosed inX. Since everyg∗srclosed set isαgclosed . Hencef−1(F)isαgclosed inX. Hence

f isαgcontinuous.

Remark 5.3. The converse of the above theorem need not be true in general as shown in the following example.

Example 5.3. If a mapf :X →Y isαgcontinuous but notg∗srcontinuous. LetX ={a, b, c}

Y ={a, b, c}

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τc ={X,{b, c},{c}, ϕ}

σ={ϕ,{a},{a, b}, X}

σc = {X,{b, c},{c}, ϕ}Let f : (X, τ) → (Y, σ) be the inverse mapping. g∗sr -closed sets of(X, τ) are

{ϕ, X,{b, c}. αgclosed sets of(X, τ)are{ϕ, X,{a},{b},{c},{a, b},{a, c,},{b, c}}Let F be a closed set in

(Y, σ). LetF ={a, b}, f−1(F) =f−1(a, b) ={a, b}isαgclosed set in(X, τ). Therefore f isαgcontinuous

thenf−1(F) =f−1(a, b)is notg∗srcontinuous. Hence here f isαgcontinuous but notg∗srcontinuous.

Theorem 5.4. If a mapf :X →Y isg∗srcontinuous then it is rg continuous.

Proof. Letf :X →Y beg∗srcontinuous. Let F be any closed set inY. The inverse image off−1(F)

isg∗srclosed inX. Since everyg∗srclosed set is rg closed . Hencef−1(F)is rg closed inX. Hence

f is rg continuous.

Remark 5.4. The converse of the above theorem need not be true in general as shown in the following example.

Example 5.4. If a mapf :X→Y is rg continuous but notg∗srcontinuous. LetX={a, b, c}

Y ={a, b, c}

τ ={ϕ,{a},{a, b}, X}

τc ={X,{b, c},{c}, ϕ}

σ={ϕ,{a},{a, b}, X}

σc = {X,{b, c},{c}, ϕ}Let f : (X, τ) → (Y, σ) be the inverse mapping. g∗sr- closed sets of(X, τ) are

{ϕ, X,{b, c}}. rg closed sets of(X, τ)are{ϕ, X,{a},{b},{c},{a, b},{a, c},{b, c}}Let F be a closed set in

(Y, σ)LetF ={a, b}, f−1(F) =f−1(a, b) = {a, b}is rg closed set in(X, τ). Therefore f is rg continuous

thenf−1(F) =f−1(a, b)is notg∗srcontinuous. Hence here f is rg continuous but notg∗srcontinuous.

Theorem 5.5. If a mapf :X →Y isg∗srcontinuous then it is gp continuous.

Proof. Letf :X →Y beg∗srcontinuous. Let F be any closed set inY. The inverse image off−1(F)

isg∗srclosed inX. Since everyg∗srclosed set is gp closed . Hencef−1(F)is gp closed inX. Hence

f is gp continuous.

Remark 5.5. The converse of the above theorem need not be true in general as shown in the following example.

Example 5.5. If a mapf :X→Y is gp continuous but notg∗srcontinuous. LetX={a, b, c, d}

Y ={a, b, c, d}

τ ={ϕ,{a},{c},{d},{a, d},{a, c},{c, d},{a, c, d}X}

τc ={X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ}

σ={ϕ,{a},{c},{d},{a, d},{a, c},{c, d},{a, c, d}X}

σc = {X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ} Let f : (X, τ) (Y, σ) be the inverse

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{ϕ, X,{b, c, d},{a, b, d},{b, d},{b, c},{a, b},{b}, ϕ}. Let F be a closed set in (Y, σ) Let

F ={a, b, c}f−1(F) =f−1(a, b, c) ={a, b, c}is rg closed set in(X, τ). Therefore f is gp continuous then

f−1(F) =f−1(a, b, c)is notg∗srcontinuous. Hence here f is gp continuous but notg∗srcontinuous.

Theorem 5.6. If a mapf :X→Y isg∗srcontinuous then it is gs continuous.

Proof. Letf :X →Y beg∗srcontinuous. Let F be any closed set inY. The inverse image off−1(F)

isg∗srclosed inX. Since everyg∗srclosed set is gs closed . Hencef−1(F)is gs closed inX. Hence

f is gs continuous.

Remark 5.6. The converse of the above theorem need not be true in general as shown in the following example.

Example 5.6. If a map f : X → Y is gs continuous but not g∗sr continuous. Let

X ={a, b, c, d} Y ={a, b, c, d}

τ ={ϕ,{b},{d},{b, d},{a, b},{b, c},{a, b, c},{b, c, d},{a, b, d}, X}

σ ={ϕ,{a},{c},{d},{a, d},{a, c},{c, d},{a, c, d}, X}

σ = {X,{b, c, d},{a, b, d},{a, b, c},{b, d},{b, c},{a, b},{b}, ϕ} Let f : (X, τ) → (Y, σ) be the inverse

mapping. g∗sr- closed sets of(X, τ)are{ϕ,{d},{a, c, d}, X}. gs closed sets of(X, τ)are

{ϕ,{a},{c},{d},{c, d},{a, c},{a, d},{a, c, d},{a, b, c}, X}. Let F be a closed set in (Y, σ) Let

F ={a, b, c}f−1(F) = f−1(a, b, c) ={a, b, c}is gs closed set in(X, τ). Therefore f is gs continuous then

f−1(F) =f−1(a, b, c)is notg∗srcontinuous. Hence here f is gs continuous but notg∗srcontinuous.

References

[1] D.Andrijevic,Mat. Vesnik,38(1986),24-32.

[2] S.P.Arya and T.Nour,Indian J.Pure Appl. Math., 21(1990),717-19.

[3] P.Bhattacharyaa and B.K.Lahiri,Indian J. Math.,29(1987),375-82.

[4] K.Balachandran and Arokia Rani, On generalized preclosed sets, preprint.

[5] W.Dunham,Kyungpook Math J.,17(1977),161-69.

[6] J .Dontchev,Mem. Fac. Sci. Kochi Univ. Ser. A, Math.16(1995),35-48.

[7] D.S. Jankovic,Ann. Soc.Sci. Bruxelles Ser.I.,97(1983), 59-72.

[8] D.S. Jankovic and I.L.Reily,Indian J.Pure Appl. Math.,16(1985), 957-64.

[9] D.S. Jankovic,Acta Math. Hungar.,46(1985), 83-92.

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[11] N.Levine,Math. Monthly,70(1963),36-41.

[12] N.Levine,Rend. Circ. Mat. Palermo,19(2)(1970),89-96.

[13] A.S Mshour, M.E Abd El-Monsef and S.N. El-Deeb,Proc. Math. Phys.Soc. Egypt,53(1982),47-53.

[14] H.Maki, R.Devi and K.Balachandran,Bull, Fukuoka Univ. Ed. Part III,42(1993),13-21.

[15] H.Maki, R.Devi and K.Balachandran,Mem. Fac. Sci. Kochi Univ. Ser. A, Math.,15 (1994),51- 63.

[16] O.Nijastad ,Pacific J. Math.,15 (1965),961-70.

[17] T.Noiri,J. Austral. Math. Soc,(Series A), 51(1991),300-304.

[18] N.Palaniappan and K.C. Rao,Kyungpook Math, J.,33(1993),211-19.

Received: May 15, 2015;Accepted: June 23, 2015

UNIVERSITY PRESS

References

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