A new form of generalized closed sets via regular local function in ideal topological spaces

A new form of generalized closed sets via regular local function in ideal

topological spaces

C. Janakia,∗ and A. Jayalakshmib

a_{L.R.G. Govt. Arts College for Women, Tirupur, Tamil Nadu, India.}

b_{Sri Krishna College of Engineering and Technology, Coimbatore, Tamil Nadu, India.}

Abstract

In this paper we define a regular local functionA∗R(I, τ) by using regular open sets in an ideal topological spaces (X, τ, I). Some properties and characterizations of a regular local functions are derived. Also, a new form of generalized closed sets via regular local function is introduced and its

basic properties are discussed.

Keywords: Regular-local function, I-grαclosed.

2010 MSC:54A05, 54A10. c2012 MJM. All rights reserved.

1

Local functions in topological spaces via ideals was introduced by kuratowski[6]. More

importance was given to the topic ideal topological spaces, by Vidyanathasamy[15]. In 1990,

Jankovic and Hamlettt [4] gave a brief account of all results established earlier and proved some

new results in ideal topological spaces. Compatibility of the topology τ with an ideal I was first

defined by Njastad [12] in 1996.Jankovic and Hamlett [4] have studied some properties and

charecteristics of compatible spaces . In this paper we define a regular-local function by using

regular-open sets with its properties and we define regular -compatible spaces. Also by using it a

new form of generalized closed called I-grαclosed set is derived with its basic properties.

∗

Corresponding author.

2

An ideal I on a topological space (X,τ) is non-empty collection of subsets of X which satisfies the

following properties.

(1) A∈I and B⊆A implies B∈I,

(1) A∈I and B∈I implies A∪B∈I.

An ideal topological spaces is a topological space(X,τ) with an ideal I on X and is denoted by (X,τ,I).

For an ideal topological space (X,τ,I) , a set operator(.)∗ : P(X) → P(X), which will be said the local function ofA ⊆ X,A∗(I, τ) = {x ∈ X : A∩U /∈ I for everyU ∈ τ(x)}whereτ(x) = {U ∈

τ : x ∈ U}[6]. We simply writeA∗ in case there is no chance for confusion. A kuratowski closure operatorcl∗(I, τ)called the * topology which is finer thanτ is defined bycl∗(A) =A∪A∗[6]. X∗ is often a proper subset ofX.The hypothesisX =X∗[3] is equivalent to hypothesisτ∩I =ϕ.

Definition 2.1 (5). Let (X,τ, I) be an ideal topological space and A a subset of X .ThenA∗S(I, τ) = {x ∈

X|A∩U /∈Ifor everyU ∈SO(X, x)}is called the regular-local function of A with respect to I andτ, where

SO(X, x) ={U ∈SO(X)|x∈U}. This semi-local functionA∗S(I, τ)is simply denoted byA∗S.

Definition 2.2. A subset A of a space (X,τ) is called a regular open set [13] if A = int(cl(A)) and a regular

closed if A = cl(int(A)). The collection of all regular open sets is denoted by RO(X). This collection does

not form a topology, since arbitrary union of regular open sets may not be a regular open set in general. The

regular-closure of A in (X,τ) is defined by the intersection of all regular-closed sets containing A and is denoted

by rcl(A).

Definition 2.3. A subset A of a space (X,τ) is called

1. regularα-open [14] if there is a regular open setU such thatU ⊂A⊂αcl(U). 2. semi-open[7] ifA⊂cl(int(A)).

Definition 2.4. A subset A of (X,α,I) is said to be

1. ∗-closed[4] ifA∗ ⊂A.

2. ∗s_{-closed[5] ifA}∗s ⊂A.

3. Irg-closed[9] ifA∗⊂U wheneverA⊂U andU is regular-open inX.

4. Igb-closed[1] ifA

∗_{⊂U wheneverA⊂U andU is semi-open inX.}

5. RgI-closed[11] ifA∗S ⊂U wheneverA⊂U andU is regular open inX.

6. Ig-closed [10] ifA∗ ⊂U wheneverA⊂U andU is open inX.

3

Definition 3.1. Let (X,τ, I) be an ideal topological space and A a subset of X .Then A∗R(I, τ) = {x ∈

X|A∩U /∈ I for everyU ∈ RO(X, x)} is called the regular-local function of A with respect toI andτ , whereRO(X, x) ={U ∈RO(X)|x∈U}. This regular-local functionA∗R(I, τ)is simply denoted byA∗R.

Theorem 3.1. Let (X,τ, I) be an ideal topological space and A a subset of X. ThenA∗(I, τ)⊆A∗R(I, τ)for every subset A of X.

Proof. Letx∈A∗,we claim thatx∈A∗R. If not, then there is a regular open set U containing x, such

thatU∩A∈I, since every regular open is open, U is open . Then by hypothesis,U∩A /∈I which is

a contradiction. Thereforex∈A∗R. This implies thatA∗(I, τ)⊆A∗R(I, τ). Remark 3.1. Let (X,τ, I) be an ideal topological space and A a subset of X. Then

(i). A∗(I, τ) =A∗R(I, τ)ifRO(X, x) =τ. (ii). If A∈I, thenA∗R =ϕ.

(iii). NeitherA⊂A∗RnorA∗R ⊂Ain general.

Example 3.1. LetX ={a, b, c, d}, τ ={ϕ, X,{a},{b, c},{a, b, c}}, I ={ϕ,{b}}. HereA ={a, b}and

A∗R ={a, c}implies that A⊂A∗R andA∗R ⊂A.

Theorem 3.2. Let (X,τ, I) be an ideal topological space and A, B subsets of X. Then the following properties

hold:

(i). (ϕ)∗R=ϕ

(ii). If A⊂B, thenA∗R ⊂B∗R.

(iii). A∗R =rcl(A∗R)⊂rcl(A)andA∗R is an regular closed subset ofrcl(A). (iv). (A∗R)∗R ⊂A∗R.

(v). (A∪B)∗R =A∗R ∪B∗R. (vi). (A∩B)∗R ⊂A∗R ∩B∗R.

(vii). A∗R−B∗R = (A−B)∗R−B∗R ⊂(A−B)∗R.

(viii). IfU ∈RO(X, x), thenU∩A∗R =U ∩(U ∪A)∗R ⊂(U∪A)∗R. (ix). ifU ∈I, then(A−U)∗R ⊂A∗R = (A∪U)∗R.

Remark 3.2. Let (X,τ,I) be an ideal topological space andA⊂X. Thencl∗R(A) =A∪A∗R is a Kuratowski closure operator. Therefore,τ∗R =

n

X−A/cl∗R(A) =A

o

is a topology on X finer than RO(X).

Theorem 3.3. Let (X,τ,I) be an ideal topological space and A, B subsets of X. Then for a local function the

following properties hold:

(i). For another idealJ ⊃IonX, A∗R(J)⊂A∗R(I). (ii). A∗R(I1∩I2) =A∗

R

(I1)∪A∗ R

(I2).

Proof. Obvious.

Theorem 3.4. Let (X,τ,I) be an ideal topological space, Thencl∗R(A) =A∪A∗Rand A, B subsets of X. Then (i). cl∗R(ϕ) =ϕ.

(ii). A⊂cl∗R(A).

(iii). cl∗R(A∪B) =cl∗R(A)∪cl∗R(B).

(iv). For anyA⊆X, cl∗R(cl∗R(A)) =cl∗R(A∪A∗R). Proof. Obvious.

Remark 3.3. The following example shows thatτ∗R-closed andτ-closed are independent of each other .

Example 3.2. LetX ={a, b, c, d},τ ={ϕ, X,{a},{b},{a, b},{a, b, c}},I ={ϕ,{a}}. Here

τ -closed ={ϕ, X,{d},{c, d},{a, c, d},{b, c, d}} andτ∗R-closed ={ϕ, X,{a},{c, d},{a, c, d},{b, c, d}}.

HereA={d} ∈τ -closed, butA /∈τ∗R -closed andB ={a} ∈τ∗R-closed, butB /∈τ -closed.

Remark 3.4. (i). Everyτ∗-open is finer thanτ∗R-open.

(ii). Everyτ∗S-open is finer thanτ∗R-open.

Example 3.3. LetX ={a, b, c, d},τ ={ϕ,{a},{b},{a, b}, X},I ={ϕ,{a}}.

τ∗={ϕ, X,{a},{b},{a, b},{b, c, d}}.

τ∗S =nϕ, X,{a},{b},{c},{d},{a, b},{a, c},{a, d},{b, c},{b, d},{c, d},{a, b, c},{a, c, d},

{a, b, d},{b, c, d}oandτ∗R ={ϕ, X,{a},{b},{a, b},{b, c, d}}.

4

Definition 4.1. Let (X,τ, I) be an ideal topological space. Thenτ is said to be a R-compatible with respect to

I, denoted byτ RIif and only if, for every x∈A there exists U∈RO(X, x) such thatU∩A∈I, then A∈I.

(i). τ RI.

(ii). If A subset A of X has a cover of regular open sets each of whose intersection with A is in I, then A∈I.

(iii). For everyA⊂X, if A∩A∗R =ϕ, A∈I.

(iv). For everyA⊂X, A−A∗R ∈I.

(v). For every A⊂X, if A contains no non-empty subset B withB ⊂B∗R,thenA∈I.

Proof. Obvious.

Theorem 4.2. Let (X, τ, I) be an ideal topological space. If τ is R-compatible with I, then the following

properties are equivalent.

(i). For everyA⊂X, A∩A∗R =ϕimplies thatA∗R =ϕ

(ii). For everyA⊂X,(A−A∗R)∗R =ϕ.

(iii). For everyA⊂X,(A∩A∗R)∗R =A∗R.

Proof. Obvious.

Theorem 4.3. Let (X,τ, I) be an ideal topological space, then the following properties are equivalent.

(i). τR∩I =ϕ.

(ii). ifJ ∈I, then rint(I) =ϕ.

(iii). For every G∈τR, G⊆G∗R.

(iv). X =X∗R.

Proof. Obvious.

Theorem 4.4. Let (X,τ, I) be an ideal topological space andτ be R-compatible with I. Then for everyU ∈τ∗R

and any subset A of X,(U ∩A)∗R = ((U∩A)∗R)∗R =rcl(U ∩A∗R).

5

Definition 5.1. Let (X,τ, I) be an ideal topological space. A subset A of (X,τ, I) is said to be I-grαclosed if

A∗R ⊆U wheneverA⊆U andU is regularα-open.

Definition 5.2. Let (X,τ, I) be an ideal topological space andA ⊆X. Then A is said to be a I-grαopen if

X−Ais I-grαclosed.

Theorem 5.1. In an ideal space (X,τ,I), the union of I-grαclosed sets is an I-grαclosed set.

Proof. Let A and B be I-gr α closed sets in (X, τ, I). Suppose A ∪B ⊆ U and U is regular α -open. Then A⊆ U andB ⊆ U. By hypothesisA∗R ⊆ U andB∗R ⊆ U. Therefore,cl∗R(A∪B) =

cl∗R(A)∪cl∗R(B)⊆U. Then(A∪B)∗R =A∗R∪B∗R.

Remark 5.1. Intersection of two I-grαclosed sets need not be I-grαclosed as seen in the following example.

Example 5.1. LetX ={a, b, c, d}, τ ={ϕ, X,{a},{b},{a, b},{a, b, c}},I ={ϕ,{a}}.

τ∗R -closed ={ϕ, X,{a},{c, d},{a, c, d},{b, c, d}}and

I−grαclosed ={ϕ, X,{a},{a, b},{c, d},{a, b, c},{a, b, d},{a, c, d},{a, b, d}}. LetA = {a, b}andB =

{b, c, d}, butA∩B ={b}∈/ I−grαclosed.

Theorem 5.2. Everyτ∗R -closed is I-grαclosed.

Proof. Obvious.

Theorem 5.3. (i). EveryI-grαclosed set is RgI-closed.

(ii). EveryI-grαclosed set isIrg-closed.

Proof. Obvious.

Remark 5.2. Converse of the above need not be true as shown in the following example.

Example 5.2. LetX ={a, b, c, d},τ ={ϕ, X,{a},{b},{a, b}},I ={ϕ,{a}}.

τ∗R-closed ={ϕ, X,{a},{c, d},{a, c, d},{b, c, d}},

Irg

-closed={ϕ, X,{a},{c},{d},{a, b},{a, c},{a, d},{b, c},{b, d},{c, d},{a, b, c},{a, b, d},{a, c, d},{b, c, d}},

RgI-closed={ϕ, X,{a},{b},{c},{d},{a, b},{a, c},{a, d},{b, c},{b, d},{c, d},{a, b, c},{a, b, d},{a, c, d},{b, c, d}}

andI-grαclosed={ϕ, X,{a},{a, b},{c, d},{a, b, c},{a, b, d},{a, c, d},{b, c, d}}.

Remark 5.3. (i). The concept ofIg-closed andI-grαclosed are independent of each other.

Example 5.3. LetX ={a, b, c, d},τ ={ϕ, X,{a},{b},{a, b}},I ={ϕ,{a}}.

Ig-closed ={ϕ, X,{a},{d},{a, d},{b, d},{c, d},{a, b, d},{a, c, d},{b, c, d}},

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

Theorem 5.4. Let (X,τ, I) be an ideal space and A⊆X. If A is I-grα closed, thencl∗RA−A contains no

non-empty regularα-open set.

Proof. Obvious

Theorem 5.5. Let (X,τ, I) be an ideal space and A∈I , A is I-grαclosed.

Proof. Let A⊆U, where U is regularα-open. SinceA∗R =ϕfor every A∈I, thencl∗RA=A∪A∗R =

A⊆U. Therefore A is I-grαclosed.

Theorem 5.6. Let (X,τ, I) be an ideal space, thenA∗R is always I-grαclosed for every subset A of X.

Proof. Obvious.

Theorem 5.7. Let (X,τ, I) be an ideal space, then every I-grαclosed, regularα-open isτ∗R closed.

Proof. Obvious.

Theorem 5.8. Let (X,τ , I) be an ideal space and A be an I-grαclosed set. Then the following are equivalent.

(i). A isτ∗R closed.

(ii). cl∗RA−Ais regularα-closed.

(iii). A∗R−Ais regularα-closed.

Proof. Obvious.

Theorem 5.9. Let (X,τ, I) be an ideal space and A⊆X. Then A is I-grαclosed if and only if A =F - N where

F isτ∗R-closed and N contains no nonempty regularα-closed set.

Proof. Obvious.

Theorem 5.10. Let (X,τ, I) be an ideal space . If A and B are subsets of X such thatA⊆B ⊆cl∗RAand A

is I-grαclosed, then B is I-grαclosed.

Proof. Since A is I-grα closed, cl∗RA−A contains no nonempty regularα -closed set, B is I-gr α

closed.

Theorem 5.11. Let (X,τ , I) be an ideal space and A⊆X. Then A is I-grαopen if and only if F⊆int∗R(A)

Proof. Obvious.

Theorem 5.12. Let (X,τ, I) be an ideal space and A⊆X. If A is I-grαopen andint∗R(A)⊆B ⊆A, then B is I-grαopen.

Proof. Since A is I-grαopen, thenX−Ais I-grαclosed.cl∗R(X−A)−(X−A)contains no nonempty regularα-closed set. Sinceint∗R(A)⊆int∗R(B), which implies that B is I-grαclosed.

Theorem 5.13. Let (X,τ, I) be an ideal space and A⊆X. Then the following are equivalent.

(i). A is I-grαclosed.

(ii). A∪(X−A∗R)isI-grαclosed. (iii). A∗R −AisI-grαopen.

Proof. Obvious.

Theorem 5.14. Let (X,τ, I) be an ideal space, then every subset of X is I-grα closed if and only if every

regularα-open set isτ∗R closed.

Proof. Suppose every subset of X is I-grαclosed. If U⊆X is regularα-open, then U is I-grαclosed

and soU∗R ⊆U. Hence U isτ∗R closed. Conversely, suppose that every regularα-open set isτ∗R

closed. If U is regular α -open set such that A ⊆ U ⊆X, thenA∗R ⊆ U∗R ⊆ U and so A is I-grα

closed.

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Received: April 12, 2015;Accepted: May 23, 2015

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