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On πg*β-Closed Sets in Topological Spaces
ISSN: 2168-9679

Journal of Applied & Computational Mathematics
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On πg*β-Closed Sets in Topological Spaces

Devika A and Vani R*
Department of Arts and Science, Faculty of Arts and Science, PSG College of Arts and Science, Avinashi Road, Civil Aerodrome Post, Coimbatore, Tamil Nadu, India
*Corresponding Author: Vani R, Department of Arts and Science, Faculty of Arts and Science, PSG College of Arts and Science, Avinashi Road, Civil Aerodrome Post, Coimbatore, India, Tel: 0422 430 3300, Email: [email protected]

Received Date: May 30, 2017 / Accepted Date: Jul 18, 2018 / Published Date: Jul 23, 2018

Keywords: πg*β-Open set; πg*β-Closed set; g-Neighbourhoods; πg*β-Interior; πg*β-Closure 2010 AMS Mathematics subject classification: 54A05

Introduction

The study of g-closed sets in a topological space was initiated by Andrijevi [1]. Arya and Nour [2] introduced g*-closed sets. Aslim [3] introduced the concepts ofπ-closed sets. Dontchev [4] and Dontchev and Noiri [5] introduced πg-closed sets. Gnanambal [6] and Janaki [7] introduce and study the πgβ-closed sets. The aim of this paper, is to introduce and study the concepts of πg*β-closed sets [8-10], πg*β- open sets in topological spaces and obtain some of their properties [11-15]. Also, we introduce πg*β-neighbourhood (briefly πg*β-nbhd) in topological spaces by using the notion of πg*β-open sets. Further we have prove that every nbhd of x in X is g*β-nbhd of x but not conversely [16-20].

Preliminaries

Let us recall the following definitions which we shall require in sequal.

Definition

A subset A of a topological space (X, τ) is called

1. A pre-open set [16] if A⊆int(cl(A)) and a pre-closed set if cl(int(A)) ⊆A.

2. A semi-open set [9] if A⊆ cl(int(A))and a semi-closed set if int(cl(A)) ⊆A.

3. An-open set [11] if A⊆ int(cl(int(A))) and an-closed set if cl(int(cl(A))) ⊆A.

4. A semi-pre open set(β-open) [1] if A⊆cl(int(cl(A))) and a semi-pre closed set(=β-closed) if int(cl(int(A))) ⊆A.

5. A regular open set [17] if A=int(cl(A)) and a regular closed set if A=cl(int(A)).

6. π-closed [20] if A is the union of regular closed sets.

The intersection of all semi-closed (resp.pre-closed, semi-preclosed, regular-closed and-closed) sets containing a subset A of (X,τ) is called the semi-closure (resp.pre-closure, semi-pre-closure, regular-closure and α-closure) of A and is denoted by scl(A) (resp. pcl(A),spcl(A), rcl(A) and cl(A)).

Definition

A subset A of a topological space(X,τ) is called

1. A regular generalized closed set (briefly rg-closed) [13] if cl(A)⊆U whenever A⊆U and U is regular open in (X,τ).

2. Aπ generalized closed set (briefly πg-closed) [5] if cl(A) ⊆U whenever A⊆U and U is-open in (X,τ).

3. A π generalized α closed set (briefly πgα-closed) [7] if αcl(A) ⊆U whenever A⊆U and U is π-open in (X,τ).

4. A π generalized regular closed set (briefly πgr-closed) [8] if rcl(A)⊆U whenever A⊆U and U is π-open in (X,τ).

5. A generalized preclosed set (briefly πgp-closed) [14] if pcl(A)⊆U whenever A⊆U and U is-open in (X,τ).

6. A π generalized semi-closed set(briefly πgs-closed) [3] if scl(A)⊆U whenever A⊆U and U is π-open in (X,τ).

7. A π generalized β closed set(briefly πgβ-closed) [15] if β cl(A)⊆U whenever A⊆U and U is π-open in (X,τ).

8. A generalized preregular closed set(briefly gpr-closed) [6] if pcl(A)⊆U whenever A⊆U and U is regular open in (X,τ).

9. A αgeneralized regular closed set(briefly αgr-closed) [19] if αcl(A)⊆U whenever A⊆U and U is regular open in (X,τ).

10. A regular generalized β closed set(briefly rgβ-closed) [15] if βcl(A)⊆U whenever A⊆U and U is regular open in (X,τ).

11. A regular w generalized closed set(briefly rwg-closed) [11] if cl(int(A))⊆U whenever A⊆U and U is regular open in (X,τ).

πg*β-Closed Sets

In this section, we introduce a new class of sets called πg*β-open sets, πg*β-closed sets and study some of its properties.

Definition

A subset A of a topological space (X,τ) is called πg*β-closed set if βcl(A)⊆U whenever A⊆U and U is πg-open in (X,τ).

Theorem

Every r-closed set is πg*β-closed.

Proof: Let A be r-closed set in X. Let U be a πg-open set such that A⊆U. Since A is r-closed, we have rcl(A)=A⊆U. But, βcl(A)⊆rcl(A)⊆U. Therefore βcl(A)⊆U. Hence A is a πg*β-closed set in X.

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

Example: Let X={a; b; c} and τ ={θ,{a}},{a,b},{{a,c},X}.Let πg*β- closed set={θ,{b},{c},{{b,c},X} and γ-closed set={θ,X}.Let A={b}. Then the subset A isπg*β-closed but not a γ-closed set.

Remark: The following diagram shows the relationship of πg*sclosed set with other known existing sets (Figure 1).

applied-computational-mathematics-conversely

Figure 1: A→B represents A implies B but not conversely.

Example: Let X={a,b,c} with τ={θ,{b},{c},{b,c},X}. Then πg*β- closed set={θ,{a},{b},{c},{a,c},{a,b},X},πg-closed, πgα-closed and πgγ- closed={θ,γ-closed, γg-closed, πgp-closed, πgs-closed and πgs-closed and πgβ-closed set={θ,{a},{b},{c},{a,b},{b,c},{a,c},X}. Let A={a}. Then the subset A is gs-closed, sg-closed, gp-closed, gsp-closed, gr-closed, gp-closed, gs-closed and πgβ-closed set but not πg*s-closed set.

Example: Let X={a,b,c} with τ={θ,{a},{b},{a,b}X}.Then πg*s-closed set={θ,{a},{b},{c},{a,c},{b,c},X}, π-closed, γg-closed, αg-closed, πgclosed, πgα-closed={θ,{c},{b,c},{a,c}X} and γwg={θ,{a,b},b,c},{a,c} X}. Let A={a}. Then the subset Ais πg*s-closed but not π-closed, γ-closed,αg-closed,πgα-closed and γwg-closed set.

Theorem

Union of two πg*β-closed subset is πg*β closed.

Proof: Let A and B be any two πg*β-closed sets in X such that A⊆U and B⊆U where U isπg-open in X and so A∪B⊆U. Since A and B are πg*β-closed. A⊆βcl(A) and B⊆βcl(A) and hence A∪B⊆βcl(A)∪βcl(B)⊆βcl(A∪B). Thus, A∪B is πg*β-closed set in (X,τ).

Example: Let X={a;b;c} and τ={θ,{c},{a,c}{b,c},X}. Let A={a} and B={b} then A∪B ={a}∪{b}={a;b} is πg*β closed set.

Theorem

Intersection of two πg*β-closed subset is πg*β closed.

Proof: Let A and B be any two πg*β-closed sets in X such that A⊆U and B⊆U where U is πg-open in X and so A∩B⊆U. Since A and B are πg*β-closed. A⊆cl(A) and B⊆cl(A) and hence A∩B⊆βcl(A)∩βcl(B)⊆ βcl(A∩B). Thus, A∩B is πg*β-closed set in (X,τ).

Example: Let X={a;b;c} and τ={θ,{a},{c},{a,c},X} . Let A={a;b}and B={b;c} then A∩B ={a;b}∩{b} is a πg*β closed set.

Theorem

A subset A of X is πg*β-closed if f βcl(A)-A contains no non-empty closed set in X.

Proof: Let A be a πg*β-closed set. Suppose F is a non-empty closed set such that F⊆βcl(A)-A. Then F ⊆βcl(A)∩Ac, since βcl(A)-A= βcl(A)∩Ac. Therefore F⊆βcl(A) and F⊆Ac. Since F⊆Ac is open, it is πg-open. Now, by the definition πg*-closed set, βcl(A)⊆ Fc, That is F⊆[βcl(A)]c. Hence F⊆cl(A)∩[βcl(A)]c=θ. That is F=θ, which is a contradiction. Thus, βcl(A)-A contains no non-empty closed set in X.

Conversely, assume that βcl(A)-A contains no non-empty closed set. Let A⊆U, where U is πg-open. Suppose that βcl(A) is not contained in U, then βcl(A)∩Uc is a non-empty closed subset of βcl(A)-A, which is a contradiction. Therefore βcl(A)⊆U and hence A is πg*β-closed.

Theorem

For any element xεX. The set X is πg*β closed set or πg-open.

Proof: Suppose X{x} is not πg-open, then X is the only πg-open set containing X{x}.This implies βclX{x}⊂X. Hence X{x} is πg*β closed or πg-open set in X.

Theorem

If A is an πg*β closed subset of X such that A⊂ B⊂ βcl(A) then B is an πg*β closed set in X.

Proof: Let A be an πg*β closed set of X such that A⊂ B⊂ βcl(A). Let U be a πg-open set of X such that B⊂U, then A⊂U. Since A is πg*β- closed, we have βcl(A)⊂U. Now, βcl(B)⊂βcl(β cl(A))⊂U, therefore B is an πg*βclosed set in X.

Definition

A subset A of a topological space (X,τ) is called πg*β-open set if and only if Acis πg*β-closed in (X,τ).

Theorem

Let A⊂X is πg*β-open if and only if F⊂int(A),where F is πg-open and F⊆A.

Proof: Let A be a πg*β-open set in X. Let F be πg-closed set and F⊂A. Then X-A⊂X-F, where X-F is πg-open, since X-A is πg*β closed, βcl(X-A)⊂X-F. Therefore βcl(X-F)=X-int(A)⊂X-int(A)⊂X-F, i.e.,) F⊂ int(A). Conversely, suppose F is πg-closed and F⊂A implies F⊂int(A). Let X-A⊂U, where U is πg-open. Then X-U⊂A, where X-U is πgclosed, By hypothesis, X-U ⊂int(A), i.e.,) X-int(A)⊂U since βcl(XA)= X-int(A), βcl(A) U, where U is πg-open this implies X-A is πg*β- closed and hence A is πg*β-open.

Theorem

If int(A)⊂B⊂A and A is πg*β-open then B is also πg*β open.

Proof: We know that if A is πg*β-closed and A⊂B⊂βcl(A) then B is also πg*β-closed. Here X-A is πg*β-closed, then X-B is also πg*β-closed. Hence B is πg*β-open.

Theorem

If A⊂X is πg*β-closed then βcl(A)-A is πg-open.

Proof: Let A be a πg*β-closed set in X. Let F be a πg-closed set such that F⊂βcl(A)-A. Then βcl(A)-A does not contain any non-empty πgclosed set. Therefore F=θ, so F⊂int(βcl(A)-A). This shows βcl(A)-A is πg-open. Hence A is πg*β-closed in (X,τ).

Theorem

If int(B)⊆B⊆A and if A is πg*β-open in X, then B is πg*β-open in X.

Proof: Suppose that int(B)⊆B⊆A and A is πg*β-open in X then Ac⊆Bc⊆cl(Ac). Since Ac is πg*β-closed in X. we have B is πg*β-open in X.

Theorem

If A is γwg-open and πg*β-closed then A is πg-closed.

Proof: Let A be a γwg-open and πg-closed set in X. Let A⊂A where A is wg-open. Since A is πg*β closed; βcl(A)⊂A whenever A⊂A and A is wg-open. the implies βcl(A)=γwg. Hence A is πg-closed.

Theorem

If A is wg-open and πg*β-closed then A is πg-closed.

Proof: Let A be a wg-open and πg*β-closed set in X. Let A⊂A where A is wg-open. Since A is πg*β-closed; βcl(A)⊂A whenever A⊂A and A is wg-open. the implies βcl(A)=γwg. Hence A is πg-closed.

g-Neighbourhoods

Definition

Let X,τ) be a topological space and let xεX, A subset N of X is said be πg*β-neighbourhood of x if there exists an πg*β-open set G such that x εG⊆N. The collection of all πg*β-neighbourhood of xεX is called πg*β neighbourhood system at x shall be denoted by πg*β-N(X).

Theorem

Every neighbourhood N of xεX is πg*β-neighbourhood of X.

Proof: Let N be a neighbourhood of point x εX, To prove that N is a πg*β-neighbourhood of x by definition of neighbourhood, there exists an open set G, such that x εG⊆ N. Hence N is πg*β-neighbourhood of X.

Remark: In general, a πg*β-neighbourhood N of xεX need not be a nbhd of x in X as seen from the following example.

Example: Let X={a;b;c} with topology τ={θ ;X; {a};{a;c}}. Then πg*β-o(X)={θ;X;{b}; {c}; {b; c}}. The set {a;b} is πg*β-nbhd of point b, since the πg*β-open set {b} is such that bε{b}⊂{a;b}. However the set {a; b} is not a nbhd of the point b, since no open set G exists such that bεG ⊂{a;b}.

Theorem

If a subset N of a space X is πg*β-open, then N is πg*β-nbhd of each of its points.

Proof: Suppose N is πg*β-open. Let xεN. We claim that N is πg*β- nbhd of x. For N is a πg*β-open set such that xεN ⊆N. Since x is an arbitrary point of N, it follows that N is a πg*β-nbhd of each of its points.

Theorem

Let X be a topological space. If F is a πg*β-closed subset of X, and xεF c: P rove that there exists a πg*β-nbhd N of x such that N∩F=θ .

Proof: Let F be πg*β-closed subset of X and xεFc: T hen Fc is πg*β- open set of X. So by Theorem 4.5 Fc contains a πg*β-nbhd of each of its points. Hence there exists a πg*β-nbhd N of x such that N⊂Fc: That is N∩F=θ.

πg*β-Interior

Definition

Let A be a subset of X. A point xεX is said to be πg*β-interior point of A if A is a πg*β-nbhd of x. The set of all πg*β-interior points of A is called the πg*β-interior of A and is denoted by πg*β-int(A).

Theorem

If A be a subset of X. Then πg*β-int(A)=∪{G:G is πg*β-open,G⊆A}.

Proof : Let A be a subset of X:xε πg*β-int(A)⇔x is a πg*β-interior point of A.

image A is a πg*β-nbhd of point x.

image There exists πg*β-open set G such that x ε G⊆A.

image x ε∪{G:G is πg*β-open, G⊆A.

Hence πg*β-int(A)=∪{G: G is πg*β-open, G⊆A}:

Theorem

Let A and B be subsets of X. Then

1. πg*β-int(X)=X and πg*β-int(θ)=θ .

2. πg*β-int(A)⊆A.

3. If B is any πg*β-open set contained in A, then B⊆πg*β-int(A).

4. If A⊆B, then πg*β-int(A)⊆ πg*β-int(B).

5. πg*β-int(π πg*β-int(A))=π πg*β-int(A).

Proof: 1. Since X and θ are πg*β-open sets, by Theorem πg*β- int(X)= ∪{G:G is πg*β-open, G⊆X}=X∪{all fall πg*β-open sets}=X. That is πg*β-int(A)=X. Since θ is the only πg*β-open set contained in θ, πg*β-int(θ)=θ .

2. Let xε πg*β-int(A))⇒x is a πg*β-interior point of A.

image A is a πg*β-nbhd of x.

image XεA. Thus x ε πg*β-int(A))⇒ x εA. Hence πg*β-int(A) A.

3. Let B be any πg*β-open sets such that B⊂A. Let x εB, then since B is a πg*β-open set contained in A. x is a πg*β-interior point of A. That is xε πg*β-int(A). Hence B⊆πg*β-int(A).

4. Let A and B be subsets of X such that A⊆B. Let xε πg*β-int(A). then x is a πg*β-interior point of A and so A is πg*β-nbhd of x. Since B⊃A, B is also a πg*β-nbhd of x. This implies that x ε πg*β-int(B). Thus we have shown that x ε πg*β-int(A)) xε πg*β-int(B). Hence πg*β- int(A)⊂ πg*β-int(B).

5. From (2) and (4) πg*β-int(πg*β-int(A)) ⊆πg*β-int(A). Let x ε πg*β-int(A) this implies A is a neighbourhood of x, so there exists a πg*β-open set G such that x εG⊆A. so every element of G is an πg*β- interior of A, hence xεG⊆πg*β-int(A) which means that x is an πg*β- interior point of πg*β-int(A) that is πg*β-int(A)⊆ πg*β-int(πg*β-int(A)). That is πg*β-int(πg*β-int(A))= πg*β-int(A). Let A be any subset of X. By the definition of πg*β-interior πg*β-int(A)⊂A, by πg*β-int(πg*β- int(A)) ⊂ πg*β-int(A). Hence πg*β-int(πg*β-int(A))⊂∩{F:A⊂Fε πg*β- C(X)}=πg*β-cl(A).

Theorem

If a subset A of space X is πg*β-open, then πg*β-int(A)=A.

Proof: Let A be πg*β-open subset of X. πg*β-int(A)⊂A. Also, A is πg*β-open set contained in A. From (3) A⊂πg*β-int(A). Hence πg*β- int(A)=A.

Remark: The converse of the above theorem need not be true, as seen from the following example.

Example: Let X={a;b;c} with topology τ={θ, {c}; {b;c}; X}. πg*β- closed set is {θ, {a}, {b},{a;b}, X}. πg*β-O(X) is πg*β-open sets in X={θ, {c}; {b;c}; {a;c}; X} πg*β-int(A)=πg*β-int({a;b})={a}∪{b}={a;b}; but {a;b} is not g-open set.

Theorem

If A and B are subsets of X, then πg*β-int(A)∪ πg*β-int(B)⊂ πg*β- int(A∪B).

Proof: Theorem πg*β-int(A)⊂ πg*β-int(A∪B) and πg*β-int(B)⊂ πg*β-int(A∪B). This implies that πg*β-int(A)∪ πg*β-(B)⊂ πg*β- int(A∪B).

g-Closure in a Space X

Definition

Let A be a subset of a space X. The πg*β-closure of A is de ned as the intersection of all πg*β-closed sets containing A. πg*β-cl(A)=∩{F:A⊂Fε πg*βC(X)}.

Theorem

If A and B are subsets of a space X. Then

(1) πg*β-cl(X)=X and πg*β-cl(θ)=θ .

(2) A⊂πg*β-cl(A).

(3) If B is any πg*β-closed set containing A, then πg*β-cl(A)⊂B.

(4) If A⊂B, then πg*β-cl(A)⊂ πg*β-cl(B).

(5) πg*β-cl(A)= πg*β-cl(πg*β-cl(A)).

Proof: (1) By the definition of πg*β-closure, X is the only πg*β- closed set containing X. Therefore πg*β-cl(X)=Intersection of all the πg*β-closed sets containing X=∩{X}=X: That is πg*β-cl(X)=X. By the definition of πg*β-closure, πg*β-cl(θ)=Intersection of all the πg*β closed sets containing θ=∩ any πg*β-closed sets containing θ=θ . That is πg*β- cl(θ)=θ .

2. By the definition of πg*β-closure of A, it is obvious that A⊂πg*β- cl(A).

3. Let B be any πg*β-closed set containing A. Since πg*β-cl(A) is the intersection of all g-closed sets containing A, πg*β-cl(A) is contained in every πg*β-closed set containing A. Hence in particular πg*β-cl(A)⊂B.

4. Let A and B be subsets of X such that A⊆B. By the definition of πg*β-closure, πg*β-cl(B)=∩{F: B ⊂επg*βC(X)g. If B ⊂F ε πg*β C(X), then πg*β-cl(B) F. Since A⊂B, A⊂B ⊂Fε πg*β C(X), πg*β-cl(A)⊂F. Therefore πg*β-cl(A) ⊂∩{F: B⊂Fε πg*β C(X)}= πg*β-cl(B). That is πg*β-cl(A)⊂ πg*β-cl(B).

5. Let A be any subset of X. By the definition of πg*β-closure, πg*β- cl(A)= ∩{F: A⊂ Fε πg*β C(X)g, If A ⊂F ε πg*β C(X), then πg*β-cl(A) F. Since F is πg*β-closed set containing πg*β-cl(A), by (3) πg*β-cl(πg*β- cl(A))⊂F. Hence πg*β-cl(πg*β-cl(A)) ⊂∩{F: A⊂ Fε πg*β C(X)}= πg*β-cl(A). that is πg*β-cl(πg*β-cl(A))=(A).

Theorem

If A⊂X is πg*β-closed, then πg*β-cl(A)=A.

Proof: Let A be πg*β-closed subset of X. By the definition of πg*β- cl(A), A⊂πg*β-cl(A). Also A⊂A and A is πg*β-closed. By Theorem πg*β-cl(A)⊂A. Hence πg*β-cl(A)=A.

Remark: The converse of the above theorem need not be true as seen from the following example.

Example: Let X={a;b;c} with topology τ={θ, {a}, {a;b}, {a;c}, X}. πg*β-closed set is {θ, {b}, {c}, {b; c}, X} and πg*β-O(X)= πg*β-open sets in X={θ, {a} ; {a; b} ; {a; c} ; X}. πg*β-cl(A)=πg*β-cl({a}={a; b}∩{a; c}={a} but {a} is not πg*β-closed set.

Theorem

If A and B are subsets of a space X, Then πg*β-cl(A∩B)⊂ πg*β- cl(A)∩ πg*β-cl(B).

Proof: Let A and B be subsets of X. clearly A∩B⊂C and A∩B⊂B. by Theorem, πg*β-cl(A∩B)⊂πg*β-cl(A) and πg*β-cl(A∩B)⊂πg*β-cl(B). Hence πg*β-cl(A∩B)⊂ πg*β-cl(A)∩ πg*β-cl(B).

Conclusion

This paper is to introduced and study the concepts of πg*β-closed sets and πg*β-neighbour hood in topological spaces. We had proved that the defined set was properly contains πgβ-closed and contained in πg-closed set. Further the defined set satisfies the union and intersection property. Hence we conclude that the defined set forms a topology which results this work may be extend widely.

References

Citation: Devika A, Vani R (2018) On πg*β-Closed Sets in Topological Spaces. J Appl Computat Math 7: 413. DOI: 10.4172/2168-9679.1000413

Copyright: © 2018 Devika A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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