Hi v M*

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Journal of Taibah University for Science 7 (2013) 114-119

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More on y-generalized closed sets in topology

Ahmed I. El-Maghrabi *

Department of Mathematics, Faculty of Science, Kafr EL-Sheikh University, Kafr EL-Sheikh, Egypt

Available online 19 June 2013

Abstract

The aim of this paper is to introduce and study a new class of sets called y-generalized regular weakly closed (briefly, ygrw-closed) set. This new class of sets lies between the class of regular weakly closed (briefly, rw-closed) sets and the class of y-generalized closed (briefly, yg-closed) sets. Also, we study the fundamental properties of this class of sets. ©2013 Taibah University. Production and hosting by Elsevier B.V. All rights reserved.

MSC: 54A05; 54C10; 54D10

Keywords: Regular closed sets; y-regular closed sets; Weakly closed sets; y-generalized regular weakly closed sets and regular semi-kernal

1. Introduction and preliminaries

Throughout this paper (X, t) and (Y, a) (or simply, X and Y) represent the non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned. For a subset A of X, cl(A), int(A) and Acor X — A represent the closure of A, the interior of A and the complement of A, respectively. Every topological space can be defined either with the help of axioms sets. So, one can imagine that, how important the concept of closed sets is in the topological spaces. In 1970, Levine [14] initiated the study of so-called generalized closed sets. By definition of a subset A of a topological space X is called generalized closed (briefly, g-closed)

* Current address: Department of Mathematics, Faculty of Science, Taibah University, AL-Madiah AL-Munawarah, P.O. Box 30002, Saudi Arabia. Tel.: +966 560801219.

E-mail addresses: aelmaghrabi@yahoo.com, amaghrabi@taibahu.edu.sa

Peer review under responsibility of Taibah University.

1658-3655 © 2013 Taibah University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jtusci.2013.04.010

set if cl(A) c H whenever A c H and H is open in (X, t). This notion had been studied extensively in the recent years by many topologists since generalized closed sets are not only the generalization of closed sets. Moreover, they also suggest separation axioms weaker than T1 and some of them found to be useful in computer science and digital topology. Furthermore, the study of generalized closed sets also provides new characterizations of some known classes of spaces, for example, extremely disconnected spaces by Cao et al. [5]. In 2007, the notion of regular weakly closed set was defined by Benchalli et al. [3] and they proved that this class lies between the class of all ^-closed sets given by Sundaram et al. [22] and the class of all regular generalized closed sets defined by Palaniappan et al. [19]. In the present paper, we introduce and study a new class of sets called a y-generalized regular weakly closed (briefly, ygrw-closed) set in topo-logical spaces which is properly placed between the regular weakly closed sets and /-generalized regular closed sets.

Now, we recall the following definitions which are useful in the sequal.

Definition 1.1. A subset A of a space X is called:

(i) regular open [21] if A = int(cl(A)) and regular closed [21] if A = cl(int(A)),

(ii) a-open [18] if A c int(cl(int(A))) and a-closed if cl(int(cl(A)) c A,

(iii) preopen [17] if A c int(cl(A)) and preclosed [17] if cl(int(A)) c A,

(iv) semi-open [13] if A ccl(int(A)) and semi-closed [13] if int(cl(A)) c A,

(v) y-open [9] or b-open [1] or sp-open [8] if A c int(cl(A)) U cl(int(A)) and y-closed [9] or b-closed [1] or sp-closed [8] if cl(int(A)) n int(cl(A)) c A,

(vi) 5-closed [23] if S-cl(A)= A, where 5-cl(A) = {x e X: int(cl(H)) n A / H er and x e H}.

(vii) regular semi-open (briefly, rs-open)[4] if there is a regular open set H such that H c A c cl(H),

Definition 1.2. A subset A of a space X is called:

(i) generalized closed (briefly, g-closed)[14] if cl(A) c H whenever A c H and H is open in (X, t ),

(ii) generalized a-closed (briefly, ga-closed)[15] if a-cl(A) c H whenever A c H and H is a-open in (X, t ),

(iii) a-generalized closed (briefly, ag-closed)[15] if a-cl(A) c H whenever A c H and H is open in (X, t ),

(iv) regular generalized closed (briefly, rg-closed)[19] if cl(A) c H whenever A c H and H is regular open in (X, t),

(v) 5-generalized closed (briefly, 5g-closed)[7] if 5-cl(A) c H whenever A c H and H is open in (X, t ),

(vi) weakly closed (briefly, w-closed)[20] if cl(A) c H whenever A c H and H is semi-open in (X, t),

(vii) regular weakly generalized closed (briefly, rw-closed)[3] if cl(A) c H whenever A c H and H is regular semi-open in (X, t),

(viii) y-generalized closed (briefly, Yg-closed)[10] if y-cl(A) c H whenever A c H and H is open in (X, t ).

(vxi) y-generalized regular closed (briefly, Ygr-closed) if y-cl(A) c H whenever A c H and H is regular open in (X, t).

The family of all regular semi-open sets in a space X

is denoted by RSO(X).

Lemma 1.1 ([11]). For a space (X, t) and A is a subset ofX, then:

(i) every regular closed set, regular open set and clopen set is regular semi-open in X,

(ii) if A is a regular semi-open set, then X - A is also regular semi-open in X.

Lemma 1.2 ([3]). Let X be a topological space and Y be an open subspace of X such that A c Y c X. Then A e RSO(Y), if A eRSO(X).

Lemma 1.3 ([3]). If Y is regular open in X and H is a subset ofY, then His regular semi-open in X if and only if His regular semi-open in the subspace Y.

Definition 1.3 ([3]). The intersection of all regular semi-open subsets of (X, t) containing A is called the regular semi-kernel of A and is denoted by rs ker(A).

Lemma 1.4 ([12]). Ifx is a point of (X, t), then {x} is either nowhere dense or preopen.

Remark 1.1 ([6]). In the notation of Lemma 1.4, we may consider the following decomposition of a given topological space (X, t), namely X=Xi UX2, where Xi = {x e X: {x} is nowhere dense } andX2 = {x e X: {x} is preopen }.

Lemma 1.5 ([3]). For any subset A of (X, t), we have A c rs ker(A).

2. y-Generalized regular weakly closed sets

In this section, we introduce and study some basic properties of a new class of sets called y-generalized regular weakly closed (briefly, Ygrw-closed) sets.

Definition 2.1. Let (X, t) be a topological space and A be a subset of X. Then A is said to be:

(i) a Y-generalized regular weakly closed (briefly, Ygrw-closed) setif y-cl(A) c H whenever A c H and H is regular semi-open in (X, t),

(ii) a y-generalized regular weakly open (briefly, Ygrw-open) set if Ac is Ygrw-closed in X.

The family of all Ygrw-closed sets in a space X is denoted by yGRWC(X).

Theorem 2.1. For a topological space (X, t), thefol-lowing statements are hold:

(i) every closed (resp. 5-closed, w-closed, rw-closed, Y-closed) set of a topological space (X, t) is Ygrw-closed,

(ii) every ygrw-closed set of a topological space (X, t) is Ygr-closed.

Proof.

(i) We prove this point for the case of rw-closed. Let A be an arbitrary rw-closed in (X, т ) such that A ÇH and H be regular semi-open. Then by definition of rw-closed, we have cl(A) ç H. Since every closed set in a topological space (X, т) is preclosed, then pcl(A) Çcl(A). So, we have, pcl(A) Çcl(A) çH. Hence, A is ygrw-closed.

(ii) Let (X, т) be a topological space and A be a ygrw-closed subset of X such that A ç H, where H is regular open. Since every regular open set is regular semi-open, then by definition of ygrw-closed, we have pcl(A) ç H. Hence A is ygr-closed.

The converse of the above theorem may be not true as is shown by the following examples.

closed —»

closed —» it;—closed i

a—closed —» ga-

Example 2.1. Let X = {x, y, z, u} with topology т = {X, V, {У}, {x У}, {x y, z}}. Then:

(i) a subset A = {x, y} of a space X is ygrw-closed but it is neither closed nor ¿-closed,

(ii) a subset B = {z} of a space X is ygrw-closed but not w-closed,

(iii) a subset C = {x, y, u} of a space X is ygrw-closed but not y-closed,

(iv) a subset D = {x, z} of a space X is ygr-closed but not ygrw-closed.

Example 2.2. Let X = {x, y,z,u, v} with topology т = {X, v, {x, y}, {z, u}, {x, y, z, u}}. Then a subset A = {x} of a space X is ygrw-closed but not rw-closed.

Remark 2.1. We can see from the following example that a ygrw-closed set is independent of a-closed (resp. ga-closed, ag-closed, rg-closed, ¿g-closed, yg-closed).

1. the u},

2. the u},

3. the {x,

4. the u},

5. the u},

6. the u},

7. the u},

8. the {x,

closed sets in (X, т) are {X, ф, {u}, {z, u}, {x, z,

{y, z,

rg-closed sets in (X, т) are {X, ф, {u}, {z, u}, {x, {y, u}, {x, y, u}, {x, z, u}, {y, z, u}}, a-closed sets in(X, т) are {X, ф, {u}, {z}, {z, u}, z, u}, {y, z, u}},

ga-closed sets in (X, т) are {X, ф, {u}, {z}, {z, {x, z, u}, {y, z, u}},

ag-closed sets in (X, т) are {X, ф, {u}, {z}, {z, {x, u}, {y, u}, {x, y, u}, {x, z, u}, {y, z, u}}, ¿g-closed sets in (X, т) are {X, ф, {u}, {z, u}, {x, {y, u}, {x, y, u}, {x, z, u}, {y, z, u}}, yg-closed sets in (X, т) are {X, {u}, {z}, {z, {x, u}, {y, u}, {x, y, u}},

ygrw-closed sets in (X, т) are {X, {u}, {x, y}, y, z}, {x, y,, u}, {x, z, u}, {y, z, u}}.

According to the above discussion, we have the following diagram.

^—closed —> rg—closed vw—closed —» ^grw—closed ^gr—closed

closed ag—closed jg—closed

A ^ B means A implies B but not conversely, A ^B means A and B are independent of each other.

Remark 2.2.

(1) The intersection of two ygrw-closed sets of a topological space (X, т) need not be ygrw-closed. Let X = {x, y, z} with topology т = {X, ф, {x}, {y}, {x, y}}. Then A = {x, y} and B = {y, z} are ygrw-closed sets but their intersection A OB = {y} is not a ygrw-closed set.

(2) The union of two ygrw-closed sets of a topological space (X, т) need not be ygrw-closed. In Example 2.2, the two subsets A = {z} and B = {u} are ygrw-closed sets but their union A UB = {z, u} is not a ygrw-closed set,

(3) The difference of two ygrw-closed sets of a topological space (X, т) need not be ygrw-closed. In Remark 2.2 (1), the two subsets D = {y, z} and E = {z} are ygrw-closed sets but the difference D — E = {y} is not a ygrw-closed set.

Theorem 2.2. Let (X, т) be a topological space and A be a ygrw-closed subset ofX. Then ycl(A) — A does not contain any non empty regular semi-open set ofX.

Example 2.3. In Example 2.1, we have the following Proof. Suppose that H be a non empty regular families: semi-open set of X such that HCycl(A) —A. Hence

HCX — A or A CX — H, then by Theorem 1.2, X — H is regular semi-open. But, A is a ygrw-closed subset of X, hence ycl(A) c X — H this implies that HCX — ycl(A) and we know that Hc ycl(A). Therefore, H c [ycl(A) n (X — ycl(A))] = y this shows that H is empty set which is a contradiction. Then ycl(A) — A does not contain any non empty regular semi-open set of X.

In the following example, we show that the converse of the above theorem is not true. □

Example 2.4. In Remark 2.2 (1), if we take A = {x}, then ycl(A) — A = [(Xn {x, z}) — {x}] = {z} does not contain any non-empty regular semi-open, where A = {x} is not a ygrw-closed set of X.

Proposition 2.1. If (X, t) is a topological space and A is a Ygrw-closed subset ofX, then ycl(A) — A does not contain any non empty regular open set ofX.

Proof. The proof follows directly from the fact that every regular open set is regular semi-open. □

Proposition 2.2. If A is a Ygrw-closed subset ofX and A c B c y-cl(A), then B is a Ygrw-closed set ofX.

Proof. Let A be an Ygrw-closed subset of X such that A c B c y-cl(A) and H be a regular semi-open set of X such that B c H. Then A c H. But A is Ygrw-closed, then y-cl(A) c H, hence, y-cl(B) c y-cl(y-cl(A)) = y-cl(A) c H. Therefore B is ygrw-closed in X. □

Remark 2.3. The converse of Proposition 2.1 is not true. In Example 2.1, if A = {u} and B = {z, u}, then A, B are ygrw-closed sets subsets of X and A c B which is not subset in y-cl(A).

Theorem 2.3. Let (X, t) be a topological space and A be a ygrw-closed subset of X. Then the following statements are equivalent:

(i) A is y-closed,

(ii) ycl(A) — A is regular semi-open.

(i) =^(ii). Since A is y-closed, y-cl(A) = A and so ycl(A) — A = y which is regular semi-open in X.

(ii) ^(i). Suppose that ycl(A) — A is regular semi-open in X. But A is a ygrw-closed set of X, then by Proposition 2.1, ycl(A) — A does not contain any non empty regular semi-open set of X. Hence, ycl(A) — A = y, so, A is y-closed.

Theorem 2.4. Let (X, t) be a topological space. Then for x eX,the setX — {x} is ygrw-closed or regular semi-open.

Proof. If the set X — {x} is ygrw-closed or regular semi-open, then we are done. Now, suppose that X — {x} is not regular semi-open. Then X is the only regular semi-open set containing X — {x} and hence ycl(X — {x}) c X that is the biggest set containing all of its subsets. Therefore, X — {x} is a ygrw-closed set of X. □

Proposition 2.3. In a topological space (X, t),for each x e X, the singleton set {x} is either ygrw-open or regular semi-open.

Proof. Obvious from Theorem 2.4. □ 3. Some properties of ygrw-closed sets.

Theorem 3.1. Let A be a regular open and a ygrw-closed set of (X, t). Then A is y-clopen.

Proof. Suppose that A is a regular open and ygrw-closed set of (X, t). Since every regular open set is regular semi-open and A c A, y-cl(A) c A. Also, A c y-cl(A). Therefore, A = y-cl(A) this means that A is y-closed. But, A is regular open, hence, A is y-open. Therefore, A is y-clopen. □

Theorem 3.2. Let Abe a regular open and a rg-closed set of (X, t). Then A is a ygrw-closed.

Proof. Let H be a regular semi-open set of X such that A c H. But, A is a regular open and rg-closed set of (X, t), then by Theorem 3.1, y-cl(A) c A. Hence, y-cl(A) ch. So, A is a ygrw-closed. □

Proposition 3.1. If A is a regular semi-open and a Ygrw-closed set of (X, t), then A is a y-closed.

Proof. Obvious. □

Theorem 3.3. Let A be a regular semi-open and a Ygrw-closed set of (X, t). Suppose that F is a closed set ofX. Then A n F is a ygrw-closed set ofX.

Proof. Suppose that A is a regular semi-open and a Ygrw-closed set of (X, t). Then by Proposition 3.1, A is y-closed. But, F is closed set of X, hence, A n F is a

y-closed and therefore A n F is a ygrw-closed set of X. □

Theorem 3.4. Suppose that A is both regular open and Ygrw-closed set of (X, t). If B c A c X and B is a ygrw-closed set relatives to A, then B is a ygrw-closed set relative to X.

Proof. Let B c H and H be a regular semi-open set of X. But, B c A c X, then B c A n H. We need to prove that A n H is regular semi-open in A. Firstly, we prove that A n H is regular semi-open in X. Since A is open and H is semi-open in X, hence A n H is semi-open in X. Also, A is both regular open and ygrw-closed set of (X, t), then by Theorem 3.1, A is semi-closed in X. Since, every regular semi-open set is semi-closed, hence, H is semi-closed in X. Therefore, A n H is semi-open in X. Thus, A n H is both semi-open and semi-closed in X and hence A n H is regular semi-open in X. Further A n H c A c X and A is open subspace of X, then by Lemma 1.2, A n H is regular semi-open in A. Since, B is a ygrw-closed set. (ii). Hence, from (i) and (ii), it follows that A n y-cl(B) c A n H. Consequently, A n y-cl(B) c H. Since, A is both regular open and ygrw-closed sets, hence by Theorem 3.1, y-cl(A) = A and so y-cl(B) c A. We have A n y-cl(B) = y-cl(B). Thus Y-cl(B) c H and hence B is a ygrw-closed set relative to X. □

Theorem 3.5. If A is a ygrw-closed set of (X, t) and A c Y cx, then A is ygrw-closed in Y, if Y is regular open in X.

Proof. Let A be ygrw-closed in X and Y be regular open subspace of X. If H is any regular semi-open set of Y such that A c H, but A c Y c X, then by Lemma 1.3, H is regular semi-open in X. Since A is a ygrw-closed set of X, then y-cl(A) c H. Hence, Y n y-cl(A) c Y n H=H. So, y-clY(A) c H. Therefore, A is ygrw-closed in Y. □

Proposition 3.2. If A is both open and yg-closed sets ofX, then A is ygrw-closed in X.

Proof. Suppose that A c H and H is regular semi-open in X. Since A is both open and yg—closed in X and A c A, then y-cl(A) c A this implies that y-cl(A) c H. Hence A is ygrw-closed in X. □

Remark 3.1. If A is both open and ygrw-closed in X, then A need not be yg-closed in X. In Example 2.4, the subset {x, y} is both open and ygrw-closed in X but not Yg-closed in X.

Theorem 3.6. For a topological space (X, t), if RSO(X) = {X, y}, then every subset ofX is ygrw-closed in X.

Proof. Let A be any subset of X and RSO(X) = {X, y}. If A = y, then A is ygrw-closed in X. Assume that A == y. Then X is the only regular semi-open containing A and so, y — cl(A) c X. Therefore A is ygrw-closed in X. □

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

Example 3.1. If X = {x, y, z, u} with the topology t = {X, y, {x, y}, {z, u}}, then every subset of X is Ygrw-closed in X, but RSO(X) = {X, y, {x, y}, {z, u}}.

Theorem 3.7. For a topological space (X, t), the following statements are equivalent:

(i) RSO(X) c{F c X: F is y-closed }.

(ii) every subset ofX is ygrw-closed.

Proof.

(i) ^ (ii). Let A be any subset of X such that A c H, H is regular semi-open in X and suppose that RSO(X) c {Fcx: F is y-closed }. Then H e {F c X: F is y-closed }, thus H is y-closed, that is y-cl(H) = H and hence y-cl(A) c H. Therefore A is ygrw-closed. (ii)^(i). Suppose that every subset of X is ygrw-closed and HeRSO(X). Since HcH and H is ygrw-closed, then y-cl(H) c H and hence H e {F c X: F is y-closed }. Therefore RSO(X) c{Fcx: Fis y-closed }.

4. Applications

In this section, we introduce and study some applications on the concept of ygrw-closed sets of a space X.

Theorem 4.1. Let (X, t) be a regular space in which every regular semi-open subset of X is open and A be a compact subset ofX. Then A is y grw-closed in X.

Proof. Assume that A c H and H is regular semi-open in X. Then by hypothesis, H is open. But, A is a compact subset in the regular space (X, t), hence there exists an open set G such that A c G c cl(G) c H. Then Y-cl(A) c y-cl(G) c H, that is, y-cl(A) c H. Thus A is Ygrw-closed in X. □

Lemma 4.1. For any subset A of (X, t), we have X2 n y -cl(A) c rs ker(A).

Theorem 4.2. If A is a subset of a topological space(X, t), then the following statements are equivalent:

(i) A is ygrw-closed in X,

(ii) y-cl(A) c rs ker(A).

Proof.

(i) ^ (ii). Since A is Ygrw-closed in X, y-cl(A) c H,

whenever A c H and H is regular semi-open in X. If x e y-cl(A) and suppose that x e rs ker(A), then there is a regular semi-open set H containing A such that x / H. But, A is Ygrw-closed in X, then y-cl(A) c H. We have x e Y-cl(A) which is a contradiction, hence x e rs ker(A) and so y-cl(A) c rs ker(A).

(ii) ^(i). Suppose that y-cl(A) crs ker(A). If H

is any regular semi-open set containing A, hence rs ker(A) c H, that is, Y-cl(A) c rs ker(A) c H. Therefore A is Ygrw-closed in X.

Theorem 4.3. For any subset A of a topological space (X, t) andX1 n y-cl(A) c A, then A is Ygrw-closed in X.

Proof. Suppose that X1 n y-cl(A) c A and A any subset of (X, t). We need to prove that A is ygrw-closed in X, then y-cl(A) c rs ker(A). But by Lemma 1.5, A c rs ker(A) hence y-cl(A) = X nY-cl(A) = (X1 UX2) nY-cl(A), that is, Y-cl(A) = (X1 n y-cl(A)) U (X2 n y-cl(A)) c rs ker(A). Since X1 nY-cl(A)crsker(A) and by Lemma 4.1, Y-cl(A) crsker(A). Therefore by Theorem 4.2, A is Ygrw-closed in X. □

Remark 4.1. The converse of the above theorem need not be true in general as shown by the following example.

Example 4.1. In Example 2.1, if X1 = {z, u}, X2 = {x, y} and A = {x, y, z}, hence A is Ygrw-closed in X. But X1 n y-cl(A) = {z, u} n X = {z, u} which is not a subset of A.

5. Conclusion

The new class of subsets suggested in this paper can be applied in the field of rough set theory approximations and gmular computing [2] which are widely applied in many lines of applications.

Acknowledgments

The authors are grateful for the support of the Deanship of Scientific Research Taibah University who financially supported this work, under contracted research project (434/3098).

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