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On [alpha]-generalized regular weakly closed sets in topological spaces.

[section]1. Introduction

Levine [13] introduced generalized closed (brief g-closed) sets in 1970. Regular open sets and regular semi-open sets have been introduced and investigated by Stone [22] and Cameron [6] respectively. Benchalli and Wali [4] introduced the concept of regular weakly closed sets in 2007. Sanjay Mishra et.al [20] defined generalized pre regular weakly closed sets in 2012. Many researchers [21, 19, 23, 24, 9] during the last decade have introduced and studied the sets like [omega]-closed sets, mildly generalized closed sets, [g.sup.*]-closed sets, semi-closed sets and semi-pre closed sets and [pi]g-closed sets.

In this paper, we define and study the properties of [alpha] generalized regular weakly closed sets ([alpha]grw-closed) in topological space which is properly placed between the regular weakly closed sets and generalized pre regular weakly closed sets.

[section]2. Preliminaries

Definition 2.1. A subset A of a topological space (X, [tau]) is called

i) a preopen set [16] if A [subset or equal to] int (cl(A)) and a preclosed set if cl(int(A)) [subset or equal to] A,

ii) a semi-open set [12] if A [subset or equal to] cl(int(A)) and a semi-closed set if int (cl(A)) [subset or equal to] A,

iii) an [alpha]-open set [18] if A [subset or equal to] int (cl(int(A))) and a [alpha]-closed set if cl(int(cl(A))) [subset or equal to] A,

iv) a semi-preopen set [2] ([beta]-open [1]) if [Asubset or equal to] cl(int(cl(A))) and a semi-preclosed ([beta] closed [1]) if int (cl(int(A))) [subset or equal to] A,

v) regular open set [22] if A = int(cl(A)) and a regular closed set [21] if A = cl(int(A)),

vi) [theta]-closed set [25] if A = c[l.sub.[theta]](A), where c[l.sub.[theta]] (A) = {x [member of] X : cl(U) [Intersection] A [not equal to] [phi], U [member of] [tau] and x[member of]U},

vii) [delta]-closed set [25] if A = c[l.sub.[delta]](A), where c[l.sub.[delta]](A) = {x [member of] X : int (cl(U)) [Intersection] A [not equal to] [phi], U[member of][tau] and x[member of]U},

viii) [pi]-open set [9] if A is a finite union of regular open sets.

The [alpha]-closure (resp. semi-closure, semi-preclosure and pre-cosure) of a subset A of X denoted by [alpha]cl(A) (resp. scl(A), spcl (A) and pcl(A)) is defined to be the intersection of all [alpha]-closed sets (resp. semi-closed sets, semi-preclosed sets and pre-closed sets) containing A.

Definition 2.2. A subset A of a topological space (X, [tau]) is called regular semi-open [6] if there is a regular open set U such that U [subset or equal to] A [subset or equal to] cl(U). The family of all regular semi-open sets of X is denoted by RSO(X).

Definition 2.3. A subset A of a topological space (X, [tau]) is called

i) a generalized closed set (briefly g closed) [12] if cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is open,

ii) a semi generalized closed set (briefly sg-closed) [5] if scl(A) [subset or equal to] U whenever A [subset or equal to] U and U is semi-open,

iii) a generalized semi closed set (briefly gs-closed) [3] if scl(A)[subset or equal to]U whenever A [subset or equal to]U and U is open,

iv) a weakly closed set (briefly [omega]-closed) [15] if cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is semi-open,

v) a weakly generalized closed set (briefly wg-closed) [17] if cl(int(A)) [subset or equal to] U whenever A [subset or equal to] U and U is open,

vi) a [alpha]-generalized closed set (briefly [alpha]g closed) [14] if [alpha]cl(A) [subset or equal to] U whenever A U and U is open,

vii) a generalized semi-preclosed set(briefly gsp-closed) [10] if spcl (A) [subset or equal to] U whenever A [subset or equal to] U and U is open,

viii) a generalized preclosed set (briefly gp-closed) [15] if pcl(A) [subset or equal to] U whenever A [subset or equal to] U and U is open,

ix) a regular weakly closed set (briefly rw-closed) [4] if cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is regular semi-open,

x) a generalized pre regular weakly (briefly gprw-closed) [20] if pcl(A) [subset or equal to] U whenever A [subset or equal to] U and U is regular semi-open,

xi) a mildly generalized closed set (briefly mildly g-closed) [19] if cl(int(A)) [subset or equal to] U whenever A [subset or equal to] U and U is g open,

xii) a strongly generalized closed set (briefly [g.sup.*]-closed) [23] if cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is g open,

xiii) a [sup.*]g-closed set [23] if cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is [omega]- open,

xiv) a [psi]-closed set [24] if scl(A) [subset or equal to] U whenever A [subset or equal to] U and U is sg-open,

xv) a regular weakly generalized closed set (briefly rwg-closed) [17] if cl(int(A)) [subset or equal to] U whenever A [subset or equal to] U and U is regular open,

xvi) a [theta]-generalized closed set (briefly [theta]-g-closed) [8] if c[l.sub.[theta]](A) [subset or equal to] U whenever A [subset or equal to] U and U is open,

xvii) a [delta]-generalized closed set (briefly [delta]-g-closed) [7] if c[l.sub.[delta]](A) [subset or equal to] U whenever A [subset or equal to] U and U is open,

xviii) a [pi]-generalized closed set (briefly [pi]g-closed) [9] if cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is [pi]-open.

[section]3. [alpha]-Generalized regular weakly closed sets

Definition 3.1. A subset A of a topological space (X, [tau]) is called [alpha]- generalized regular weakly closed [briefly [alpha]grw-closed if [alpha]cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is regular semi-open in (X, [tau]). We denote the set of all [alpha]grw-closed sets in (X, [tau]) by [alpha]GRWC(X).

Theorem 3.2. Every [omega]-closed set is [alpha]grw-closed.

Proof. Let A be [omega]-closed and A [subset or equal to] U where U is regular semi-open. Since every regular semi-open set is semi-open and [alpha]cl(A) [subset or equal to] cl(A), [alpha]cl(A) [subset or equal to] U. Hence A is [alpha]grw closed.

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

Example 3.3. Let X = {a, b, c, d} and [tau] = {0, {a}, {b}, {a, b}, {a, b, c}, X}. Then A = {c} is [alpha]grw-closed but not [omega]-closed in (X, [tau]).

Theorem 3.4. Every rw-closed set is [alpha]grw-closed.

Proof. Let A be rw-closed and A [subset or equal to] U where U is regular semi- open. Then cl(A) [subset or equal to] U. Since [alpha]cl(A) [subset or equal to] cl(A), [alpha]cl(A) [subset or equal to] U. Hence A is [alpha]grw-closed.

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

Example 3.5. In Example 3.3, the set A = {c} is [alpha]grw-closed but not rw- closed in (X, [tau]).

Theorem 3.6. Every [alpha]-closed set is [alpha]grw-closed.

Proof. Let A be an [alpha]-closed set and A [subset or equal to] U where U is regular semi open. Then [alpha]cl(A) = [Asubset or equal to]U. Hence A is [alpha]grw-closed.

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

Example 3.7. Let X = {a, b, c, d} and [tau] = {0, {a}, X}. Then A = {a} is [alpha]grw-closed but not [alpha]-closed in (X, [tau]).

Theorem 3.8. Every [alpha]grw-closed set is gprw-closed.

Proof. Let A be an [alpha]grw-closed set and A [subset or equal to] U where U is regular semi open. Since pcl(A) [subset or equal to] [alpha]cl(A), pcl(A) [subset or equal to] U. Hence A is gprw-closed.

The converse of the above theorem need not be true as seen from the following example. Example 3.9. Let X = {a, b, c, d} and [tau] = {0, {a}, {b, c}, {a, b, c}, X}. Then A = {b} is gprw-closed but not [alpha]grw-closed in (X, [tau]).

Theorem 3.10. Every closed set is [alpha]grw-closed.

Proof. Every closed set is rw-closed [4] and by Theorem 3.4, every rw-closed set is [alpha]grw-closed. Hence the proof.

Theorem 3.11 Every regular closed set is [alpha]grw-closed.

Proof. Every regular closed set is rw-closed [4] and by Theorem 3.4, every rw- closed set is [alpha]grw-closed. Hence the proof.

Theorem 3.12. Every [theta]-closed set is [alpha]grw-closed.

Proof. Every [theta] closed set is rw-closed [4] and by Theorem 3.4, every rw- closed set is [alpha]grw-closed. Hence the proof.

Theorem 3.13. Every [delta]-closed set is [alpha]grw-closed

Proof. Every [delta]-closed set is rw-closed [4] and by Theorem 3.4, every rw- closed set is [alpha]grw-closed. Hence the proof.

Theorem 3.14. Every [pi]-closed set is [alpha]grw-closed

Proof. Every [pi]-closed set is rw-closed [4] and by Theorem 3.4, every rw- closed set is [alpha]grw-closed. Hence the proof.

Remark 3.15. The following example shows that [alpha]grw-closed sets are independent of g-closed sets, wg-closed sets, [alpha]g-closed sets, gs-closed sets, sg-closed sets, gsp-closed sets and gp-closed sets.

Example 3.16. Let X = {a, b, c, d} and [tau] = {0, {a}, {b}, {a, b}, {b, c, d}, X}. Then

1. Closed sets in (X, [tau]) are 0, {a}, {c, d}, {a, c, d}, {b, c, d}, X.

2. [alpha]grw-closed sets in (X, [tau]) are 0, {a}, {c}, {d}, {a, b}, {a, c}, {c, d}, {a, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

3. g-closed sets in (X, [tau]) are 0, {a}, {c}, {d}, {a, c}, {b, c}, {a, d}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

4. wg-closed sets in (X, [tau]) are 0, {a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

5. [alpha]g-closed sets in (X, [tau]) are 0, {a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

6. gs-closed sets in (X, [tau]) are 0, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

7. sg-closed sets in (X, [tau]) are 0, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

8. gsp-closed sets in (X, [tau]) are 0, {a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

9. gp-closed sets in (X, [tau]) are 0, {a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

Remark 3.17. The following example shows that [alpha]grw-closed sets are independent of [g.sup.*]-closed sets, mildly g-closed sets, semi closed sets, [pi]g-closed sets, [theta]-generalized closed sets, [delta]-generalized closed sets, *g-closed sets, [psi]-closed sets and rwg- closed sets.

Example 3.18. Let X = {a, b, c, d} and [tau] = {0, {a}, {b}, {a, b}, {a, b, c}, X}. Then

1. closed sets in (X, [tau]) are 0, {d}, {c, d}, {a, c, d}, {b, c, d}, X.

2. [alpha]grw-closed sets in (X, [tau]) are 0, {c}, {d}, {a, b}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

3. [g.sup.*]-closed sets in (X, [tau]) are 0, {d}, {a, d}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, X.

4. Mildly g-closed sets in (X, [tau]) are 0, {d}, {a, d}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, X.

5. Semi closed sets in (X, [tau]) are 0, {a}, {b}, {c}, {d}, {b, c}, {b, d}, {c, d}, {a, d}, {b, c, d}, {a, c, d}, X.

6. [pi]g-closed sets in (X, [tau]) are 0, {c}, {d}, {b, c}, {b, d}, {c, d}, {a, c}, {a, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

7. [theta]g-closed sets in (X, [tau]) are 0, {d}, {c, d}, {b, d}, {a, d}, {b, c, d}, {a, c, d}, {a, b, d}, X.

8. [delta]g-closed sets in (X, [tau]) are 0, {d}, {c, d}, {b, d}, {a, d}, {b, c, d}, {a, c, d}, {a, b, d}, X.

9. *g-closed sets in (X, [tau]) are 0, {d}, {c, d}, {b, d}, {a, d}, {b, c, d}, {a, c, d}, {a, b, d}, X.

10. [psi]-closed sets in (X, [tau]) are 0, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, X.

11. rwg-closed sets in (X, [tau]) are 0, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}, X.

Theorem 3.19. If A and B are [alpha]grw-closed then [Aunion]B is [alpha]grw- closed

Proof. Let A and B be any two [alpha]grw-closed sets. Let [Aunion]B [subset or equal to] U and U is regular semi open. We have [alpha]cl(A) [subset or equal to] U and [alpha]cl(B) [subset or equal to] U. Thus [alpha]cl([Aunion]B) = [alpha]cl(A)[union][alpha]cl(B) [subset or equal to] U. Hence [Aunion]B is [alpha]grw-closed

Remark 3.20. The intersection of two [alpha]grw-closed sets of a topological space (X, [tau]) is generally not [alpha]grw-closed

Example 3.21. In Example 3.18, Then {a, b} and {b, c, d} are [alpha]grw-closed sets. But {a, b}[Intersection]{b, c, d} = {b} is not [alpha]grw-closed in (X, [tau]).

Theorem 3.22. If a subset A of X is [alpha]grw-closed then [alpha]cl(A)-A does not contain any non empty regular semi open sets.

Proof. Suppose that A is [alpha]grw-closed set in X. Let U be a regular semiopen set such that U [subset or equal to] [alpha]cl(A)-A and U [not equal to] 0. Now U [subset or equal to] X-A which implies A [subset or equal to] X-U. Since U is regular semi open, X-U is also regular semiopen in X [11]. Since A is an [alpha]grw-closed set in X, by definition we have [alpha]cl(A) [subset or equal to] X-U. So U [subset or equal to] X-acl(A). Also U[subset or equal to][alpha]cl(A). Therefore U [subset or equal to] [alpha]cl(A) [Intersection] (X-[alpha]cl(A)) = 0, which is a contradiction. Hence [alpha]cl(A)-A does not contain any non empty regular semiopen set in X.

The converse of the above theorem needs not be true as seen from the following example.

Example 3.23. Let X = {a, b, c} and [tau] = {0, {b}, {c}, {b, c}, X} and A = {b}. Then [alpha]cl(A)-A = {a, b}-{b} = {a} does not contain nonempty regular semiopen set, but A is not an [alpha]grw-closed set in (X, [tau]).

Theorem 3.24. For an element x [member of] X, the set X-{x} is [alpha]grw-closed or regular semi open.

Proof. Suppose X-{x} is not regular semi open. Then X is only regular semi open set containing X-{x} and also [alpha]cl(X-{x}) [subset or equal to] X. Hence X-{x} is [alpha]grw-closed set in X.

Theorem 3.25. If A is regular open and [alpha]grw-closed, then A is [alpha]- closed

Proof. Suppose A is regular open and [alpha]grw-closed. As every regular open set is regular semi open and [Asubset or equal to]A, we have [alpha]cl(A) [subset or equal to] A. Also A [subset or equal to] [alpha]cl(A). Therefore [alpha]cl(A) = A. Hence A is [alpha]-closed

Theorem 3.26. If A is an [alpha]grw-closed subset of X such that A [subset or equal to] B [subset or equal to] [alpha]cl(A), then B is an [alpha]grw-closed set in X.

Proof. Let A be an [alpha]grw-closed set of X, such that A [subset or equal to] B [subset or equal to] [alpha]cl(A). Let B [subset or equal to] U and U be regular semiopen set. Then A [subset or equal to] U. Since A is [alpha]grw- closed, we have [alpha]cl(A)[subset or equal to]U. Now [alpha]cl(B) [subset or equal to] [alpha]cl([alpha]cl(A)) = [alpha]cl(A) [subset or equal to] U. Therefore B is [alpha]grw- closed set.

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

Example 3.27. In Example 3.18. Let A = {c, d} and B = {b, c, d} are [alpha]grw- closed sets in (X, [tau]). Thus any [alpha]grw-closed set need not lie between a [alpha]grw- closed and its [alpha]-closure.

Theorem 3.28. Let A be an [alpha]grw-closed in (X, [tau]). Then A is [alpha]- closed if and only if [alpha]cl(A)-A is regular semi open.

Proof. Suppose A is [alpha]-closed. Then [alpha]cl(A) = A and so [alpha]cl(A)-A = 0, which is regular semi open in X. Conversely, suppose [alpha]cl(A)-A is regular semi open in X. Since A is [alpha]grw-closed, by Theorem 3.22, [alpha]cl(A)-A does not contain any non empty regular semi open in X. Then [alpha]cl(A)-A = 0. Therefore [alpha]cl(A) = A. Hence A is [alpha]-closed.

Theorem 3.29. If A is both open and [alpha]g-closed then A is [alpha]grw-closed.

Proof. Let A be an open and [alpha]g-closed. Let [Asubset or equal to]U and U be regular semiopen. Now A [subset or equal to] A and by hypothesis [alpha]cl(A)-A. Therefore [alpha]cl(A) [subset or equal to] U. Hence A is [alpha]grw-closed.

Remark 3.30. If A is both open and [alpha]grw-closed, then A need not be [alpha]g-closed as seen from the following example.

Example 3.31. In Example 3.18, the subsets {a, b} and {a, b, c} are [alpha]grw- closed and open but not [alpha]g-closed.

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N. Selvanayaki, Department of Mathematics, Akshaya College of Engineering and Technology, Coimbatore, Tamilnadu, India

Gnanambal Ilango, Postgraduate and Research Department of Mathematics, Government Arts College, Coimbatore, Tamilnadu, India

E-mails: selvanayaki.nataraj@gmail.com
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