International Journal of Mathematical Archive-6(1), 2015, Available online through ISSN

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1 International Journal of Mathematical Archive-6(1), 2015, Available online through ISSN On Pre Generalized Regular Weakly [pgrw]-closed sets in a Topological Space R. S. Wali *1 and Vijaykumari T. Chilakwad 2 1Department of Mathematics, Bhandari Rathi College, Guledagud , Karnataka State, India. 2Department of Mathematics, Government First Grade College, Dharwad, Karnataka State, India. (Received On: ; Revised & Accepted On: ) ABSTRACT This paper contains introduction and study of properties of a new class of closed sets called pgrω-closed sets. The class of pgrw- closed sets lies between the class of pgpr- closed sets and the class of gp-closed sets. Keywords: Topological space, generalized closed sets, rω -closed sets, pgrω-closed sets, T pgrw space. 1. INTRODUCTION In a topological space the concept of closed sets plays an important role. The generalization of closed sets has been studied in different ways in previous years by many topologists leading to several new ideas. In 1969 Levine [15] gave the concept and properties of generalized closed (briefly g-closed) sets. In 1982 Mashhour et.al [18] introduced and studied the concept of pre-open set. Later Maki et.al [16,17], Dontechev[10], Gyanambal[11], Arya and Nour[3], Bhattacharya and Lahiri[7] introduced and studied the concepts of gp-closed, gsp-closed, gpr-closed, gs- closed, sgclosed and αg-closed and their complements, the respective open sets. O. N. Jasted[14] introduced and studied the concept of α sets. Sundarm and Sheik John[23] defined and studied w-closed sets in topological spaces. S.S. Benchalli and R.S.Wali[4] introduced rw-closed sets. In this paper we define and study the properties of a new set called Pre Generalised Regular Weakly Closed set. 2 PRELIMINARIES Definition 2.1: A subset A of a topological space (X, T) is called 1. a semi-open set[13] if A cl(int(a)) and semi-closed set if int(cl(a)) A. 2. a pre-open set[15] if A int(cl(a)) and pre-closed set if cl(int(a)) A. 3. an α--open set [11]if A int(cl(int(a))) and α -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 [25] if A = int(cla)) and a regular closed set if A = cl(int(a)). 6. δ-closed[30] if A = clδ(a), where clδ(a) = {xϵ X : int(cl(u)) A θ, U ϵ T and x ϵ U} 7. Regular semi open [9] set if there is a regular open set U such that U A cl(u). 8. a generalized closed set (briefly g-closed)[15] if cl(a) U whenever A U and U is open in X. 9. a regular generalized closed set(briefly rg-closed)[19] if cl(a) U whenever A U and U is regular open in X. 10. a α generalized closed set(briefly αg -closed)[16] if αcl(a) U whenever A U and U is open in X. 11. a generalised α-closed (gα closed)[17] if αcl(a) U whenever A U and U is α -open in X. 12. a generalized pre regular closed set(briefly gpr-closed)[11] if pcl(a) U whenever A U and U is regular open in X. 13. a generalized semi-pre closed set(briefly gsp-closed)[10] if spcl(a) U whenever A U and U is open in X. 14. a ω-closed set [23]if cl(a) U whenever A U and U is semi-open in X. 15. a strongly generalized closed set[28] (briefly, g*-closed) if Cl(A) U whenever A U and U is g-open in X. 16. a regular generalized α-closed set[27] (briefly, rgα-closed)if αcl (A) U whenever A U and U is regular α-open in X. Corresponding Author: R. S. Wali *1 International Journal of Mathematical Archive- 6(1), Jan

2 17. an α-generalized regular closed[29] (briefly αgr-closed) set if αcl(a) U whenever A U and U is regularopen in X. 18. a ωα- closed set[6] if αcl(a) U whenever A U and U is ω-open in X. 19. a pre generalized pre regular closed set[2] (briefly pgpr-closed) if pcl(a) U whenever A U and U is rgopen in X. 20. a generalized pre closed (briefly gp-closed) set[17] if pcl(a) U whenever A U and U is open in X. 21. a α-regular w- closed set[32] if αcl(a) U whenever A U and U is rw -open in X. 22. a generalized pre regular weakly closed (briefly gprw-closed) set [13] if pcl(a) U whenever A U and U is regular semi- open in X. The complements of the above mentioned closed sets in (8) - (22), are their open sets respectively. 23. Pre-regular T 1/2 -space [30]: A topological space (X, T) is called pre-regular T 1/2 -space if every gpr-closed set is preclosed. Definition 2.2: Let (X, T) be a topological space and A X. The intersection of all closed (resp pre-closed, α-closed and semi-pre-closed ) subsets of space X containing A is called the closure (resp pre-closure, α-closure and Semi-preclosure) of A and denoted by cl(a) (resp pcl(a), αcl(a), spcl(a)). 3. PRE GENERALIZED REGULAR WEAKLY CLOSED SET In this section we define pre generalized regular weakly closed set in topological spaces and study its properties. Definition 3.1: A subset A of a topological space (X, T) is called a pre regular weakly closed set if pcl(a) U whenever A U and U is a rw-open set. The abbreviation is pgrw-closed. Set of all pgrw-closed sets in X is denoted by PGRWC(X). Example: X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}}. pgrw-closed sets are X,Φ, {c}, {b, c}, {a, c}. Theorem 3.2: Every pgpr-closed set is pgrw-closed. Proof: Let A be a pgpr closed set. Let A U where U is a rw-open set. Then as a rw-open set is a rg-open set and A is pgpr-closed, pcl(a) U. Thus pcl (A) U whenever A U and U is rw-open..'. A is pgrw closed. For example: X ={a, b, c, d}, T={X, Φ, {a}, {b}, {a, b},{c, d}}. Here {a, d} is pgrw-closed, but not pgpr-closed. Theorem 3.3: If A is rw -open and pgrw-closed, then A is pgpr-closed. Proof: A is a rw-open and pgrw closed set. Let U be a rg-open set such that A U. A A, a rw-open set and A is pgrw-closed..'. pcl(a) A U..'. A is pgpr- closed. If A is rw - open and pgpr-closed, then A is not pgrw-closed. For example: X= {a, b, c}, T={X, Φ, {a}, {a, c}}. Here {a, c} is rw- open and pgpr- closed, but not pgrw-closed. Theorem 3.4: Every pgrw-closed set is gp-closed. Proof: Let A be a pgrw-closed set. Let A U where U is open. Then A U and U is rw-open. As A is pgrw-closed, pcl (A) U. Thus pcl (A) U whenever A U and U is open..'. A is gp-closed. 2015, IJMA. All Rights Reserved 77

3 The converse of the theorem is not true. For example: X = {a, b, c,}, T={X, Φ, {a}}. Here {a, b} is gp-closed, but not pgrw-closed. Theorem 3.5: If A is open and gp-closed set, then A is pgrw-closed. Proof: A is an open and gp-closed set. Let U be a rw-open set such that A U. A A, an open set and A is gp-closed..'. pcl (A) A U. Thus every rw-open set U containing A contains pcl(a)... A is pgrw closed. If A is open and pgrw-closed, then A is not gp-closed. For example: X= {a, b, c, d}, T={X, Φ, {a}, {c, d}, {a, c, d}}. Here {a, c, d} is both open and pgrw-closed but not gp-closed. Remark: From the theorems 3.2 and 3.4 it follows that the class of pgrw- closed sets lies between the class of pgprclosed sets and the class of gp-closed sets. Theorem 3.6: A pre-closed set is pgrw-closed. Proof: Let A be a pre-closed set. Then cl(int(a)) A. pcl (A) = AU cl ( int A))=A. Let A U where U is a rw- open set. As A = pcl (A), pcl(a) U. Thus every rw-open set U containing A contains pcl (A)... A is pgrw-closed. For example: X = {a, b, c, d,}, T={X, Φ, {a}, {b}, {a, b}, {c, d}}. Here {a, b, d} is pgrw-closed, but not pre-closed. Corollary 3.7: Every α- closed set is pgrw- closed. Proof: Set A is α- closed A is pre closed. A is pgrw-closed. For example: X = {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {a, d} is pgrw- closed, but not α -closed. Corollary 3.8: Every closed set is pgrw-closed. Proof: Set A is closed A is pre closed. A is pgrw-closed. For example: X = {a, b, c, d,}, T={X, Φ, {a}, {b}, {a, b}, {c, d}}. Here {b, d} is pgrw-closed, but not closed. Corollary 3.9: Every regular closed set is pgrw-closed. Proof: Set A is regular closed A is pre closed. A is pgrw-closed. 2015, IJMA. All Rights Reserved 78

4 For example: X = {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}}. Here {a, b, d} is pgrw-closed, but not regular closed. Corollary 3.10: Every δ - closed set is pgrw- closed. Proof: Set A is δ - closed A is pre closed. A is pgrw-closed. For example: X = {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}.here {a, d} is pgrw-closed but not δ closed. Theorem 3.11: Every #rg- closed set is pgrw- closed. Proof: Let A be a #rg- closed set. Then cl (A) U whenever A U and U is rw- open. As pcl(a) cla, pcl (A) U. Thus every rw-open set U containing A contains pcl(a)..'. A is pgrw- closed. For example: X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {a, b, d} is pgrw- closed, but not #rg- closed. Theorem 3.12: Every pgrw- closed set is gsp-closed. Proof: Let A be a pgrw- closed set. Let A U, an open set. Then A U and U is rw- open. pcl (A) U. As spcl(a) pcl(a), spcl (A) U, U is open. Thus every open set U containing A contains spcl(a). A is gsp- closed. For example: X={a, b, c, d}, T={X, Φ, {a}, {b}, {a, b},{a, b, c}}. Here {a, c} is gsp- closed, but not pgrw- closed. Corollary 3.13: Every pgrw- closed set is gspr- closed. Proof: Let A be pgrw- closed.then A is gsp- closed and every gsp- closed is gspr- closed... A is gspr-closed. For example: X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {a, b} is gspr- closed, but not pgrw- closed. Corollary 3.14: Every pgrw- closed set is gpr- closed. Proof: Let A be a pgrw- closed set. Then A is gp- closed and every gp- closed is gpr- closed. By the theorem 3.4, A is gpr- closed. For example: X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {a, b} is gpr- closed, but not pgrw-closed. Theorem 3.15: If A is both w- open and wα- closed, then A is pgrw- closed. Proof: A is w-open and wα- closed. Let U be a rw open set containing A. A A, a w- open and wα- closed..'. αcl(a) A..'. αcl(a) U. And pcl(a) αcl(a)..'. pcl(a) U. Thus every rw-open set U containing A contains pcl(a)..'. A is pgrw- closed. If A is both w-open and pgrw closed then A is not wα-closed. 2015, IJMA. All Rights Reserved 79

5 For example: X = {a, b, c, d,}, T={X, Φ, {b, c}, {a, b, c}, {b, c, d}}. Here {c} is w-open and pgrw closed, but A is not wα-closed. Theorem 3.16: If A is regular open and pgrw-closed, then A is pre-closed. Proof: A is regular- open and pgrw-closed..'. A is rw- open and pgrw-closed. A A, a rw- open set and pgrw- closed..'. pcl(a) A. A pcl(a)..'. A=pcl(A). So A is pre-closed. Corollary 3.17: if A be a regular-open and pgrw-closed set and F, a pre-closed set, then A F is pgrw-closed. Proof: A is a regular-open and pgrw-closed set. F is pre- closed. A and F are pre- closed..'. A F is pre-closed and hence A F is pgrw-closed. Theorem 3.18: Every αrw-closed set is pgrw-closed. Proof: Let A be an αrw- closed set and A U, a rw- open set. Then αcl(a) U. As pcl(a) αcl(a ), pcl(a) U. Thus every rw-open set U containing A contains pcl(a). The converse of the theorem is not true. For example: X= {a, b, c}, T={X,Φ, {a}, {b, c}}. Here {a, b} is pgrw- closed, but not αrw closed. Remark 3.19: Union of two pgrw-closed sets is not pgrw-closed as shown by the following example. X= {a, b, c, d}, T={X, Φ, {a}, {b, c} {b, c, d}, {a, b, c}}.{b}, {c} are pgrw-closed. But {b} U {c} = {b, c} not pgrw-closed. Remark 3.20: Intersection of two pgrw-closed sets is not pgrw-closed as shown by the following example. X= {a, b, c, d}, T={X, Φ, {a}, {c, d}, {a, c, d}}. {a, b, c},{a, c, d} are pgrw-closed, but {a, b, c} {a, c, d}={a, c} is not pgrw - closed. Theorem 3.21: If A is pgrw-closed and A B pcl(a), then B is also pgrw-closed. Proof: A B pcl(a) pcl(a) pcl(b) pcl(pcl(a)) pcl(a) pcl(b) pcl(a) pcl(b)= pcl(a).now B U where U is rw-open. A U because A B. pcl(a) U because A is pgrw-closed. pcl(b) U because pcl(b)= pcl (A). Thus every rw- open set U containing B contains pcl(b)..'. B is pgrw-closed. Theorem 3.22: For every pgrw-closed set A, a non-empty set B [pcl(a) A] B is not rw-closed. Proof: Suppose B is a non-empty rw-closed set and B [pcl(a)-a]. Then B pcl(a) and B A B pcl(a) and A B, a rw-open set. B pcl(a) and pcl(a) B.. A is pgrw-closed. B pcl(a) B B B which is not true. Hence the theorem. Theorem 3.23: For every pgrw-closed set A, a nonempty set B [pcl(a) A] B is not w-closed. Proof: Suppose B is a non-empty w-closed set and B [pcl(a)-a]. B pcl(a) and A B, a w-open set. B pcl(a) and pcl(a) B.. A is pgrw-closed. B pcl(a) B B B 2015, IJMA. All Rights Reserved 80

6 which is not true. Hence the theorem. Corollary 3.24: Let A be pgrw-closed. Then A is pre-closed iff [pcl(a)-a] is rw-closed. Proof: 1. A is a pre-closed set..'. pcl(a)=a..'. pcl(a)-a=φ which is rw-closed. 2. A is pgrw-closed and pcl(a)-a is rw-closed. Then pcl(a)-a = Φ by the theorem [3.22]... pcl(a)=a or A is pre-closed. Theorem 3.25: In a topological space X, for each x in X, X-{x} is pgrw-closed or rw-open. Proof: x ϵx..'. pcl( X-{x}) X. X-{x} is pgrw-closed. Theorem 3.26: If A is both regular-open and rg-closed, then A is pgrw-closed. Proof: A is a regular-open and rg-closed set. Let A U where U is rw- open. A A, a regular-open set and A is rg-closed. So cl(a) A. pcl(a) cl(a)... pcl(a) A, A U..'. pcl(a) U. Thus every rw-open set U containing A contains pcl(a). Theorem 3.27: If A is both open and g-closed, then A is pgrw -closed. Proof: A is open and g-closed. Let U be a rw-open set containing A. A A, an open set. And A is g-closed... cl(a) A. pcl(a) cl(a) A U. Thus every rw-open set U containing A contains pcl(a). Theorem 3.28: If A is regular- open and gpr-closed, then it is pgrw-closed. Proof: A is regular-open and gpr-closed. Let U be a rw-open set such that A U... pcl(a) A and pcl(a) U. Thus every rw-open set U containing A contains pcl(a). Hence A is pgrw -closed. Theorem 3.29: If A is regular-open and αgr -closed, then it is pgrw -closed. Proof: A is a regular-open and αgr closed set. Let U be a rw-open set containing A. A A, a regular-open set. As A is αgr-closed, αcl(a) A. pcl(a) αcl(a)... pcl(a) A U. Thus every rw-open set U containing A contains pcl(a). Hence A is pgrw-closed. Theorem 3.30: If A is both regular semi-open and rw-closed, then A is pgrw-closed. Proof: A is a regular semi-open and rw-closed set. Let U be rw-open containing A. A A,a regular semi-open set and A is rw-closed..'. cl(a) A. As pcl(a) cl(a), pcl(a) A U. 2015, IJMA. All Rights Reserved 81

7 If A is both regular semi-open and pgrw-closed, then A is not rw-closed. For example: X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {b, d} is both regular semi -open and pgrwclosed, but not rw -closed. Theorem 3.31: If A is both pre-open and pg-closed, then A is pgrw closed. Proof: A is a pre-open and pg-closed set. Let U be rw-open containing A. A A, a pre-open set and pg-closed..'. pcl(a) A, A U..'. pcl(a) U. Thus every rw-open set U containing A contains pcl(a).. '. A is pgrw-closed. If A is both pre- open and pgrw -closed, then A need not be pg-closed. For example: X={a, b, c, d}, T={X,Φ, {a}, {b}, {a, b},{a, b, c}}. Here {a, b, d} is pre- open and pgrw-closed, but not pg-closed. Theorem 3.32: If A is both g- open and g*-closed, then A is pgrw-closed. Proof: A is a g-open and g * -closed set. Let U be a rw-open set containing A.A A, which is g-open and g* -open..'. cl(a) A. pcl(a) cl(a) A U. Thus every rw-open set U containing A contains pcl(a). If A is g-open and pgrw-closed, then A need not be g*-closed. X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {b, c} is both g- open and pgrw-closed, but not g*-closed. Theorem 3.33: If A is both semi-open and w-closed, then it is pgrw-closed. Proof: A is a semi- open and w-closed set. Let U be a rw-open set containing A.A A, a semi-open set. As A is w closed, cl(a) A. pcl(a) cl(a). So pcl(a) U. Thus every rw - open set U containing A contains pcl(a). If A is semi- open and pgrw- closed, then it is not w-closed. X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {b, c} is both semi-open and pgrw-closed, but not w-closed. Theorem 3.34: If A is both regular semi- open and gprw-closed, then it is pgrw-closed. Proof: A is a regular- semi-open and gprw-closed set. Let U be a rw-open set containing A. A A, a regular- semi open set. As A is gprw-closed, pcl(a) A. So pcl(a) U. Thus every rw - open set U containing A contains pcl(a)..'. A is pgrw -closed. If A is regular semi- open and pgrw-closed, then A need not be gprw -closed. For example, X={a, b, c, d}, T={X,Φ, {a}, {b}, {a, b},{a, b, c}}. Here {b, c} is both regular semi- open and pgrwclosed, but not gprw -closed. Theorem 3.35: If A is open and αg-closed, then it is pgrw -closed. Proof: A is an open and αg-closed set. 2015, IJMA. All Rights Reserved 82

8 Let U be a rw-open set containing A. A A, an open set. As A is αg-closed, αcl(a) A. pcl(a) αcl(a).'. pcl(a) A U. Thus every rw-open set U containing A contains pcl(a). Hence A is pgrw -closed. If A is open and pgrw-closed, then need not be αg-closed. For example: X={a, b, c, d}, T={X,Φ, {a}, {c, d}, {a, c, d}}. Here {a, c, d} is both open and pgrw-closed but not αg-closed. Theorem 3.36: If A is α -open and gα -closed, then it is pgrw -closed. Proof: A is α-open and gα closed set. Let U be the rw-open set containing A. A A, an α -open set. As A is gα -closed, αcl(a) A. pcl(a) αcl(a)... pcl(a) A U. Thus every rw-open set U containing A contains pcl(a). Hence A is pgrw-closed. If A is α-open and pgrw -closed, then need not be gα- closed. For example: X= {a, b, c, d}, T={X, Φ, {a}, {c, d}, {a, c, d}}. Here {a, c, d} is both α open and pgrw-closed, but not gα- closed. Theorem 3.37: The following statements are equivalent. 1. Every pgrw-closed set is αg-closed. 2. Every pre-closed set is αg-closed. Proof: (1) (2): A is pre-closed. A is pgrw-closed. A is αg-closed by hypothesis. (2) (1): A is pgrw-closed. pcl(a) U whenever A U & U is rw-open. Let V be an open set containing A. Then V is rw-open containing A. As A is pgrw closed, pcl(a) V. Let pcl(a) = B. Then B is pre-closed... B is αg- closed by hypothesis. And B V... αcl(b) V. As A B=pcl(A), αcl(a) αcl(b) V. Thus every open set V containing A contains αcl(a)... A is αg-closed. Theorem 3.38: The following statements are equivalent. 1. Every pgrw-closed set is αrw-closed. 2. Every pre-closed set is αrw-closed. Proof: (1) (2): A is pre-closed. A is pgrw-closed A is αrw-closed by hypothesis (1). (2) (1): A is pgrw-closed. Let A U where U is rw-open (i). Then as A is pgrw-closed, pcl(a ) U. Let B=pcl(A). Then B U. B is pre-closed..'. by hypothesis (2) B is αrw-closed... αcl(b) U..(ii). A B. αcl(a) αcl(b) U from (ii)..'. αcl(a) U.(iii). (i) and (iii) every rw-open set U containg A contains αcl(a). Hence A is αrw-closed. 4. T pgrw SPACE: Definition 4.1: A topological space X is called a T pgrw space if every pgrw-closed set in it is pre-closed. 2015, IJMA. All Rights Reserved 83

9 Example: X= {a, b, c, d}, T={X,Φ, {a}, {b}, {a, b}}. Here every pgrw-closed set is preclosed. So (X, T) is a T pgrw space. Example: X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}, {a, b, c}}. Here {a, d} is pgrw-closed, but not preclosed. So (X, T) is not T pgrw space. Theorem 4.2: A topological space X is a T pgrw space iff for each x of X, {x} is either rw-closed or pre-open. Proof: Hypothesis: X is a T pgrw - space. Let xϵx. If {x} is rw-closed, then there is nothing to prove. If {x} is not rw-closed, then X-{x} is not rw-open and so X is the only rw-open set containing X-{x} and pcl(x-{x}) X... X-{x} is pgrw-closed. X is T pgrw space (hypothesis) and X-{x} is pgrw-closed... X-{x} is pre-closed... {x} is pre-open. Thus for every x of X, a T pgrw space, {x} is either rw-closed or pre-open. Conversely, suppose for every xϵx, {x} is either rw-closed or pre-open. Let A be a pgrw-closed subset of X. Now to prove A is pre-closed, we prove pcl(a) A. Let x ϵ pcl(a). Then by hypothesis a) {x} is pre-open or b) {x} is non-empty rw-closed. If x is not in A, then A {x}, a pre-closed set. x ϵ pcl(a) and {x} is non-empty rw-closed... pcl(a) {x}. x ϵ pcl(a) and {x} pcl(a)-a by thm x ϵ {x} which is not true. x ϵ A.. x ϵ A... pcl(a) A. pcl(a) A. Thus every pgrw-closed set is pre-closed... X is a T pgrw - space. Theorem 4.3: Every pre-regular T 1/2 -space is T pgrw space. Proof: Let X be a pre-regular T 1/2 -space and A be a pgrw-closed set. As every pgrw-closed set is gpr-closed, A is gprclosed. Since X is pre-regular T 1/2 -space, A is preclosed. So every pgrw-closed set is preclosed... X is T pgrw space. Converse of the above theorem is not true. For example: X= {a, b, c, d}, T={X, Φ, {a}, {b}, {a, b}}. Here (X, T) is a T pgrw- space, but not a pre-regular T 1/2 -space. 2015, IJMA. All Rights Reserved 84

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