On $gamma$-Regular-Open Sets and $gamma$-Closed Spaces

On γ -Regular-Op en Sets and γ -Close d Spaces SABIR HUSSAIN Departmen t of Mathematics, Islamia Univ ersit y Bahaw alpur, P akistan. Presen t Address: Departmen t of Mathematics, Y an bu Univ ersit y , P . O. Bo x 31387, Y an bu Alsinaiy ah, Saudi Arabia. E. mail: sabiriub@y aho o.com. Abstract. The purp ose of this p ap er is to con tin ue studying the prop erties o f γ -regular op en sets introdu ced and explored in [6]. The concept of γ -closed sp aces ha ve also b een defin ed and discussed. AMS Sub ject C lassification: 54A05, 5 4A10, 54D 10, 54D99 . Keyw ords. γ -clo sed (op en), γ -interior(cl osure), γ -regular-open (closed), γ - θ -op en(closed), γ -extremally disconnected, γ -R-con v erge, γ -R-accumulat e, γ -closed space s. 1 In tro duction The concept of op er ation γ w as initiated b y S . K asahara [7]. He also in tro duced γ -closed graph of a function. Usin g this op eration, H. Ogata [8] introd uced the concept of γ -open sets and in v estigated the related topological prop erties of the associated top ology τ γ and τ . He fu rther in v estigated general op erator app roac hes of close graph of mappings. F urther S. Hu ssain and B. Ahmad [1-6] con tin ued studying the prop erties of γ -op en(closed) sets and generalized man y classical notions in their w ork. T he purp ose of this p ap er is to con tin ue studying the pr op erties of γ -regular op en sets introd uced and explored in [6]. The conce pt of γ -closed spaces h av e also b een defin ed and discus sed. First, we recall some defin itions and r esults u sed in this pap er. Hereafter, we shall wr ite a space 1 in place of a top olog ical space. 2 Preliminaries Throughout the present pap er, X denotes top ological spaces. Definition [7]. An op eratio n γ : τ → P(X) is a function fr om τ to th e p o w er set of X s uc h that V ⊆ V γ , for eac h V ∈ τ , wh ere V γ denotes the v a lue of γ at V. The op erations defined b y γ (G) = G, γ (G) = cl(G) and γ (G) = intcl(G ) are examples of op eration γ . Definition [7]. Let A ⊆ X . A p oint x ∈ A is said to b e γ -in terior p oint of A, if there exists an op en nbd N of x suc h that N γ ⊆ A and w e denote the set of all such p oints b y int γ (A). Thus int γ (A) = { x ∈ A : x ∈ N ∈ τ and N γ ⊆ A } ⊆ A . Note that A is γ -op en [8] iff A = int γ (A). A set A is called γ - closed [1] iff X-A is γ -op en. Definition [1]. A p oin t x ∈ X is called a γ -closure p oint of A ⊆ X, if U γ ∩ A 6 = φ , f or eac h op en n b d U of x. The set of all γ -closure p oints of A is called γ -closure of A and is d en oted by cl γ (A). A su b set A of X is called γ -closed, if cl γ ( A ) ⊆ A . Note that cl γ ( A ) is con tained in ev ery γ -closed sup er s et of A. Definition [7]. An op eration γ on τ is said b e regular, if for any op en nb ds U,V of x ∈ X, there exists an op en nb d W of x such that U γ ∩ V γ ⊇ W γ . Definition [8]. An op eration γ on τ is said to b e op en, if for an y op en n b d U of eac h x ∈ X , th ere exists γ -op en set B suc h that x ∈ B and U γ ⊇ B . 3 γ - Regular-Op en Sets Definition 3.1 [6]. A subset A of X is said to be γ -regular-op en (resp t. γ -regular-closed), if A = int γ ( cl γ ( A )) (resp t. A = cl γ ( int γ ( A ))). It is clear th at R O γ ( X, τ ) ⊆ τ γ ⊆ τ [6]. The follo wing example shows that the con v erse of ab o v e in clusion is not tru e in general. Example 3.2. Let X= { a, b, c } , τ = { φ, X, { a } , { b } , { a, b } , { a, c }} . F or b ∈ X , define an op eration γ : τ → P ( X ) by 2 γ ( A ) =    A, if b ∈ A cl ( A ) , if b 6∈ A Calculations sho ws that { a, b } , { a, c } , { b } , X, φ are γ -op en sets and { a, c } , { b } , X, φ are γ -regular- op en sets. Here set { a, b } is γ -op en b ut not γ -regular-op en. Definition 3.3[7]. A space X is calle d γ -extremall y disconnected, if for all γ -op en sub set U of X, cl γ ( U ) is a γ -op en sub set of X. Prop osition 3.4. If A is a γ -clop en set in X, then A is a γ -regular-op en set. Moreo v er, if X is γ -extremally disconnected th en the con v erse holds. Pro of. If A is a γ -clopan set, then A = cl γ ( A ) and A = int γ ( A ), and so we hav e A = int γ ( cl γ ( A )). Hence A is γ -regular-op en. Supp ose that X is a γ -extremally disconnected space and A is a γ -regular-op en set in X. Then A is γ -op en and so cl γ ( A ) is a γ -op en set. Hence A = int γ ( cl γ ( A )) = cl γ ( A ) and hence A is γ -clo sed set. This completes the pro of. The follo wing example s h o ws th at s pace X to b e γ -extremally disconnected is necessary in the con v erse of ab o v e P r op osition. Example 3.5 Let X= { a, b, c } , τ = { φ, X, { a } , { b } , { a, b }} . Define an op eration γ : τ → P ( X ) b y γ ( B ) = int ( cl ( B )). Clearly X is n ot γ -extremally disconnected s p ace. Calculations sho ws that { a } , { a, b } , { b } , X, φ are γ -op en as well as γ -regular-op en sets. Here { a } is a γ -regular-op en set but not γ -clopan s et. Theorem 3.6. Let A ⊆ X , then ( a ) ⇒ ( b ) ⇒ ( c ), where : (a) A is γ -clopan. (b) A = cl γ ( int γ ( A )). (c) X − A is γ -regular-op en. Pro of. ( a ) ⇒ ( b ). T h is is obvious. ( b ) ⇒ ( c ). Let A = cl γ ( intγ ( A )). Then X − A = X − cl γ ( intγ ( A )) = int γ ( X − int γ ( A )) = int γ ( cl γ ( X − A )), and hence X − A is γ -regular-op en set. Hence the pr o of. 3 Using Prop osition 3.4, we ha v e the follo wing T heorem: Theorem 3.7. If X is a γ -extremally disconnected space. Th en ( a ) ⇒ ( b ) ⇒ ( c ), where : (a) X − A is γ -regular-op en. (b) A is γ -regular-op en. (c) A is γ -clopan. Pro of. ( a ) ⇒ ( b ). Supp ose X is γ -extremally disconnected space. F rom Pr op osition 3.4. , X − A is a γ -op en and γ -closed set, and h ence A is a γ -op en and γ -close d set. Th us A = int γ ( cl γ ( A )) implies A is γ -regular-op en set. ( b ) ⇒ ( c ). This d irectory follo ws from P rop osition 3.4. This completes as requir ed . Com bining Theorems 3.6 and 3.7, we ha v e the follo wing: Theorem 3.8. If X is a γ -extremally disconnected space. Th en the follo wing statemen ts are equiv alen t: (a) A is γ -clopan. (b) A = cl γ ( int γ ( A )). (c) X − A is γ -regular-op en. (d) A is γ -regular-op en. Theorem 3.9. L et A ⊆ X and γ b e an op en op eration. If cl γ ( A ) is a γ -regular-op en set. Then A is a γ -op en set in X. Moreo v er, if X is extremally γ -disconnected then the con v erse holds. Pro of. Su p p ose that cl γ ( A ) is a γ -regular-op en sets. Since γ is op en, we ha ve A ⊆ cl γ ( A ) ⊆ int γ ( cl γ ( cl γ ( A ))) = int γ ( cl γ ( A )) = int γ ( A ). This implies that A is γ -op en set. Supp ose that X is γ -extremally disconnected and A is γ -op en set. Then cl γ ( A ) is a γ -op en set, and hence γ -clopan set. Thus by Theorem 3.8, cl γ ( A ) is a γ -regular-op en set. This completes the pro of. Corollary 3.10. Let X b e a γ -extremally disconn ected space. Then for eac h subset A of X, the set cl γ ( int γ ( A )) is γ -regular-open sets. 4 Definition 3.11. A p oint x ∈ X is said to b e a γ - θ -cluster p oint of a subset A of X, if cl γ ( U ) ∩ A 6 = φ for every γ -op en set U conta ining x. The set of all γ - θ -cluster p oin ts of A is called the γ - θ -closure of A and is den oted by γ cl θ ( A ). Definition 3.12. A s ubset A of X is said to b e γ - θ -closed, if γ cl θ ( A ) = A . T he complemen t of γ - θ -closed set is called γ - θ -op en sets. Clearly a γ - θ -closed ( γ - θ -op en ) is γ -closed( γ -op en) set. Prop osition 3.13. L et A and B b e sub sets of a space X. Th en the follo wing prop erties hold: (1) If A ⊆ B , then γ cl θ ( A ) ⊆ γ cl θ ( B ). (2) If A i is γ - θ -closed in X, for eac h i ∈ I , then T i ∈ I A i is γ - θ -closed in X. Pro of. (1). T his is obvious. (2). Let A i b e a γ - θ -closed in X for eac h i ∈ I . T hen A i = γ cl θ ( A i ) for eac h i ∈ I . Thus w e ha ve γ cl θ ( T i ∈ I A i ) ⊆ T i ∈ I γ cl θ ( A i ) = T i ∈ I A i ⊆ γ cl θ ( T i ∈ I A i ). Therefore, w e ha v e γ cl θ ( T i ∈ I A i ) = T i ∈ I A i and hence T i ∈ I A i is γ - θ -closed. Hence the p ro of. Theorem 3.14. If γ is an op en op eration. T hen for any subset A of γ -extremally disconnected space X, the f ollo wing hold: γ cl θ ( A ) = T { V : A ⊆ V and V is γ - θ -closed } = T { V : A ⊆ V and V is γ -regular-op en } Pro of. Let x / ∈ γ cl θ ( A ). Th en there is a γ -op en set V with x ∈ V such that cl γ ( V ) ∩ A = φ . By Theorem 3.9, X − cl γ ( V ) is γ -regular-open and hen ce X − cl γ ( V ) is a γ - θ -closed set contai ning A and x / ∈ X − γ cl θ ( V ). Thus w e hav e x / ∈ T { V : A ⊆ V and V is γ - θ -clo sed } . Con v ersely , supp ose that x / ∈ T { V : A ⊆ V and V is γ - θ -closed } . Then there exists a γ - θ -closed set V s uc h th at A ⊆ V and x / ∈ V , and s o th ere exists a γ -op en set U with x ∈ U s uc h th at U ⊆ cl γ ( U ) ⊆ X − V . Thus we ha v e cl γ ( U ) ∩ A ⊆ cl γ ( U ) ∩ V = φ implies x / ∈ γ cl θ ( A ). The pro of of the second equation follo ws similarly . Th is completes the p ro of. Theorem 3.15. Let γ b e an op en op eration. If X is a γ -extremally disconnected space and A ⊆ X . Then the follo wings h old: (a) x ∈ γ cl θ ( A ) if and only if V ∩ A 6 = φ , for eac h γ -regular-op en set V w ith x ∈ V . 5 (b) A is γ - θ -open if and only if for eac h x ∈ A there exists a γ -regular-op en set V with x ∈ V s u c h that V ⊆ A . (c) A is a γ -regular-op en set if and only if A is γ - θ -clo pan. Pro of. (a) and (b) follo ws directly from Theorems 3.8 and 3.9. (c) Let A b e a γ -regular-op en set. Then A is a γ -op en set and so A = cl γ ( A ) = γ cl θ ( A ) an d hence A is γ - θ -clo sed. Since X − A is a γ -regular-op en set, by the argument ab o v e, X − A is γ - θ -closed and A is γ - θ -op en. The con v erse is obvious. Hence th e pro of. It is obvio us that γ -regular-op en ⇒ γ - θ -open ⇒ γ -op en. But the conv erses are not necessarily true as the f ollo wing examples sho w. Example 3.16. Let X= { a, b, c } , τ = { φ, X, { a } , { b } , { a, b } , { a, c }} . F or b ∈ X , define an op eratio n γ : τ → P ( X ) by γ ( A ) =    A, if b ∈ A cl ( A ) , if b 6∈ A Calculations sho ws that { a, b } , { a, c } , { b } , X , φ are γ -op en sets as we ll as γ - θ -op en sets and γ - regular-op en sets are { a, c } , { b } , X, φ . Then the s ubset { a, b } is γ - θ -op en but not γ -regular-op en. Example 3.17. Le t X= { a, b, c } , τ = { φ, X, { a } , { b } , { a, b } , { a, c }} b e a top ology on X. F or b ∈ X , define an op eration γ : τ → P ( X ) by γ ( A ) = A γ =    cl ( A ) , if b ∈ A A, if b 6∈ A Calculations s h o w s that { φ, X, { a } , { a, c }} are γ -op en s ets and { φ, X, { a, c }} are γ - θ -open s ets. The the subset { a } is γ -op en b ut n ot γ - θ -op en. 4 γ - Closed Spaces Definition 4.1 . A filterbase Γ in X, γ -R-con v erges to x 0 ∈ X , if for eac h γ -regular-op en s et A with x 0 ∈ A , there exists F ∈ Γ su c h that F ⊆ A . Definition 4.2. A filterbase Γ in X γ -R-accum ulates to x 0 ∈ X , if for eac h γ -regular-op en s et A 6 with x 0 ∈ A and eac h F ∈ Γ, F ∩ A 6 = φ . The follo wing Theorems directly follo w from the ab o v e d efinitions. Theorem 4.3. If a filterbase Γ in X, γ -R-con ve rges to x 0 ∈ X , then Γ γ -R-accum u lates to x 0 . Theorem 4.4. I f Γ 1 and Γ 2 are filterbases in X suc h that Γ 2 sub ord inate to Γ 1 and Γ 2 γ -R- accum u lates to x 0 , then Γ 1 γ -R-acc umulate s to x 0 . Theorem 4.5. If Γ is a maximal fi lterbase in X, then Γ γ -R-accum ulates to x 0 if and only if Γ γ -R-con v erges to x 0 . Definition 4.6. A sp ace X is said to b e γ -closed, if ev ery co v er { V α : α ∈ I } of X by γ -op en sets has a finite subset I 0 of I suc h that X = S α ∈ I cl γ ( V α ). Prop osition 4.7. If γ is an op en op eration, Then the follo wing are equ iv alen t: (1) X is γ -closed. (2) F or eac h family { A α : α ∈ I } of γ -closed subsets of X suc h that T α ∈ I A α = φ , there exists a finite s ubset I 0 of I such th at T α ∈ I 0 int γ ( A α ) = φ . (3) F or eac h family { A α : α ∈ I } of γ -closed sub s ets of X, if T α ∈ I 0 int γ ( A α ) 6 = φ , for eve ry fin ite subset I 0 of I, then T α ∈ I A α 6 = φ . (4) Every fi lterbase Γ in X γ -R-accum ulates to x 0 ∈ X . (5) Every m aximal fi lterbase Γ in X γ -R-con v erges to x 0 ∈ X . Pro of. (2) ⇔ (3). T his is obvious. (2) ⇒ (1). Let { A α : α ∈ I } b e a family of γ -op en sub sets of X such hat X = S α ∈ I A α . Then eac h X − A α is a γ -clo sed s u bset of X and T α ∈ I ( X − A α ) = φ , and so there exists a fin ite subset I 0 of I su c h that T α ∈ I 0 int γ ( X − A α ) = φ , and hence X = S α ∈ I 0 ( X − int γ ( X − A α )) = S α ∈ I 0 cl γ ( A α ). Therefore X is γ -closed, since γ is op en. (4) ⇒ (2). Let { A α : α ∈ I } b e a family of γ -closed sub sets of X such that T α ∈ I A α = φ . Supp ose that for ev ery fin ite subfamily { A α i : i = 1 , 2 , ..., n } , T n i =1 int γ ( A α i ) 6 = φ . Then T n i =1 ( A α i ) 6 = φ and Γ = { T n i =1 A α i : n ∈ N , α i ∈ I } forms a filterbase in X. By (4), Γ γ -R- 7 accum u lates to some x 0 ∈ X . Th us for ev ery γ -op en set A with x 0 ∈ A and every F ∈ Γ, F ∩ cl γ ( A ) 6 = φ . Since T F ∈ Γ F = φ , there exists a F ∈ Γ suc h that x 0 / ∈ F , and so ther e exists α 0 ∈ I such that x 0 / ∈ A α 0 and hence x 0 ∈ X − A α 0 and X − A α 0 is a γ -op en set. Thus x 0 / ∈ int γ ( A α 0 ) and x 0 ∈ X − int γ ( A α 0 ), and hence F 0 ∩ ( X − int γ ( A α 0 )) = F 0 ∩ cl γ ( X − A α 0 ) = φ , whic h is a con trad iction to our hyp othesis. (5) ⇒ (4). Let Γ b e filterbase in X. Th en there exists a maximal filterbase ξ in X such that ξ sub ord inate to Γ . Since ξ γ -R-con v erges to x 0 , so b y T heorems 4.4 and 4.5, Γ γ -R-accum ulate to x 0 . (1) ⇒ (5). Su pp ose that Γ = { F a : a ∈ I } is a maximal filterbase in X whic h do es not γ -R-con ve rge to any p oin t in X. F rom Theorem 4.5, Γ do es not γ -R-accum ulates at a ny p oint in X. Thus for eve ry x ∈ X , th ere exists a γ -op en set A x con taining x and F a x ∈ Γ suc h that F a x ∩ cl γ ( A x ) = φ . Since { A x : x ∈ X } is γ -op en co v er of X, th ere exists a finite subfamily { A x i : i = 1 , 2 , ..., n } such that X = S n i =1 cl γ ( A x i ). Bec ause Γ is a filterb ase in X, there exists F 0 ∈ Γ su ch that F 0 ⊆ T n i =1 F a x i , and h en ce F 0 ∩ c l γ ( A x i )) = φ for all i = 1 , 2 , .. ., n . Hence w e ha v e that, φ = F 0 T ( S n i =1 cl γ ( A x i )) = F 0 ∩ X , and hen ce F 0 = φ . T his is a con tradiction. Hence the pro of. Definition 4.8. A net ( x i ) i ∈ D in a space X is said to b e γ -R-con ve rges to x ∈ X , if for eac h γ -op en s et U with x ∈ U , there exists i 0 suc h that x i ∈ cl γ ( U ) for all i > i 0 , where D is a dir ected set. Definition 4.9. A net ( x i ) i ∈ D in a space X is said to b e γ -R-accum ulates to x ∈ X , if for eac h γ -op en s et U with x ∈ U and eac h i , x i ∈ cl γ ( U ), where D is a directed set. The pro ofs of follo win g Prop ositio ns are easy and thus are omitted: Prop osition 4.10. Let ( x i ) i ∈ D b e a net in X. F or the filterbase F (( x i ) i ∈ D ) = {{ x i : i ≤ j } : j ∈ D } in X, (1) F (( x i ) i ∈ D ) γ -R-con v erges to x if and on ly if ( x i ) i ∈ D γ -R-con ve rges to x . (2) F (( x i ) i ∈ D ) γ -R-accum ulates to x if and only if ( x i ) i ∈ D γ -R-acc umulate s to x . Prop osition 4.11. E very filterbase F in X determines a net ( x i ) i ∈ D in X su c h that 8 (1) F γ -R-con v erges to x if and only if ( x i ) i ∈ D γ -R-con ve rges to x . (2) F γ -R-accum ulates to x if and only if ( x i ) i ∈ D γ -R-acc umulate s to x . F rom Pr op ositions 4.11 and 4.12, filterbses and nets are equiv alent in th e sense of γ -R-con v erges and γ -R-accum ulates. Th us we h av e th e follo wing T heorem: Theorem 4.13. F or a space X, the follo win g are equiv alent : (1) X is γ -closed. (2) Each net ( x i ) i ∈ D in X has a γ -R-accum ulation p oin t. (3) Each un iv ersal net in X γ -R-con ve rges. References [1] B. Ahmad and S . Hussain: Pr op erties of γ -O p er ations on T op olo gic al Sp ac es, A ligarh Bul l. Math. 22(1) (2003), 45-51. [2] B. Ahmad and S. Hussain: γ -Conver genc e in T op olo gic al Sp ac e , Southe ast Asian Bul l. Math., 29(20 05), 832-842 . [3] B. Ahmad an d S. Hussain: γ ∗ -R e gular and γ -N ormal Sp ac e , Math. T o day., 22(1)(2006 ), 37-44. [4] B. Ahmad and S. Hussain: On γ -s-Close d Subsp ac es , F ar East J r. Math. Sci., 31(2)(2008) , 279-2 91. [5] S. Huss ain and B. Ah m ad: On Minimal γ -Op en Sets, Eu r. J. Pur e A ppl. M aths., 2(3)(200 9), 338-3 51. [6] S. Huss ain and B. Ah m ad: On γ -s-Close d Sp ac es, Sci. M agna J r., 3(4)(2007), 89-93. [7] S. Kasahara: Op er ation-Comp act Sp ac e s, Math. 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