Journal of Advanced Research in Applied Mathematics Online ISSN: 1942-9649
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Vol. 5, Issue. 1, 2013, pp. 60-65 doi: 10.5373/jaram.1396.041312
ˇ gα-Closed sets in biCech closure spaces
M. Vigneshwaran∗ , R. Devi Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore, Tamil Nadu, India.
Abstract. In this paper, we introduce the concepts of (k1 , k2 )-∗ gα-closed sets, (k1 , k2 )∗ ˇ ˇ gα-open sets, ∗ gα C0 -biCech spaces and ∗ gα C1 -biCech spaces. Keywords: (k1 , k2 )-∗ gα-closed set; (k1 , k2 )-∗ gα-open set; ˇ biCech closure space.
∗ gα
ˇ C0 -biCech closure space;
∗ gα
C1 -
Mathematics Subject Classification 2010: 54A05, 54D10, 54F65, 54G05.
1
Introduction
ˇ ˇ Cech closure spaces were introduced by Cech[3] and then studied many authors [1, 2, ˇ 4, 5, 6, 9]. In Cech’s approach, the operator satisfies idempodent condition among Kuratowski axioms. This condition need not hold for every subset A of X. When this condition is also true, the operator becomes topological closure operator. Thus the concept of closure space is the generalization of a topological space. M. Vigneshwaran et.al [12] introduced the concepts of ∗ gα-closed sets and investigated their properties in topological spaces. In this paper, we introduce the concepts of (k1 , k2 )-∗ gα-closed sets, ˇ ˇ (k1 , k2 )-∗ gα-open sets, ∗ gα C0 -biCech closure spaces and ∗ gα C1 -biCech closure spaces.
2
Preliminaries
Definition 2.1. [4] Two functions k1 and k2 from power set X to itself are called ˇ biCech closure operators (briefly biclosure operator) for X if they satisfies the following properties: (a) k1 (ϕ) = ϕ and k2 (ϕ) = ϕ.
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Correspondence to: M. Vigneshwaran, Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore, Tamil Nadu, India. Email:
[email protected] † Received: 13 April 2012, accepted: 15 October 2012. http://www.i-asr.com/Journals/jaram/
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c ⃝2013 Institute of Advanced Scientific Research
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(b) A ⊂ k1 (A) and A ⊂ k2 (A) for any set A ⊂ X. (c) k1 (A ∪ B) = k1 (A) ∪ k1 (B) and k2 (A ∪ B) = k2 (A) ∪ k2 (B) for any A, B ⊂ X. A ˇ structure (X, k1 , k2 ) is called a biCech closure space. Example 2.1. Let X = {a, b, c} and let k1 and k2 be defined as: k1 ({a}) = {a}, k1 ({b}) = k1 ({c}) = k1 ({b, c}) = {b, c}, k1 ({a, b}) = k1 ({a, c}) = k1 ({X}) = X and k1 ({ϕ}) = ϕ. k2 ({a}) = {a}, k2 ({b}) = {b, c}, k2 ({c}) = k2 ({a, c}) = {a, c}, k2 ({a, b}) = k2 ({b, c}) = k2 (X) = X and k2 ({ϕ}) = ϕ. ˇ Now, (X, k1 , k2 ) is a biCech closure space. ˇ Definition 2.2. [5] A subset A of a biCech closure space (X, k1 , k2 ) is said to be (a) ki -closed if A = ki (A), i = 1, 2. (b) ki -α-closed if ki [intki (ki (A))] ⊆ A, i = 1, 2. (c) ki -α-open if A ⊆ intki (ki [intki (A)]), i = 1, 2. The smallest ki -α closed set containing A is called ki -α closure of A and it is denoted by kαi (A). The largest ki -α open set contained in A is called ki -α interior of A and it is denoted by intkαi (A), i=1,2.
3
(k1 , k2 )-∗ gα-closed sets
ˇ Definition 3.1. A subset A of a biCech closure space (X, k1 , k2 ) is said to be (a) (k1 , k2 )-gα-closed if kα2 (A) ⊆ U whenever A ⊆ U and U is a k1 -α-open set in X. (b) (k1 , k2 )-gα-open if the complement of A is (k1 , k2 )-gα-closed. ˇ Definition 3.2. A subset A of a biCech closure space (X, k1 , k2 ) is said to be (k1 , k2 )if k2 (A) ⊆ U whenever A ⊆ U and U is a (k1 , k2 )-gα-open set in X.
∗ gα-closed
Example 3.1. In Example 2.1, when U = {b, c}, A = {b} is (k1 , k2 )-∗ gα-closed. ˇ Theorem 3.3. If A and B are (k1 , k2 )-∗ gα-closed sets in a biCech closure space (X, k1 , k2 ) and so is A ∪ B. Proof. Let A and B be two (k1 , k2 )-∗ gα-closed sets. Let U be a (k1 , k2 )-gα-open set in X. Let (A ∪ B) ⊆ U . We have A ⊆ U and B ⊆ U . Then k2 (A) ⊆ U and k2 (B) ⊆ U implies (k2 (A) ∪ k2 (B)) ⊆ U . Hence k2 (A ∪ B) ⊆ U . Thus A ∪ B is (k1 , k2 )-∗ gα-closed set. ˇ Theorem 3.4. If A is a (k1 , k2 )-∗ gα-closed set in a biCech closure space (X, k1 , k2 ), then k2 (A)-A contains no non-empty (k1 , k2 )-gα-closed sets.
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∗
ˇ gα-Closed sets in biCech closure spaces
Proof. Let A be (k1 , k2 )-∗ gα-closed. Let U be (k1 , k2 )-gα-closed contained in k2 (A) − A. Now, U ⊆ k2 (A) and U ⊆ Ac .
(3.1)
Then A ⊆ U c . Since U is (k1 , k2 )-gα-closed, U c is (k1 , k2 )-gα-open. Thus, we have k2 (A) ⊆ U c . Consequently, U ⊆ [k2 (A)]c .
(3.2)
From (3.1) and (3.2), U ⊆ k2 (A) ∩ [k2 (A)]c =ϕ. Therefore U = ϕ. Hence k2 (A) − A contains no non-empty (k1 , k2 )-gα-closed sets. ˇ Theorem 3.5. If A is a (k1 , k2 )-∗ gα-closed set in a biCech closure space (X, k1 , k2 ), then kα1 (x) ∩ A ̸= ϕ holds for each x ∈ k2 (A). Proof. Let A be a (k1 , k2 )-∗ gα-closed set. Suppose that kα1 (x) ∩ A=ϕ for some x ∈ k2 (A), we have A ⊆ [kα1 (x)]c . Now, kα1 (x) is k1 -α-closed. Therefore [kα1 (x)]c is k1 -αopen. Thus [kα1 (x)]c is (k1 , k2 )-gα-open. Since A is a (k1 , k2 )-∗ gα-closed set, we have k2 (A) ⊆ [kα1 (x)]c implies k2 (A) ∩ kα1 (x)=ϕ. Then x ∈ / k2 (A) is a contradiction. Hence kα1 (x) ∩ A ̸= ϕ holds for each x ∈ k2 (A). ˇ Theorem 3.6. Let (X, k1 , k2 ) be a biCech closure space. For each x in X, {x} is c ∗ (k1 , k2 )-gα-closed or {x} is (k1 , k2 )- gα-closed. ˇ closure space. Suppose that {x} is not (k1 , k2 )-gαProof. Let (X, k1 , k2 ) be a biCech c closed, {x} is not (k1 , k2 )-gα-open. Therefore, the only (k1 , k2 )-gα-open set containing {x}c is X. Thus {x}c ⊆ X. Now k2 [{x}c ] ⊆ k2 (X)=X. Hence {x}c is a (k1 , k2 )-∗ gαclosed set. ˇ Theorem 3.7. Let A be a (k1 , k2 )-∗ gα-closed set in a biCech closure space (X, k1 , k2 ). If A is (k1 , k2 )-gα-open then A = k2 (A). Proof. Let A be both (k1 , k2 )-∗ gα-closed and (k1 , k2 )-gα-open set in X. Then k2 (A) ⊆ U whenever A ⊆ U and U is a (k1 , k2 )-gα-open set in X. Since A is (k1 , k2 )-gα-open and A ⊆ A, we have k2 (A) ⊆ A. But, always A ⊆ k2 (A). Thus A = k2 (A). Theorem 3.8. Let A ⊆ Y ⊆ X and suppose that A is (k1 , k2 )-∗ gα-closed in (X, k1 , k2 ). Then A is (k1 , k2 )-∗ gα-closed relative to Y . Proof. Let S be any (k1 , k2 )-gα-open set in Y such that A ⊆ S. Then S = U ∩ Y for some U is (k1 , k2 )-gα-open in X. Therefore A ⊂ U ∩ Y implies A ⊆ U . Since A is a (k1 , k2 )-∗ gα-closed set in X, we have k2 (A) ⊆ U . Hence Y ∩ k2 (A) ⊆ Y ∩ U = S. Thus A is a (k1 , k2 )-∗ gα-closed set relative to Y .
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(k1 , k2 )-∗ gα-open sets
ˇ Definition 4.1. A subset A of a biCech closure space (X, k1 , k2 ) is called (k1 , k2 )-∗ gαc ∗ open if A is (k1 , k2 )- gα-closed in (X, k1 , k2 ). Example 4.1. In Example 2.1, when U = {b, c}, A = {a, c} is a (k1 , k2 )-∗ gα-open set. ˇ Theorem 4.2. A subset A of a biCech closure space (X, k1 , k2 ) is (k1 , k2 )-∗ gα-open if and only if F ⊂intk2 (A) whenever F is a (k1 , k2 )-gα-closed set and F ⊆ A. Proof. Suppose that A is (k1 , k2 )-∗ gα-open in (X, k1 , k2 ). Let F be a (k1 , k2 )-gα-closed set and F ⊆ A. Then F c is (k1 , k2 )-gα-open and Ac ⊆ F c . This implies that F ⊆ [k2 (Ac )]c = intk2 (A). That is, F ⊆ intk2 (A) whenever F is a (k1 , k2 )-gα-closed set and F ⊆ A. Let V be any (k1 , k2 )-gα-open set in X such that Ac ⊆ V . Thus V c ⊆ A and V c is (k1 , k2 )-gα-closed. Therefore V c ⊆ intk2 (A). Then [intk2 (A)]c ⊆ V . Hence k2 (Ac ) ⊆ V and Ac is a (k1 , k2 )-∗ gα-closed set. Thus A is (k1 , k2 )-∗ gα-open. ˇ Corollary 4.3. If a subset A of a biCech closure space (X, k1 , k2 ) is (k1 , k2 )-∗ gα-closed ∗ set, then k2 (A) − A is (k1 , k2 )- gα-open set. Proof. Let F be a (k1 , k2 )-gα-closed set such that F ⊆ k2 (A) − A. Then, F = ϕ by Theorem 3.4. Therefore F ⊆ intk2 {k2 (A) − A}. Hence, by Theorem 4.2, k2 (A) − A is a (k1 , k2 )-∗ gα-open set. ˇ Theorem 4.4. If A and B be (k1 , k2 )-∗ gα-open sets in a biCech closure space (X, k1 , k2 ), then so is A ∩ B. Proof. Let (Ac ∪ B c ) ⊆ U where U is (k1 , k2 )-gα-open. This implies that Ac ⊆ U and B c ⊆ U gives k2 (Ac ) ⊆ U and k2 (B c ) ⊆ U . Thus (k2 (Ac ) ∪ k2 (B c )) ⊆ U . Thus k2 (Ac ∪ B c ) ⊆ U . Therefore A ∩ B is a (k1 , k2 )-∗ gα-open set. Theorem 4.5. Let A ⊆ Y ⊆ X and suppose that Y is k2 -closed in X and A is (k1 , k2 )in X, then A is a (k1 , k2 )-∗ gα-open set relative to Y .
∗ gα-open
Proof. Let S be any (k1 , k2 )-gα-closed set in Y such that S ⊆ A. Then S = U ∩ Y for some U is (k1 , k2 )-gα-closed in X. Therefore U ∩ Y ⊂ A implies U ⊆ A. Since A is a (k1 , k2 )-∗ gα-open set in X, we have U ⊆ intk2 (A). Hence S = Y ∩ U ⊆ Y ∩ intk2 (A). Thus A is a (k1 , k2 )-∗ gα-open set relative to Y .
5
∗ gα
ˇ C0 -biCech spaces and
∗ gα
ˇ C1 -biCech spaces
ˇ ˇ Definition 5.1. A biCech closure space (X, k1 , k2 ) is said to be a ∗ gα C0 -biCech space ∗ if for every (k1 , k2 )- gα-open subset U of (X, k1 ), x ∈ U implies k2 ({x}) ⊆ U .
∗
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ˇ gα-Closed sets in biCech closure spaces
Example 5.1. Let X = {a, b, c} and define a closure operator k1 on X by k1 ({ϕ}) = ϕ, k1 ({a}) = {a}, k1 ({b}) = k1 ({c}) = k1 ({b, c}) = {b, c} and k1 ({a, b}) = k1 ({a, c}) = k1 (X) = X. Define a closure operator k2 on X by k2 ({ϕ}) = ϕ, k2 ({a}) = {a}, k2 ({b}) = {b, c}, k2 ({c}) = k2 ({a, c}) = {a, c} and k2 ({a, b}) = k2 ({b, c}) = k2 (X) = X. ˇ Then (X, k1 , k2 ) is a ∗ gα C0 -biCech space. ˇ ˇ Theorem 5.2. A biCech closure space (X, k1 , k2 ) is a ∗ gα C0 -biCech space if and only ∗ if for every (k1 , k2 )- gα-closed subset F of (X, k1 ) such that x ∈ / F , k2 ({x}) ∩ F = ϕ. Proof. Let F be a (k1 , k2 )-∗ gα-closed subset of (X, k1 ) and let x ∈ / F . Since x ∈ X − F and X − F is a (k1 , k2 )-∗ gα-open subset of (X, k1 ), k2 ({x}) ⊆ X − F . Consequently k2 ({x}) ∩ F = ϕ. Conversely, let U be a (k1 , k2 )-∗ gα-open subset of (X, k1 ) and let x ∈ U . Since X − U is a (k1 , k2 )-∗ gα-closed subset of (X, k1 ) and x ∈ / X − U , k2 ({x}) ∩ (X − U ) = ϕ. ˇ Consequently k2 ({x}) ⊆ U . Hence (X, k1 , k2 ) is a ∗ gα C0 -biCech space. ˇ ˇ Definition 5.3. A biCech closure space (X, k1 , k2 ) is said to be ∗ gα C1 -biCech space if ∗ for each x, y ∈ X such that k1 ({x}) ̸= k2 ({y}), there exist a disjoint (k1 , k2 )- gα-open subset U of (X, k2 ) and a (k1 , k2 )-∗ gα-open subset V of (X, k1 ) such that k1 ({x}) ⊆ U and k2 ({y}) ⊆ V . Example 5.2. Let X = {a, b} and define a closure operator k1 on X by k1 ({ϕ}) = ϕ and k1 ({a}) = k1 (X) = X. Define a closure operator k2 on X by k2 ({ϕ}) = ϕ and ˇ k2 ({b}) = k2 (X) = X. Then (X, k1 , k2 ) is a ∗ gα C1 -biCech space. Theorem 5.4. Every
∗ gα
ˇ C1 -biCech space is a
∗ gα
ˇ C0 -biCech space.
ˇ Proof. Let (X, k1 , k2 ) be a ∗ gα C1 -biCech space. Let U be a (k1 , k2 )-∗ gα-open subset of (X, k1 ) and let x ∈ U . If y ∈ / U , then k2 ({x}) ̸= k1 ({y}) because x ∈ / k1 ({y}). Then there exists a (k1 , k2 )-∗ gα-open subset Vy of (X, k2 ) such that k1 ({y}) ⊆ Vy and x ∈ / Vy , which ˇ implies y ∈ / k2 ({x}). Consequently, k2 ({y}) ⊆ U . Hence (X, k1 , k2 ) is a ∗ gα C0 -biCech space. The converse of the above theorem need not be true. It can be seen by the following example. Example 5.3. Let X = {a, b} and define a closure operator k1 on X by k1 ({ϕ}) = ϕ and k1 ({a}) = k1 (X) = X. Define a closure operator k2 on X by k2 ({ϕ}) = ϕ, k2 ({a}) = {a} ˇ and k2 ({b}) = k2 (X) = X. Then (X, k1 , k2 ) is a ∗ gα C0 -biCech space, but it is not a ˇ ∗ gα C1 -biCech space. ˇ ˇ Theorem 5.5. A biCech closure space (X, k1 , k2 ) is a ∗ gα C1 -biCech space if and only if every pair of points x, y of (X, k1 , k2 ) such that k1 ({x}) ̸= k2 ({y}), there exist a (k1 , k2 )-∗ gα-open subset U of (X, k1 ) and a (k1 , k2 )-∗ gα-open subset V of (X, k2 ) such that x ∈ V , y ∈ U and U ∩ V = ϕ.
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ˇ space. Let x, y be points of (X, k1 , k2 ) Proof. Suppose that (X, k1 , k2 ) is a ∗ gα C1 -biCech such that k1 ({x}) ̸= k2 ({y}). There exist a disjoint (k1 , k2 )-∗ gα-open subset U of (X, k1 ) and a (k1 , k2 )-∗ gα-open subset V of (X, k2 ) such that x ∈ k1 ({x}) ⊆ V and y ∈ k2 ({y}) ⊆ U. Conversely, suppose that there exist a (k1 , k2 )-∗ gα-open subset U of (X, k1 ) and a (k1 , k2 )-∗ gα-open subset V of (X, k2 ) such that x ∈ V , y ∈ U and U ∩ V = ϕ. Since ˇ ˇ every ∗ gα C1 -biCech space is a ∗ gα C0 -biCech space, k1 ({x}) ⊆ V and k2 ({y}) ⊆ U . References ˇ ˇ [1] C. Boonpok. C0 -biCech spaces and C1 -biCech spaces. Acta Math. Acad. Paedagog. Nyhazi. (N. Si), 2009, 25(2): 277 - 281. ˇ [2] C. Boonpok. Generalized biclosed sets in biCech spaces. Int. J. Math. Anal. (Ruse), 2010, 4(1-4): 89 -9 7. ˇ [3] E. Cech. Topological Spaces, Interscience Publishers. John Wiely and Sons, New York, 1966. [4] K. Chandrasekhara Rao, R. Gowri. On biclosure spaces. Bull. Pure Appl. Sci. Sect. E Math. Stat., 2006, 25: 171 - 175. [5] K. Chandrasekhara Rao, R. Gowri. Regular generalized closed sets in biclosure spaces. J. Inst. Math. Comput. Sci. Math. Ser., 2006, 19(3): 283 - 286. ˇ [6] K. Chandrasekhara Rao, R. Gowri, V. Swaminathan. αgs closed sets in biCech spaces. Int. J. Contemp. Math. Sci., 2008, 3(21-24): 1165 - 1172. [7] J. Chvalina. On homeomorphic topologies and equivalent set-systems. Arch. Math. (Brno), 1976, 12(2): 107 - 115. [8] J. Chvalina. Stackbases in power sets of neighbourhood spaces preserving the continuity of mappings. Arch. Math. (Brno), 1981, 17(2): 81 - 86. ˇ [9] G.M. Pandya, C. Janaki, I. Arockiarani. πgα-separation axioms in biCech spaces. Int. Math. Forum, 2011, 6(21-24): 1045 - 1052. [10] L. Skula. Systeme von stetigen Abbildungen. Czechoslovak Math. J., 1967, 17(92): 45 - 52. ˇ [11] J. Slapal. Closure operations for digital topology. Theoret. Comput. Sci., 2003, 305(1-3): 457 - 471, doi: 10.1016/S0304-3975(02)00708-9. [12] M. Vigneshwaran, R. Devi. On GαO-kernel in the digital plane. International Journal of Mathematical Archive, 2012, 3(6): 2358 - 2373
