On θ-Semigeneralized Pre Closed Sets in Topological Spaces

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Issue 3, March 2015, PP 74-80 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org On θ-Semigeneralized Pre Closed Sets in Topological Spaces Md. Hanif PAGE V.T. Hosamath Asst. Professor, Department of Mathematics, B.V.B. College of Engg. and Technology, Hubli-580031, Karnataka, India. [email protected] Asst. Professor, Department of Mathematics, K.L.E. Institute of Technology, Hubli-580030, Karnataka, India. [email protected] Abstract: This paper introduces new class of sets called θ-semigeneralized pre closed set in topological spaces.Basic properties of this new generalized closed sets are analysed. 2010 Mathematics Subject Classification: 57N505 Keywords: θ-sgp-closed, θ-sg-closed, θ-g-closed set, sg-closed set, sgp-closed set, semi-open set, pre-θclosed set. 1. INTRODUCTION General topology is important in many fields of applied sciences as well as in all branches of mathematics. The concept of generalized closed sets introduced by Levine[13] plays important role in general topology. This notion has been extensively studied in recent years by many topologists. Bhattacharyya and Lahiri [2] continued the work of Levine and offered another notion analogous to Levine’s g-closed sets called semi-generalized closed set (briefly sg-closed) by replacing the closure operator in Levine’s g-closed set by semi-closure operator and by replacing its open super set by semi-open super set. Recently, Dontchev and Maki [9] gave another new generalization of Levin’s g-closed set by utilizing θ-closure operator called θ-gclosed set. The concept of θ-g-closed set was applied to the digital line. In 2003, Caldas and Jafari defined θ-semigeneralized closed set using semi-θ-closure operator. In section three, we introduce a new form of generalized closed set called θ-semigeneralized pre closed set (briefly, θ-sgp-closed set) by utilizing pre-θ-closure operator. We investigate its relation to θ-g-closed sets, θ-sg-closed sets and other generalized closed sets. We have proved that the class of θ-sg-closed sets and the class of θ-sgp-closed sets are independent. 2. PRELIMINARIES Throughout this paper (X, τ) and (Y, σ) (or simply X and Y) denote topological spaces on which no separation axioms are assumed unless explicitly stated. If A is any subset of space X, then Cl(A) and Int(A) denote the closure of A and the interior of A in X respectively. The following definitions are useful in the sequel. Definition 2.1: A subset A of space X is called (i) a semi-open set [12] if A ⊆ Cl(Int(A)). (ii) a semi-closed set [5] if Int(Cl(A)) ⊆ A. (iii) a pre-open set[15] if A ⊆ Int(Cl(A)). (iv) a pre-closed set[15] if Cl(Int(A)) ⊆ A. (v) an α-closed set[16] if Cl(Int(Cl(A))) ⊆ A. (vi) a regular open set[21](resp. a regular closed set[21]) if A = Int(Cl(A))(resp. A = Cl(Int(A))). ©ARC Page | 74 Md. Hanif PAGE & V.T. Hosamath Definition 2.2: A subset A of a topological space X is called (i) a generalized-closed (briefly g-closed) set[13] if Cl(A) ⊆ U and U is open in X. (ii) a semi-generalized closed set (briefly sg-closed)[2] if sCl(A) ⊆ U and U is semi-open in X. The complement of a sg-closed set is called a sg-open set. (iii) a semi-generalized pre closed set (briefly sgp-closed)[17] if pCl(A) ⊆ U whenever A ⊆ U and U is semi-open in X. (iv) a generalized preregular closed set(briefly gpr-closed)[11] if pCl(A) ⊆ U whenever A ⊆ U and U is regular open in X. (v) an α-generalized semi-closed set(briefly αgs-closed)[20] if αCl(A) ⊂ U whenever A ⊂ U and U is semi-open in X. (vi) a generalized preclosed set(briefly gp-closed)[14] if pCl(A) ⊆ U whenever A ⊆ U and U is open in X. (vii) a generalized semi-preclosed set(briefly gsp-closed)[8] if spCl(A) ⊆ U whenever A ⊆ U and U is open in X. (viii) θ-generalized closed set(briefly θ-g-closed)[9] if Clθ(A) ⊆ U whenever A ⊆ U and U is open in X. (ix) θ-generalized semi-closed set(briefly θ-gs-closed)[18] if sClθ(A) ⊂ U whenever A ⊂ U and U is open in X. (x) θ-semigeneralized closed set(briefly θ-sg-closed)[4] if sClθ(A) ⊂ U whenever A ⊂ U and U is semi-open in X. Definition 2.3: The semi-closure [5] of a subset A of X is the intersection of all semi-closed sets that contain A and is denoted by sCl(A). Definition 2.4: The pre-closure [6] of a subset A of X is the intersection of all pre-closed sets that contain A and is denoted by pCl(A). Definition 2.5: The θ-closure [22] of a set A is denoted by Clθ(A) and is defined by Clθ(A) = {x  X : Cl(U) ∩ A ≠ Ø,U ε τ, x ε U} and a set A is θ-closed if and only if A = Clθ(A). Definition 2.6: A point x  X is called a semi-θ-cluster point of A [7] if sCl(U) ∩ A ≠ Ø, for each semi-open set U containing x. Definition 2.7: A point x  X is called a pre-θ-cluster point of A[19] if pCl(U) ∩ A ≠ Ø, for each pre-open set U containing x. Definition 2.8: The semi-θ-closure [7] denoted by sClθ(A), is the set of all semi-θ-cluster points of A. A subset A is called semi-θ-closed set [7] if A = sClθ(A). The complement of semi-θ-closed set is semi-θ-open set. Definition 2.9: The pre-θ-closure denoted by pClθ(A), is the set of all pre-θ-cluster points of A. A subset A is called pre-θ-closed set [19] if A = pClθ(A). The complement of pre-θ-closed set is preθ-open set. Definition 2.10: The set {x  X ǀ sCl(U) ⊂ A for some U ε SO(X, x)} is called the semi-θinterior of A and is denoted by sIntθ(A). A subset A is called semi-θ-open[10] if A = sIntθ(A). Definition 2.11: A topological space X is a pre-θ-R0 space[1] if every pre-θ-open set contains pre-θ-closure of each of its singletons. Definition 2.12: Let A be subset of a topological space X. The pre-θ-kernal[1] of A ⊂ X, denoted by pKerθ(A), is defined to be the set ∩{O : O  PθO(X, τ) and A ⊂ O}. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 75 On θ-Semigeneralized Pre Closed Sets in Topological Spaces Lemma 2.13[3]: For any subset A of a topological space X, pCl(A) ⊂ pClθ(A). 3. θ- SEMIGENARALIZED PRE CLOSED SETS We introduce the following definition. Definition 3.1: A subset A of a topological space X is called θ-Semigeneralized pre closed set (briefly, θ-sgp-closed set) if pClθ(A) ⊂ U whenever A ⊂ U and U is semi-open in X. The complement of θ-Semigeneralized pre closed set is called θ-Semigeneralized pre open set (briefly, θ-sgp-open). Remark 3.2: The concept of θ-sgp-closed sets and closed sets are independent of each other as seen from the following examples. Example 3.3: Let X = {a, b, c} and τ = {X, Ø, {a, b}}. Then the subset A = {a, c} is θ-sgp-closed set but it is not closed set in X. Example 3.4: Let X = {a, b, c} and τ = {X, Ø, {a}, {a, b}, {a, c}}. Then the subset A = {b, c} is closed set but it is not θ-sgp-closed set in X. Theorem 3.5: Every pre-θ-closed set is θ-sgp-closed set but not conversely. Proof: Let A ⊂ U be pre-θ-closed. Then A = pClθ(A). Let A ⊂ U and U is semi-open in X. It follows that pClθ(A) ⊂ U. This means that A is θ-sgp-closed set. Example 3.6: Let X = {a, b, c} and τ = {X, Ø, {a}, {b}, {a, b}, {a, c}}. Then the subset A = {b, c} is θ-sgp-closed set but it is not pre-θ-closed set in X. Theorem 3.7: Every θ-sgp-closed set is sgp-closed set but not converse. Proof: It is true that pCl(A) ⊂ pClθ(A) for every subset A of X. Example 3.8: Let X = {a, b, c} and τ = {X, Ø, {a}, {a, b}}. Set A = {b} and U = {a, b}. But pClθ(A) = X which is not a subset of U, where U is semi-open in X. Hence A = {b} is not θ-sgpclosed set. But it is sgp-closed set. Theorem 3.9: Every θ-sgp-closed set is gp-closed set. Proof: Let A be an θ-sgp-closed set in a topological space X. Let U be an open set and so it is semi-open such that A ⊆ U. Then pCl(A) ⊆ U. Hence A is gp-closed set. The converse of the above theorem need not be true as seen from the following example. Example 3.10: Let X = {a, b, c} and τ = {X, Ø, {a}, {c}, {a, c}}. Then a subset A = {a, b} is gpclosed set but it is not θ-sgp-closed set. Theorem 3.11: Every θ-sgp-closed set is gsp-closed set. Proof: Let A be a θ-sgp-closed set in X. Let A ⊆ U, where U is open and so it is semi-open set in X. Then pClθ(A) ⊆ U. But spCl(A) ⊆ pCl(A) ⊆ pClθ(A). Therefore spCl(A) ⊆ U. Hence A is gspclosed set in X. The converse of the above theorem need not be true as seen from the following example. Example 3.12: Let X = {a, b, c} and τ = {X, Ø, {a}}. Then a subset A = {a, c} is gsp-closed set but it is not θ-sgp-closed set. Remark 3.13: The concept of θ-sgp-closed sets and θ-gs-closed sets are independent of each other as seen from the following examples. Example 3.14: Let X = {a, b, c} and τ = {X, Ø, {a}, {c}, {a, c}}. Then the subset A = {a, b} is θgs-closed set but it is not θ-sgp-closed set in X. Example 3.15: Let X = {a, b, c} and τ = {X, Ø, {b}, {b, c}, {a, c}}. Then the subset A = {c} is θsgp-closed set but it is not θ-gs-closed set in X. Remark 3.16: The notion of θ-sgp-closed sets and α-closed sets are independent of each other as seen from the following examples. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 76 Md. Hanif PAGE & V.T. Hosamath Example 3.17: Let X = {a, b, c} and τ = {X, Ø, {a, b}}. Then the subset A = {a, c} is θ-sgpclosed set but it is not α-closed set in X. Example 3.18: Let X = {a, b, c} and τ = {X, Ø, {a}, {a, b}, {a, c}}. Then the subset A = {b, c} is α-closed set but it is not θ-sgp-closed set in X. Remark 3.19: The concept of θ-sgp-closed sets and αgs-closed sets are independent of each other as seen from the following examples. Example 3.20: Let X = {a, b, c} and τ = {X, Ø, {b}, {b, c}, {a, c}}. Then a subset A = {c} is a θsgp-closed set but it is not αgs-closed set. Example 3.21: Let X = {a, b, c} and τ = {X, Ø, {a}, {c}, {a, c}}. Then a subset A = {a, b} is αgs-closed set but it is not θ-sgp-closed set. Theorem 3.22: Every θ-g-closed set is θ-sgp-closed set. Proof: Let A be a θ-g-closed set in X. Let A ⊆ U, where U is open set in X. Then Clθ(A) ⊆ U. But pClθ(A) ⊆ Clθ(A). Therefore pClθ(A) ⊆ U. Hence A is θ-sgp-closed set in X. The converse of the above theorem need not be true as seen from the following example. Example 3.23: Let X = {a, b, c} and τ = {X, Ø, {b}, {b, c}, {a, c}}. Then a subset A = {c} is θsgp-closed set but it is not θ-g-closed set. Remark 3.24: The notion of θ-sgp-closed sets and θ-sg-closed sets are independent of each other as seen from the following examples. Example 3.25: Let X = {a, b, c} and τ = {X, Ø, {a}, {c}, {a, c}}. Then a subset A = {a, b} is θsg-closed set but it is not θ-sgp-closed set. Example 3.26: Let X = {a, b, c} and τ = {X, Ø, {b}, {b, c}, {a, c}}. Then a subset A = {c} is θsgp-closed set but it is not θ-sg-closed set. Theorem 3.27: Every θ-sgp-closed set is gpr-closed set. Proof: Let A be a θ-sgp-closed set in X. Let A ⊆ U, where U is regular-open and so it is semiopen set in X. Then pClθ ⊆ U. Hence A is gpr-closed set in X. The converse of the above theorem need not be true as seen from the following example. Example 3.28: Let X = {a, b, c} and τ = {X, Ø, {a}, {c}, {a, c}}. Then a subset A = {a, b} is a gpr-closed set but it is not θ-sgp-closed set. Remark 3.29: Union of θ-sgp-closed sets need not be a θ-sgp-closed set as seen from the following example. Example 3.30: Let X = {a, b, c} and τ = {X, Ø, {a}, {b}, {a, b}, {a, c}, {b, c}}. Then the subsets {a} and {b} are θ-sgp-closed sets but their union {a} ∪ {b} = {a, b} is not a θ-sgp-closed set in X. Remark 3.31: Intersection of θ-sgp-closed sets need not be a θ-sgp-closed set as seen from the following example. Example 3.32: Let X = {a, b, c} and τ = {X, Ø, {a}, {b}, {a, b} {a,c}}. Then the subsets {a, b} and {a, c} are are θ-sgp-closed sets but their intersection {a, b} ∩ {a, c} = {a} is not a θ-sgpclosed set in X. Theorem 3.33: A set A ⊂ X is θ-sgp-open set if and only if F ⊂ pIntθ(A) whenever F is semiclosed set in X and F ⊂ A. Proof: Necessity. Let A be θ-sgp-open set and F ⊂ A, where F is semi-closed set. It is obvious that Ac (complement of A) is contained in Fc. This implies that pClθ(Ac) ⊂ Fc. Hence pClθ(Ac) = (pIntθ(A))c ⊂ Fc, i.e. F ⊂ pIntθ(A). Sufficiency. If F is a semi-closed set with F ⊂ pIntθ(A) whenever F ⊂ A, then it follows that Ac ⊂ Fc and (pIntθ(A))c ⊂ Fc i.e. pClθ(Ac) ⊂ Fc. Therefore Ac is θ-sgp-closed set and therefore A is θ-sgp-open set. Lemma 3.34: Let A be a θ-sgp-closed subset of X. Then, International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 77 On θ-Semigeneralized Pre Closed Sets in Topological Spaces (i) pClθ(A)\A does not contain a nonempty semi-closed set. (ii) pClθ(A)\A is θ-sgp-open set. Proof: (i). Let F be semi-closed set such that F ⊂ pClθ(A)\A. Since Fc is semi-open set and A ⊂ Fc, it follows that pClθ(A) ⊂ Fc, i.e. F ⊂ (pClθ(A))c. This implies that F⊂(pClθ(A))c∩pClθ(A) = Ø. (ii) If A is θ-sgp-closed set and F is a semi-closed set such that F ⊂ pClθ(A)\A, then by (i), F is empty and therefore F ⊂ pIntθ(pClθ(A)\A). By theorem 3.33, pClθ(A)\A is θ-sgp-open set. Lemma 3.35: For any subset A of a topological space X, pClθ(A) is pre θ-closed set. Lemma 3.36: If A is a θ-sgp-closed set of a topological space X such that A ⊂ B ⊂ pClθ(A) then B is also a θ-sgp-closed set of X. Proof: Let O be a semi-open set of X such that B ⊂ O. Then A ⊂ O. Since A is θ-sgp-closed set, pClθ(A) ⊂ O. By using Lemma 3.35, pClθ(B) ⊂ pClθ(pClθ(A)) = pClθ(A) ⊂ O. Therefore B is also a θ-sgp-closed set of X. Lemma 3.37: Let X be a topological space and x ϵ X. The following two statements are equivalent: (i) y  pKerθ({x}); (ii) x  pClθ({y}). Proof: Let y  pKerθ({x}). It follows that there exists a semi θ-open set U containing x such that y  U. This means that x  pClθ({y}). The converse can be proved by the same taken. Lemma 3.38: The following statements are equivalent for any points x and y in a topological space X: (i) pKerθ({x}) ≠ pKerθ({y}); (ii) pClθ({x}) ≠ pClθ({y}). Proof: (i)→(ii): Let pKerθ({x}) ≠ pKerθ({y}). Then there exists a point z in X such that z  pKerθ({x}) and z  pKerθ({y}). By z  pKerθ({x}), it follows that {x} ∩ pClθ({z}) ≠ Ø. This implies x  pClθ({z}). By z  pKerθ({y}), we obtain {y} ∩ pClθ({z}) = Ø. Since x  pClθ({z}), pClθ({x}) ⊂ pClθ({z}) and {y} ∩ pClθ({x}) = Ø. Hence it follows that pClθ({x}) ≠ pClθ({y}). Now pKerθ({x}) ≠ pKerθ({y}) implies that pClθ({x}) ≠ pClθ({y}). (ii)→(i): Let pClθ({x}) ≠ pClθ({y}). Then there exists a point z in X such that z  pClθ({x}) and z  pClθ({y}). This means that there exists a pre-θ-open set containing z and therefore x but not y, i.e., y  pKerθ({x}). Hence pKerθ({x}) ≠ pKerθ({y}). Theorem 3.39: A topological space X is a pre-θ-R0 space if and only if for x and y in X, pClθ({x}) ≠ pClθ({y}) implies pClθ({x}) ∩ pClθ({y}) = Ø. Proof: Suppose that X is pre-θ-R0 and x, y  X such that pClθ({x}) ≠ pClθ({y}). Then, there exist z  pClθ({x}) such that z  pClθ({y}) (or z  pClθ({y}) such that z  pClθ({x})). There exists V  SO(X, τ) such that y  V and z  V; hence x  V. Therefore, we have x  pClθ({y}). Thus x  X \ pClθ({y}), which implies pClθ({x}) ⊂ X \ pClθ({y}) and pClθ({x}) ∩ pClθ({y}) = Ø. The proof for otherwise is similar. Sufficiency. Let V be pre-θ-open set and let x  V. We will show that pClθ({x}) ⊂ V. Let y  V, i.e., y  X \ V. Then x ≠ y and x  pClθ({y}). This shows that pClθ({x}) ≠ pClθ({y}). By assumption, pClθ({x}) ∩ pClθ({y}) = Ø. Hence y  pClθ({x}). Therefore pClθ({x}) ⊂ V. Theorem 3.40: A topological space X is a pre-θ-R0 space if and only if for any points x and y in X, pKerθ({x}) ≠ pKerθ({y}) implies pKerθ({x}) ∩ pKerθ({y}) = Ø. Proof: Suppose that X is pre-θ-R0 space. Thus by Lemma 3.38, for any points x and y in X if pKerθ({x}) ≠ pKerθ({y}) then pClθ({x}) ≠ pClθ({y}). Now we prove that pKerθ({x}) ∩ pKerθ({y}) = Ø. Asuume that z  pKerθ({x}) ∩ pKerθ({y}). By z  pKerθ({x}) and Lemma 3.37, it follows that x  pClθ({z}). Since x  pClθ({x}), by Theorem 3.39, pClθ({x}) = pClθ({z}). International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 78 Md. Hanif PAGE & V.T. Hosamath Similarly, we have pClθ({y}) = pClθ({z}) = pClθ({x}). This is a contradiction. Therefore, we have pKerθ({x}) ∩ pKerθ({y}) = Ø. Conversely, let X be a topological space such that for any points x and y in X, pKer θ({x}) ≠ pKerθ({y}) implies pKerθ({x}) ∩ pKerθ({y}) = Ø. If pClθ({x}) ≠ pClθ({y}), then by Lemma 3.38, pKerθ({x}) ≠ pKerθ({y}). Because z  pClθ({x}) implies that x  pKerθ({z}) and therefore pKerθ({x}) ∩ pKerθ({z}) ≠ Ø. By hypothesis, we therefore have pKerθ({x}) = pKerθ({z}). Then z  pClθ({x}) ∩ pClθ({y}) implies that pClθ({x}) = pClθ({z}) = pClθ({y}). This is a contradiction. Hence, pClθ({x}) ∩ pClθ({y}) = Ø and by Theorem 3.39, X is a pre-θ-R0 space. 3.41 Remark: The “Implication Diagram” about θ-sgp-closed set. Closed set θ-gs-closed set θ-g-closed set α-closed set αgs-closed set θ-sg-closed set θ-sgp-closed set pre-θ-closed set gpr-closed set gsp-closed set sgp-closed set gp-closed set where A B (resp. AB) represents A implies B but not conversely (resp.A and B are independent). 4. CONCLUSION In the class of θ-sgp-closed sets defined using semi-open sets lies between the class of θ-g-closed sets and the class of sgp-closed set. The θ-sgp-closed set can be used to derive a new decomposition of continuity and new separation axioms. This concept can be extended to bitopological and fuzzy topological spaces. REFERENCES [1]. A. A. El-Atik, Some more results on pre-θ-open sets, Antarctica J. Math. 2(1)(2005), 111121. [2]. P.Bhattacharyya and B.K.Lahiri, Semi-generalized closed sets in topology, Indian J. Math. 29(3)(1987), 375-382. [3]. 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