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Ultra Scientist Vol. 28(3)A, 162-171 (2016). www.ultrascientist.org ISSN 2231-3478 (Print) 2319-8052 (Online) Investigation on Weak form of Generalized Closed sets in Ideal Topological Spaces G.JAYAPARTHASARATHY Department of Mathematics St. Jude’s College, Thoothoor Kanya Kumari-62 176, Tamil Nadu (India) E-mail : [email protected] (Acceptance Date 11th July, 2016) Abstract This paper proposes a new class of closed sets in ideal spacescalled -closed set and shows that this class of closed sets lies betweenthe class of -closed sets and the class of -closed sets. Furthermore, it shows that the necessary and sufficient conditions for a set to be a -closed set and investigatesthe properties of - open sets in idealtopological spaces. Key words: semi-*-closed set, , -closed set, -closed set, -closed sets, -closed sets. 2010 Mathematics subject classification: 54A05, 54D15. 1. Introduction The concept of ideals in topological spaces was initiated byKuratowski 7 and Vaidyanathaswamy14. In 1990, Jankovic and Hamlett4 also studied the properties of ideal topological spaces. After that -closedset was introduced by Noiri et al.6,11. Also Khan and Noiri5 introducedand studied the properties of -closed sets in ideal topological spaces. Weakform of open sets called semi-open sets and also the first step of generalizingclosed sets was done by Levine 9,10 . Recently, Jayaparthasarathy et al. 8 introduced a new class of closed sets called -closed sets and its propertiesalso discussed. In this paper we introduce and characterize the properties of -closed set, and this class of closed sets G. Jayaparthasarathy 163 lies between the classes of the class of -closedsets and vi) -closed set13 if whenever and is #gs-openin (X, τ). vii) -closed set8 if -closed sets. We also introduce -open sets andprove the necessary and sufficient conditions of -open sets. Further,, weinvestigate the relationship between closed sets and other existing sets inideal topological spaces. 2. Preliminaries : This section discuss some basic properties about ideal topological spaces and weak form of open sets in topological spaces which are useful in sequel. Definition 2.1 9 Let (X, τ) be a topological space. A subset A of X is called semi-open if . The complement of semi-open set iscalled semi-closed. Definition 2.2 A subset A of a topological space (X, τ) is called a i) generalized closed (briefly g-closed) set9 if wheneverand and U is open in (X, τ). ii) semi-generalized closed (briefly sg-closed) set3 if whenever and is semi-open in (X, τ). -closed set (= ω-closed)12 if whenever and issemi-open in (X, τ). iv) -closed set13 if whenever iii) and is -open in(X, τ). v) #g-semi-closed set (briefly #gs-closed)13 if whenever and is -open in (X, τ). and is whenever -openin (X, τ). The complement of the above mentioned sets are called their respective generalized open sets. Definition 2.37,14 An idealIon a topological space (X, τ) is a non-emptycollection of subsets ofXsatisfying the following two conditions: (i) If AI and BA, then BI (ii) If AI and BI, then ABI. Let (X, τ) be a topological space andIan ideal of subsets of X. An idealtopological space is a topological space (X, τ)with an idealIonXand is denoted by (X, τ, I). Definition 2.41,4 For a subset A of X, (i) A*(I, ) = {xX: UAI for every U τ(X, x)} is called the local function of A with respect to ideal I and topology τ, where τ (X, x) ={Uτ: xU}. (ii) (I, τ) ={x X: U A I for every USO(X, x)} is called the semi-local function of A with respect to ideal I and topology τ, where SO(X,x)={USO(X): xU}. In 1,4 very ideal topological space (X, τ, I), there exists a topology τ* finer than τ, defined by τ*={UX:Cl*(XU)=XU} which is generated by the base β(I, τ) ={UJ: Uτ 164 Investigation on Weak form of---Topological Spaces. andJI} and Cl*(A) =AA* is a Kuratowski closure operator for the topology τ*. A subset A of an ideal topological space (X, τ, I) is said to be *-closed4 if A*A and Int*(A) denote the interior of the set A in (X, τ*, I). Also we 5 define a topology by (A) A∪ and there exists (I) finer than τ and τ*, defined by (I) ={UX: (XU) =XU} which is generated by the base s(I, τ) ={UJ: Uτ and JI}. A subset A of an ideal topological space (X, τ, I) is said to be semi-*-closed5 if A and is said to be *-semi dense5 if A = = . Also note that1 =sCl( = )A* (A) and (A) denote the interior of the set A in (X, Theorem 2.8 5 Let AandBbe two subsets of an ideal space (X, τ, I). Then . 3. Weak open sets in ideal topological spaces : This section introduces a weak form of closed sets in ideal spacesand investigates some properties of it. Definition 3.1 A subsetAof an ideal topological space (X, τ, I) is saidto be closed if = U whenever and -open in X. The complement of set is called is -closed -open set. ,I). Definition 2.5 A subsetAof an ideal topological space (X, τ, I) is said to be (i) Igeneralized closed (briefly Ig-closed)6 if whenever and is open in X. (ii) semi generalized I (briefly sgI)-closed5 if = U whenever and is semi- open in X. (iii) -closed2 if and whenever is semi-open in X. Theorem 2.65 Let {Aa : aΩ} be a locally finite family of sets of an ideal space =( (X, τ, I). Then Example 3.2 Let X = {a, b, c, d}, τ O(X) ={, X,{a},{b, c},{a, b, c}} and I={, {a}}. Then -closed sets are , X, {a}, {d}, {a, d}, {b, c},{b, d},{c, d},{a, b, c}, {a, b, d}, {a, c, d}and{b, c, d}. Remark 3.3 In an ideal topological space (X, τ, I), then the following are trivially true. (i) If =∅ for every A X, then A is closed. =is an (ii) -closed for every A X. Theorem 3.4 Let (X, τ, I) be an ideal topological space. Then the following are true. . (i) Every Theorem 2.75 Let (X, τ, I) be an ideal space and . If , then and Bis *-semi dense in itself. -closed set is -closed, but not converse. (ii) Every -closed set is -closed, but not converse. G. Jayaparthasarathy 165 (iii) Every *-closed set is -closed, but not converse. (iv) Every semi-*-closed set is -closed, but not converse. (v) Every -closed is , but not converse. (vi) If is an element of I, then A is -closed. Proof : (i) Let A be an (ii) (iii) (iv) (v) (vi) =∅⊆A⊆U. Hence A is closed and -closed set. Theorem 3.6 Let (X, τ, I) be an ideal topological space and AX. Then the following statements are equivalent: (i) A is -closed. for every (iii) For all x ∈ -closed. -closed set, but it is not -open set U })∩A≠∅. (A), sCl({x}) (iv) (A)A contains no non empty -closed set. = (v) A contains no non empty -closed set. Proof (i) (ii): Let A be an Then = U whenever -closed set. and is - open in X and implies whenever and is -open in X. (ii) (iii): Let and suppose Then where , is semi-open set. Since every semi-openset is -open and by hypothesis, . This contradicts the fact that . Hence, - -closed. Again consider X={a, b, c},τ O(X) ={∅ , X,{a}} and I={∅ ,{a},{b}, {a, b}}. Then {a, c} is -closed set, but it is not *-closed. Also consider X = {a, b, c, d}, τ O(X) = {∅ , X,{a},{b, c},{a, b, c}} and I ={∅ ,{a}}. Then {a, b, d} is - containing A. -closed set and let Example 3.5 Consider X={a, b, c}, τ O(X) ={∅ , X,{a},{b},{a, b}}and I={∅ ,{a}}. Then {b} is closed set, but it is not (ii) where U is open set. Since and every open set is -open, then A is -closed. Proof is similar to part (i). Let A be a*-closed set, then A*A. Let where is -open. Hence , whenever and is open. Proof is trivially by part (i) and part (iii). Let A be an -closed and where is semi-open. Since every semi-open set is -open, then we have, , whenever and is semi-open. Let where is -open. If AI, then but it is not semi-*-closed. Also {a, b} is -closed set, (iii)(iv): Suppose F is a , where -closed set containinga pointx. Since and {x}F, we have sCl({x}F and and by hypothesis, is a contradiction. . Since This 166 Investigation on Weak form of---Topological Spaces. every semi-openset is (iv)(v): Since -open and by hypothesis, ,this contradictsthe Therefore empty contains no non F is a and Therefore be a is a Therefore . and {x}F, we have sCl({x}) F and . Since and by hypothesis, . Therefore, . Since , where -closed set containinga point x. Since -open. and set, so contained in and hence Hence (iii)(iv): Suppose -closed set. (v)(i): Let fact that is semi-closed -closed set This is a contradiction. (iv)(v): Assume that F is , where -closed set and This gives This contr adicts the Theorem 3.7 Let (X, τ, I) be an ideal topological space and . Then the following statements are equivalent: and U be a −A. Since set, so (iv) contains no non empty - closed set. is semi-closed -closed set conta- −A. Therefore U. -closed set. Theorem 3.8 Let (X, τ, I) be an ideal topological space. Then every -closed and Proof (i)(ii): Let A be an whenever -closed set. and is -open in X. By Definition 2.4, we have the desired result. (ii)(iii): Let ined in is a and hence contains no non empty Then -open. . Therefore -open set (iii) For all where (v)(i): Let Therefore XUXA and (i) A is -closed. (ii) for every U containing A. (v) hypothesis. and suppose . Then , is semi-open set. Since -open set is semi-*-closed. Proof By hypothesis, A A and A is *-closed. A whenever -open set. Hence A is semi- Theorem 3.9 Let locally finite family of be a -closed sets of an G. Jayaparthasarathy 167 ideal topological space (X, τ, I). Then Ui Ai is -closedset. Proof If A is theorem 3.7, N=A*s -A -closed, then by contains no non-empty -closed set. If Proof Let Ui Ai U where U is -open. Since each Ai is for eachi i Ω, U and Ui hence Ui Ai is U and -closed set. Conversely, suppose A=FN where F is semi- Example 3.10 Intersection of two -closed sets need not be -closed. Consider X={a, b, c, d}, τ O(X) = {∅, X,{a},{b, c}, {a, b, c}} and I ={∅ ,{a}}. Then {b, c} -closed sets, but {b, c} {c, d} = {c} is not a -closed set. Theorem 3.11 LetAbe an set andB be -closedset. Proof Let U be a -open subset of X containing AB. Then AU(X-B). Since A is -closed, then and Now A F and so then and By hypothesis, since is -closed, and so Hence A is -closed. . , because B is -closed. Proof Since A is -closed, then by sCl( )BsCl( )B contains no nonempty set, by theorem 3.7. Hence B is Theorem 3.12 Let (X, τ, I) be an ideal - closed if and only if A=FN where F is semi*-closed and N contains no nonempty closed set. closed, then B is -closed set. Since sCl( - -closed. topological space and AX. Then A is Theorem 3.13 Let (X, τ, I) be an ideal topological space and A and B be subsets of X such that and A is - theorem 3.7, sCl(A*s)A contains no nonempty . By theorem 2.8, is *-closed and Ncontains no nonempty -closed set. Let U be a -open set such that AU. Then FN U implies . -closed -closed set of an ideal topological space (X, τ, I). Then AB is closed. Thus * -closed such that -closed set, then U. By theorem 2.6, (U i and {c, d}are then F is semi- )A, -closed -closed. Theorem 3.14 Let (X, τ, I) be an ideal topological space. Then every -closed, but not conversely.. -closed is Proof Assume A is -closed and 168 Investigation on Weak form of---Topological Spaces. since , then whenever A U and U is open. By theorem 3.7, we get thedesired result. Example 3.15 Consider X={a, b, c}, τ O(X) = {∅ , X, {a},{b},{a, b}} and I={∅ ,{a}}. Then {a, b} is closed. -closed set, but it is not Proof SupposeA is sCl( 3.13, we get B is -closed. Since A B , then = Theorem 3.19 Let (X, τ, I) be an ideal topological space and A X. Then A is - closed if and only if - -closed. Proof Assume A is -closed and be a Ais *-semi dense initself, A A contains no non-empty and implies )U whenever A U and U -open. Hence Ais -closed. Theorem 3.17 Let (X, τ, I) be an ideal space where I={∅ }. Then A is -closed if and only if A is -closed. Proof Since I={∅ }, This implies Ais * -semi dense and so it is -closed. Converse is theorem 3.14. -closed. Let U -open set such that Then is is closed. *-semi dense set. Let A U where U is open, then theorem 3.7 (ii), sCl( )U. Since sCl(A)sCl( ), by Theorem . Then Aand B are * -semi dense and both are -closed, by theorem 3.16. - Theorem 3.16 Let (X, τ, I) be an ideal topological space and A X is -closed and *-semi dense set in itself, then A is Since . ))= Since A is -closed, by Theorem 3.7 (v) A. -closed set. This implies . Thus X is the only -open set containing Thus Thus is -closed. Conversely, assume is -closed and let U be a such that A U. Let F be any such that . Since we have and since X F is -open set -closed set -open. Therefore, Theorem 3.18 Let (X, τ, I) be an ideal topological space and A and B be subsets of X such that A B and A is -closed. and hence Then A and B are empty -closed set. By theorem 3.7(v), A is -closed. -closed. Proof Let A and B be subsets of X such that A B A*s and A is -closed. implies But . Thus contains no non Theorem 3.20 Let (X, τ, I) be an ideal G. Jayaparthasarathy 169 space and A X. Then -closed if and only if is is -closed. Proof Proof is trivially from the fact that Proof (i)(ii): If A is semi-*-closed, then and so Hence is a -closed set. (ii)(iii): Since Theorem 3.21 Let (X, τ, I) be an ideal topological space and if every -open set in X is semi-*-closed. Then every subset of X is -closed. Proof Assume that every -open set is semi-*-closed and let A X and U be -open set such that A U. Then Hence Ais -closed. Theorem 3.22 Let A be a -closed set and B be -closed set of an ideal topological space (X, τ, I). Then A B is closedset. so is and -closed set. (iii)(i): If is a -closed set, since A is -closed set, then by theorem 3.6, =∅ and so A is semi-*-closed. 4. Properties of -open sets : This section is to investigate the properties of -open sets in idealtopological spaces. Theorem 4.1 Let (X, , I) be an ideal topological space and A B. Then is -open if and only if whenever F is -closed and F A. Proof Let U be a X containing A B. Then Since A is -closed, then and -open subset of . . By theorem 2.7, Proof Suppose A is -open. If F is -closed and F A, then XF XF and so . Then because B is -closed. Thus A B is -closed. Theorem 3.23 Let (X, τ, I) be an ideal topological space and AX be an -closed set, then the following are equivalent. (i) Ais a semi-*-closed set. (ii) (iii) A*A is a is a -closed set. -closed set. and so . Conversely, let U be a -open set such that . Then and so . Therefore and implies Therefore, XA is -closed. Hence A is open. Theorem 4.2 Let (X, , I) be an ideal 170 Investigation on Weak form of---Topological Spaces. topological space and -open , then B is and is . If A is -open. Proof Since A is -open, then XA -closed. By theorem 3.6, contains no nonempty set. Since -closed which implies that and XF is -open. Since and so it follows that . Since . Hence A is -closed. (iii) (i): Since and so Hence B is -open. Conclusion Theorem 4.3 Let (X, τ, I) be an ideal topological space and A X, then the followingare equivalent. (i) A is -closed. (ii) A is (iii) is -closed. This paper introduces new weak closed sets in ideal topological spaces and investigate its basic properties. This set can be extended in different research fields such as extended topology, Fuzzy topology, Intuitionistic topology, digital topology etc. -open. References Proof (i)(ii): Suppose A is If U is any -closed. -open set such that , then . Since A is 3.7(v),we get -closed, by theorem Therefore and implies Hence is (ii)(iii): Suppose closed. If F is any is -closed set such that , then and -closed. and . Therefore 1. Abd El-Monsef. M.E., Lashien. E.F. and Nasef. A.A., Some topological operators via ideal, Kyungpook Mathematical Journal, 32(2), 273-284 (1992). 2. Antony Rex Rodrigo.J, Ravi. O. and Nalinirmalatha. 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