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534 Int. J Comp Sci. Emerging Tech Vol-2 No 4 August, 2011 Closed and closed set in supra Topological Spaces I. Arockiarani * and M.Trinita Pricilla** *Department of Mathematics,Nirmala College for Women, Coimbatore – 641 046. **Department of Mathematics,Jansons Institute of technology, Karumathampatti,India [email protected] Abstract - The aim of this paper is to define and investigate a weaker class of supra closed set in supra topological spaces. New characterizations of supra closed set are established. We do a comparative study of supra closed set with the existing sets together with closed set, c1 µ closed and supra suitable examples. We further discuss the concept of supra and supra continuity and obtained their applications. Keywords and Phrases: Definition: 1.2 [8] The supra closure of a set A is denoted by C1µ(A),and defined as The supra interior of a set A is denoted by intµ(A),and defined as Intµ closed set, closed set and closed set 1. Introduction and Preliminaries Introduction Definition: 1.3 [8] Let ( ) be a topological space and µ be a supra topology on we call µ a supra topology associated with The notion of closed sets was introduced by T.Noiri and O.R.Syed [ 9].Sr.I.Arockiarani and Jeenu korian [1] introduced and closed sets in topological spaces.In 1983, A.S.Mashhour et al [8] introduced the supra topological spaces and studied S-S continuous functions and S* - continuous functions. In 2010, O.R.Sayed and Takashi Noiri [10] introduced supra b - open sets and supra b continuity on topological spaces. In this paper, we use closed and closed set as a tool to introduce the concept of supra supra closed and closed set. We discuss the concept of supra continuity and supra continuity and also obtained their applications. 1.1. Preliminaries Definition: 1.1 [ 8] A subfamily µ of x is said to be a supra topology on X if i) ii) If for all i J, then .( ,µ) is called a supra topological space. The elements of µ are called supra open sets in ( µ) and complement of supra open set is called supra closed sets and it is denoted by µc. ___________________________________________________________________________________ International Journal of Computer Science & Emerging Technologies IJCSET, E-ISSN: 2044 - 6004 Copyright © ExcelingTech, Pub, UK (http://excelingtech.co.uk/) Definition: 1.4 Let ( µ) be a supra topological space. A set A is called supra semi - open set if A C1µ (Int µ(A) ). The complement of supra semi - open set is supra semi - closed set. Definition: 1.5 Let µ) be a supra topological space. A set A of X is called supra generalized - closed set (simply gµ - closed) if C1µ(A) U whenever A U and U is supra open. The complement of supra generalized closed set is supra generalized open set. Definition: 1.6 Let µ) be a supra topological space. A set A of X is called supra semi - generalized closed set (simply sgµ - closed) if SC1µ (A) U and U is supra semi - open. The complement of supra semi generalized closed set is supra semi - generalized open set. Definition: 1.7 Let µ) be a supra topological space. A set A of X is called supra generalized- semi closed set (simply gsµ - closed) if SC1µ (A) U whenever A U and U is supra - open. The complement of supra generalized- semi closed set is supra generalized semi - open set. 535 Int. J Comp Sci. Emerging Tech 2. Supra Vol-2 No 4 August, 2011 Closed and Supra Closed Set Definition: 2.1 A Subset A of (X, ) is said to be supra - closed in (X,) if Scl (A) Int (U) whenever A U and U is supra - open. Consider F AC then F Cl (F) X– SCl (A). Therefore F SCl (A) [X– SCl (A)] = .Hence F = . Proposition: 2.9 A subset A is supra S – closed then SCl (A) – A does not contain non-empty supra -open and supra - closed set. Proof: Similar to theorem 2.8 Definition: 2.2 A Subset A of (X,) is said to be supra S closed in (X,) if Scl (A) Int cl (U) whenever A U and U is supra - open. Definition: 2.3 A Subset A of (X,) is said to be supra closed in (X, ) if Scl (A) Int (U) whenever A U and U is supra - open. Definition: 2.4 A Subset A of (X,) is said to be supra S - closed in (X,) if Scl (A) Int cl (U) whenever A U and U is supra - open. Definition: 2.5 A Subset A of (X, ) is said to be supra regular open if A = Int(Cl (A)) and supra regular closed if A = cl(Int (A)) . The finite union of supra regular open set is said to be supra - open. Theorem: 2.6 If D (E) DS (E) for each subset E of supra topological space (X,) then the union of two supra - closed sets is supra - closed. Proof: Let A and B be two supra - closed subsets in (X, ) then SCl (A) Int (U) ; SCl (B) Int (U). Let U be supra - open such that A B U. Then we have SCl (A B) = SCl (A) SCl (B). Thus SCl (A B) Int (U). Hence A B is supra - Closed. Proposition: 2.7 Every supra open and supra semi-closed subset (X,) is supra - closed. Proof: Let A be supra open and supra semi-closed subset of (X,) where A U and U is supra open. Then SCl (A) Int (U), Since A is Supra – open and supra semi – closed. Therefore A is supra - closed. Proposition: 2.8 A subset A is supra - Closed then SCl (A) – A does not contain a non empty supra closed subset. Proof: Let A be supra - closed set. Suppose F is a supra - closed set of SCl (A) – A and F SCl (A) – A. This implies F SCl (A) and F AC. Proposition: 2.10 If a subset A of (X,) is supra - open and supra - closed then it is supra semi-closed. Proof: Let A be supra - open and supra - closed then SCl (A) A. Therefore A = SCl (A). Hence A is supra semi-closed. Theorem: 2.11 A subset A is supra regular open iff A is supra - open, supra -open and supra - closed. Proof: Suppose A is supra - open and supra closed then by proposition 2.10, A is supra semi – closed, so Int Cl (A) A. Then A Int (Cl(A)). This implies that A = Int Cl (A). Therefore A is supra regular open. Conversely, Let A be supra regular open then A is supra - open, supra - open and supra closed. Remark: 2.12 Every supra - closed set is supra S – closed. Proof: Let A U and U is supra - open.Let A be supra - closed then SCl (A) Int Cl (A). Therefore A is supra S – closed. Remark: 2.13 But the converse is not true by the following example: Let X = {a, b, c,d}; {, , {a},{b},{a,b}} . {a,b} is supra closed but it is not supra closed. Remark: 2.14 We have the following relationship between supra closed set and supra closed set and other related sets. 536 Int. J Comp Sci. Emerging Tech Vol-2 No 4 August, 2011 4. closed closed Every supra s - continuous function is supra s - continuous function. Proof: It is obvious. closed closed gsµ - closed gµ - closed 3. Supra - Continuity and Supra s Continuity Let f: (X, ) (Y, ) be a function from a supra topological space (X, ) into a supra topological space (Y,). Example: 3.6 Let X = {a, b, c,d}; Hence f is supra S – continuous but it is not supra - continuous and supra - continuous. Also f is supra S – continuous but it is not supra continuous. Remark: 3.7 We have the following relationship between supra continuity; supra continuity and other related sets. continuity continuity continuity Definition: 3.2 A function f : (X, ) (Y, ) is said to be supra - irresolute (resp supra S – irresolute) if f-1 (V) is supra - closed (resp supra S – closed) in (X, ) for every supra - closed (resp supra S – closed) set V of (Y, ). Definition: 3.4 A function f: (X, ) (Y, ) is said to be supra - irresolute (resp supra S – irresolute) if f-1 (V) is supra - closed (resp supra S – closed) in (x, ) for every supra - closed (resp Supra s – closed) set V of (Y, ). Proposition: 3.5 1. Every supra - continuous function is supra S – continuous function. 2. Every supra - continuous function is supra s - continuous function. 3. Every supra - continuous function is supra - continuous function. , {a},{b},{a,b}}; Y = {p,q,r}; σ={,Y , {p},{q},{p,q}} Define a function f: (X,) (Y,) such that f(a) = p; f(b) = r ; f(c) = q = f(d). Definition: 3.1 A function f : (X, ) (Y, ) is said to be supra - continuous (resp supra S - continuous) if f-1 (V) is supra - closed (resp supra S – closed) in (X, ) for every supra closed set V of (Y, ). Definition: 3.3 A function f : (X, ) (Y, ) is said to be supra - continuous (resp supra S –continuous) if f-1 (V) is supra - closed (resp supra S – closed) in (X, ) for every supra closed set V of (Y,). {, continuity gsµ - continuity sgµ - continuity gµ - continuity 4. Applications Definition: 4.1 A supra topological space (X, µ) is 1. Supra πΩ – T 1 if every supra πΩs-closed set is 2 2. supra semi-closed in (X, µ) Supra πΩ – Ts if every supra πΩs–closed set is supra closed in (X, µ). Proposition: 4.2 Let (X, µ) be a supra topological space. 1. For every x εX, { x } is supra π-closed or its complement X-{ x } is supra πΩ –closed in (X, µ). 2. For every x εX, { x } is supra open and supra πclosed or its complement X – { x } is supra πΩSclosed in (X,µ) 537 Int. J Comp Sci. Emerging Tech Proof: 1. Suppose { x } is not supra π-closed in (X, µ) then X–{ x } is not supra π-open and the only supra πopen set containing X–{ x } is X. Since supra πclosed set is supra semi-closed then Sclµ {X– { x }} X= Intµ (x). Therefore X–{ x } is supra πΩ-closed in (X,µ). 2. Suppose { x } is not supra open and let U be supra πopen set such that X–{ x } U.If U = X then Sclµ {X – { x }} Intµ (C1µ (U)) = U. If U = X – { x } then Intµ ( Clµ (U)) = Int (C1µ (X –{ x }] = Intµ { x } = X.Hence SC1µ {X –{ x }} Intµ (C1(U)). Therefore X–{ x } is supra πΩSclosed. Proposition: 4.3 Let (X, µ) be a supra topological space 1. For every x εX, { x }is supra semi-closed or its complement X–{ x } is supra πΩ –closed in (X,µ). 2. For every x εX, { x } is supra open and supra semiclosed or its complement X – { x } is supra πΩSclosed in (X,µ). Proof: It is obvious. Theorem: 4.4 For a supra topological space (X, µ) if every supra πΩ-closed set is supra semi-closed in(X, µ) then for each x εX, { x } is supra semi-open or supra semi-closed in (X µ). Proof: Suppose that for a point x εX, { x }is not supra semi-closed in (X, µ).By proposition 4.3, X– { x } is supra πΩ-closed in (X, µ).By assumption X– { x } is supra semi-closed in (X, µ)and hence { x } is supra semi-open. Therefore each singleton set is supra semi-open or supra semi-closed in (X, µ). Vol-2 No 4 August, 2011 Theorem: 4.6 Let f: (X, µ) (Y, σ) & g: (Y, σ) (Z,γ) be two function then 1. gof is supra πΩ-continuous (resp Supra πΩS – continuous) if g is continuous and f is supra πΩ-continuous (resp supra πΩScontinuous). 2. gof is supra πΩ-irresolute (resp supra πΩSirresolute) if f and g are supra πΩ-irresolute (resp πΩS-irresolute). 3. got is supra πΩ-continuous if g is supra πΩcontinuous (resp supra πΩS-contiuous) and f is supra Ω-irresolute (resp supra πΩSirresolute). 4. Let (Y, σ) be a supra πΩ-TS space, then gof is continuous if f is continuous and g is supra πΩS-continuous. 5. Let f be supra πΩS-continuous then f is continuous (resp supra semi-continuous) if (X, µ) is supra πΩ-TS (resp supra πΩ-T1/2). Proof: It is obvious. Theorem: 4.7 Let f: (X, µ) (Y, σ) be a function. 1. Let f be an supra πΩS-irresolute and closed surjection if (X, µ) is an supra πΩ-TS space then (Y, σ) is also supra πΩ-TS 2. Let f be an supra πΩS-irresolute and semiclosed surjection. If (X, µ) is an supra πΩ-TS space then (Y, σ) is also supra πΩ-T1/2 3. Let f be an supra πΩS-irresolute and pre semi-closed surjection. If (X, µ) is an supra πΩ-T1/2 space then (Y, σ) is also supra πΩT1/2 Proof: It is obvious. References: [1] I.Arockiarani and Jeenu kurian, on Theorem: 4.5 For a topological space (X, µ) the following properties hold: 1. If (X, µ)is supra πΩ-TS then for each x εX, the singleton { x } is supra open or supra semiclosed. 2. If (X, µ) is supra πΩ-TS then it is supra πΩ-T1/2 Proof: 1. Suppose that x εX, { x }is not supra semiclosed. By proposition4.3, then X–{ x } is supra πΩ-closed in (X, µ). Hence X–{ x } is supra πΩs-closed in (X, µ).Then X–{ x } is supra closed in (X, µ). Thus { x } is supra open in (X, µ). 2. Let X be supra πΩ-TS space then every supra πΩS-closed set is supra closed. Thus every supra πΩS- closed set is supra semiclosed. Since every supra closed set is supra semi-closed. Therefore X is supra πΩ-T1/2 space. 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