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Journal of Computer and Mathematical Sciences, Vol.7(8), 412-419, August 2016 (An International Research Journal), www.compmath-journal.org ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) On Supra R-open sets and Supra R-continuity Anuradha N.1 and Baby Chacko2 1 Assistant Professor in Mathematics, Government Engineering College, Kozhikode-5, Kerala, INDIA. email: [email protected]. 2 Associate Professor in Mathematics, St. Joseph’s College, Devagiri, Kozhikode-8, Kerala, INDIA. (Received on: August 18, 2016) ABSTRACT In this paper a new class of sets and maps between topological spaces called supra r-open sets and supra r- continuous maps respectively are introduced. Properties of supra r-continuous map and its relation with other types of functions are studied. Furthermore, the concept of supra r-open maps and supra r-closed maps is introduced and several properties of them are studied. Separation axioms in terms of supra r-open sets are defined. Supra r-closed graph is defined and some of its properties are studied. Mathematics Subject Classification: 54C08, 54C10, 54D10, 54D15. Keywords: Supra r-open sets, supra r–continuous map, supra r-closed graph. 1. INTRODUCTION In 1983, A.S. Mashhour8 introduced supra topological space and studied s- continuous maps and s*-continuous maps. In 1996, D. Andrijevic2 introduced and studied a class of generalized open sets in a topological space called b-open sets. This class of sets is contained in the class of β-open sets1 and contains all semi open sets6 and all pre open sets. In 2008, R. Devi4 introduced and studied a class of sets and maps between topological spaces supra α-open sets and supra α -continuous maps respectively. O.R. Sayed and Takashi Noiri9 introduced and studied supra b-open sets and supra b-continuity on topological spaces. S. Sekhar and P. Jayakumar10 studied supra-I open sets and supra-I continuous maps. In this paper, supra r-open sets, supra r-continuous maps, supra r-open maps (resp. supra r-closed maps), Supra r- closed graph and strongly supra r-closed graphs are introduced and their properties are discussed. 2. PRELIMINARIES Throughout the paper (X, ), (Y, ), and (z,) denote topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of (X, ), the closure 412 Anuradha N., et al., Comp. & Math. Sci. Vol.7 (8), 412-419 (2016) and interior of A in X are denoted by Cl (A) and Int (A) respectively. The complement of A is denoted by X - A. A subset A is said to be regular open if A = Int (Cl (A)) and regular closed if A = Cl (Int(A)). A subcollection * is called a supra topology [8] on X, if X, * and * is closed under arbitrary union. (X, *) is called a supra topological space. If (X, ) is a topological space and *, then * is known as supra topology associated with . The elements of a supra topological space are known as supra open sets. The complement of a supra open set is supra closed. 2.1. Definition : A map f:(X, ) (Y, ) is called totally continuous5 if inverse image of each open set of Y is clopen in X. 2.2. Definition: A map f: (X,) (Y,) completely continuous3 if inverse image of each open set of Y is regular open in X. 2.3. Definition: A map f: (X,) (Y,) almost completely continuous6 if inverse image of each regular open set of Y is regular open in X. 2.4. Definition: A map f: (X,) (Y,) almost perfectly continuous11 if inverse image of each regular open set of Y is clopen in X. 3. SUPRA R-OPEN SETS 3.1 Definition: Let (X, *) be a supra topological space. A set A is called Supra r-open if A=Supra Int(Cl(A)), where Supra Int(Cl(A)) denotes Int(Cl(A)) in *. The complement of a supra r-open set is called a supra r-closed set. 3.2 Example: Let (X, *), where X = {a, b,c, d},*={X, , {a},{b},{a,b}} be a supra topological space.Then {a} is supra r-open. 3.3 Remark: Every regular open set is supra r-open. 3.4 Theorem: Every supra r-open set is supra open. Proof: Since every regular open set is open, supra r-open set is supra open. 3.5 Remark: Converse of the above theorem need not be true. 3.6 Example: Let (X, *), where X = {a, b, c, d},*={X, , {a}, {b},{a, b}} be a supra topological space. Then {a, b} is a supra open set, but not a supra r-open set. 3.7 Theorem: If supra topology equals discrete topology, then every supra open set is supra ropen. 3.8 Theorem: Supra r-open sets possess the following properties: (i) Finite union of supra r-open sets may fail to be supra r-open. 413 Anuradha N., et al., Comp. & Math. Sci. Vol.7 (8), 412-419 (2016) (ii) Finite intersection of supra r-open sets is supra r-open. Proof: (i) Let X = {a, b, c},*={X, , {a}, {b},{a, b}}.Then {a} and {b} are supra r-open. But their union {a, b} is not supra r-open. (ii) Obvious. 3.9 Theorem: Supra r-closed sets possess the following properties: (i) Finite union of supra r-closed sets is supra r-closed. (ii) Arbitrary intersection of supra r-closed sets may fail to be supra r-closed. Proof: (i) Let V1 and V2 be supra r-closed. Then (X -V1) ∩ (X -V2) are supra r-open. Or X-(V1 V2) is supra r-open. Hence V1 V2 is supra r-closed. (ii) Let X = {a, b, c},*={X, , {a}, {b}, {a, b}}.Then {a ,c} and {b, c}are supra r-closed. But their intersection {c} is not supra r-closed. 3.10 Definition: Supra r-closure of a set A denoted by Supra rCl (A) is the inter-section of all supra r-closed sets containing A. 3.11 Example: Let X = {a, b, c},*={X, , {a}, {b}, {a, b}}. Then Supra rCl ({a}) = {a, c}. 3.12 Definition: Supra r-interior of a set A denoted by Supra rInt (A) is the union of all supra r-open sets contained in A. 3.13 Example: Let X = {a, b, c},*={X, , {a}, {b}, {a, b}}. Then Supra rInt({a})={a}. 3.14 Remark: Supra rInt (A) and Supra rCl (A) satisfy the following properties: (i) Supra rInt (A) is a supra r-open set. (ii) Supra rCl (A) is a supra r-closed set. 3.15 Theorem: The following results hold for Supra rInt and Supra rCl of a set A. (i) Supra rInt (A) A and equality holds if and only if A is a supra r-open set. (ii) A Supra rCl (A) and equality holds if and only if A is a supra r-closed set. 3.16 Theorem: Complements of Supra rInt and Supra rCl of a set A satisfy the following properties: (i) X - Supra rInt(A) = Supra rCl (X - A). (ii) X - Supra rCl (A) = Supra rInt (X -A). 414 Anuradha N., et al., Comp. & Math. Sci. Vol.7 (8), 412-419 (2016) 3.17 Theorem : The following result hold for Supra rInt and Supra rCl of a set A : (i) Supra rInt (A) ∩ Supra rInt (B) = Supra rInt(A ∩ B). (ii) Supra rCl(A) Supra rCl(B) = Supra rCl (A B). 3.18 Remark: Union of a Supra r-open set and a supra open set is a supra open set. 3.19 Remark: Intersection of a Supra r-open set and a supra open set need not be a supra open set. 3.20 Example: Let X = {a, b, c}, = {X, ,{a}},*={X, , {a}, {b, c},{a, c}}. Then {b, c} is supra r-open.{a, c} is supra open. But their intersection {c} is not supra open. 4. SUPRA R-CONTINUOUS FUNCTIONS 4.1 Definition: Let (X,) and (Y,) be topological spaces and * be an associated supra topology of . A map f: (X, *) (Y, ) is said to be supra r-continuous if inverse image of each open set of Y is supra r-open in X. 4.2 Example: Let X = {a, b, c}, = {X,, {a}},*={X, , {a},{b},{a, b}}and f: (X, *) (X, ) be defined by f(a)=b, f(b)=a, f(c)=b.Then f is supra r-continuous. 4.3 Theorem: Every completely continuous function is supra r-continuous. Proof: Since regular open sets are supra r-open, the result holds. 4.4 Remark: Converse of the above theorem need not be true. 4.5 Example: Let X = {a, b, c}, = {X,, {a}},*={X, , {a},{b},{a, b}}and f: (X, *) (X, ) be defined by f(a)=b, f(b)=a, f(c)=c. Then f is supra r-continuous, but not completely continuous. 4.6 Theorem: Let (X,) and (Y,) be topological spaces and * be the associated supra topology of .Let f: (X, *) (Y,) be a bijective map. Then the following are equivalent: (i) f is a supra r-continuous map. (ii) Inverse image of a closed set in Y is supra r-closed in X. (iii) Supra rCl (f -1(A)) f -1(Cl (A)) for every A Y. (iv) f(Supra rCl(A)) Cl(f(A)) for every A X. (v) f -1(Int(B)) Supra rInt(f -1(B)) for every B Y . Proof: (i). (i) (ii). Let V be closed in Y. Then Y - V is open. Since f is supra r-continuous, f -1 (Y -V ) is supra r-open. Or X – f -1(V) is supra r-open. That is f -1(V) is supra r-closed. (ii). (ii) (iii). Let A Y. Then Cl(A) is closed in Y . By (ii), f -1(Cl(A)) is supra r-closed. So Supra rCl(f -1(Cl(A))) = f -1(Cl(A)). Now f -1(A) f -1(Cl(A)). So f -1(Cl(A)) = Supra rCl(f -1(Cl(A))) Supra rCl(f -1(A)). That is Supra rCl(f -1(A)) f -1(Cl(A)). 415 Anuradha N., et al., Comp. & Math. Sci. Vol.7 (8), 412-419 (2016) (iii). (iii) (iv). Let A X. Then f (A) Y. By (iii), Supra rCl(f -1(f(A))) f -1(Cl(f(A))). That is Supra rCl(A) f -1(Cl(f(A)). Or f (Supra rCl (A)) Cl(f(A)). (iv). (iv) (v). By (iv), f (supra rCl(A)) Cl(f(A)). Then X-Supra rCl(A) X-f -1(Cl(f(A))). Or Supra rInt(X-A) f rInt(f -1(B)) f -1(Int(B)) where B = f(X - A). -1 (Int(f(X-A))). Or Supra (v) .(v) (i). Let A be open in Y. Then by (v), Supra rInt(f -1(A)) f -1(Int(A)). This implies Supra rInt (f -1(A)) f -1 (A), since A is open. But Supra rInt(f -1(A)) f -1(A). Hence Supra rInt(f -1(A)) = f -1(A). So f -1(A) is supra r-open. So (i) holds. 4.7 Theorem: Let (X, ), (Y, ), and (Z,) be topological spaces. Let * be an associated supra toplogy of . If a map f: (X; *) (Y,) is supra r-continuous and g : (Y, ) (Z, ) is continuous, then g f : (X, *) (Z, ) is supra r-continuous. 4.8 Theorem: Let (X, ) and (Y, ) be topological spaces. Let * be an associated supra topology of . Then f : (X, *) (Y, ) is supra r-continuous, if one of the following holds: (i) f -1(Supra rInt(B)) rInt(f -1(B)) for every B Y . (ii) rCl (f -1(B)) f -1(Supra rCl (B)) for every B Y . (iii) f (rCl(A)) Supra rCl(f(A)) for every A X. Proof: (i) Let V be any open set of Y. If (i) holds, f - 1(Supra rInt (V)) rInt (f -1(V)). Or f 1 (V) rInt (f -1(V)). But f -1(V) rInt (f -1(V)). So f -1(V) is regular open and so supra r-open. Hence f is supra r-continuous. (ii) Let V be open in Y. By (ii), rCl (f -1(V) f -1(supra rCl(V )) for every V Y . Then rInt f 1 (Y -V) f -1(Supra rInt(Y - V)). Then by (i) f is supra r-continuous. (iii) Let V be open in Y. By (iii), f (rCl(f -1(V )) Supra rCl(V ). So by (ii), f is supra rcontinuous. 4.9 Theorem: Every totally continuous function is supra r-continuous. Proof: Since clopen sets are regular open and regular open sets are supra open, the result follows. 4.10 Remark: Converse of the above theorem need not be true. 4.11 Example: Let X = {a, b, c}, = {X,, {a}},*={X, , {a},{b},{a, b}}and f: (X, *) (X, ) be defined by f(a)=b, f(b)=a, f(c)=c. Then f is supra r-continuous, but not totally continuous. 416 Anuradha N., et al., Comp. & Math. Sci. Vol.7 (8), 412-419 (2016) 4.12 Theorem: If X is a discrete space, every supra r-continuous function is totally continuous. 4.13 Theorem: Every almost perfectly continuous function into a discrete space, is supra rcontinuous. 4.14 Theorem: Every almost completely continuous function into a discrete spaceis supra rcontinuous. 4.15 Definition: Let (X,) and (Y,) be topological spaces and * and * be associated supra topologies of and respectively. Then f: (X, *) (Y, *) is supra* r- continuous, if inverse image of each supra r-open set is supra r-open. 5. SUPRA R-OPEN MAPS AND SUPRA R-CLOSED MAPS 5.1 Definition: Let (X,) and (Y,) be topological spaces and * and * be associated supra topologies of and respectively. A map f: (X,) (Y, *) is supra r-open (resp. supra r-closed) if image of each open (resp. closed) set in X is supra r-open (resp. supra r-closed) in (Y, *). 5.2 Example: Let X = {a, b, c}, ={X,, {a}},*={X,, {a}, {b},{a, b}}. Let f: (X,) (X,*) be defined by f (b)=a, f(a)=b, f(c)=c. Then f is supra r-open. 5.3 Theorem: A map f: (X,) (Y, *) is supra r-open if and only if f (Int A) Supra Int (f (A)) for each A X. Proof: Suppose f is supra r-open. Since f (Int A) is a supra r-open set contained in f (A) and Supra rInt(f(A)) is the largest supra r- open set contained in f(A), f(IntA) Supra rInt (f (A)) for each set A X. Converse part is obvious. 5.4 Theorem: A map f: (X,) (Y, *) is supra r-closed if and only if Supra rCl (f (A)) f (Cl(A)) for each A X. Proof: Suppose f is supra r-closed. Since f (Cl (A)) is a supra r-closed set containing f (A) and Supra rCl (f (A)) is the smallest supra r-closed containing f (A), Supra rCl (f (A)) f (Cl(A)), for each A X. Converse is obvious. 5.5 Theorem: Let (X,), (Y,), and (z,) be topological spaces. Let * and * be associated supra topologies of and respectively. Then (i) if gof : (X, ) (z, *) is supra r-open and f : (X, ) (Y, ) is a continuous surjection, then g: (Y, ) (z,*) is a supra r-open map. (ii) if gof : (X, ) (z, ) is open and g : (Y, *) (z,*) is a supra r-continuous injection, then f : (X, ) (Y, *) is a supra r-open map. 5.6 Theorem: Let (X,) and (Y,) be topological spaces. Let * be the associated supra topology of . Let f: (X,) (Y, *) be a bijective map. Then the following are equivalent: 417 Anuradha N., et al., Comp. & Math. Sci. Vol.7 (8), 412-419 (2016) (i) f is a supra r-open map. (ii) f is a supra r-closed map. (iii) f-1 is a supra r-continuous map. 6. SEPARATION AXIOMS IN SUPRA R-TOPOLOGICAL SPACES 6.1 Definition: Let (X,) be a topological space and * be an associated supra topology of . Then the space (X, *) is called (i) Supra rT0 if for every two distinct points of X, there exists a supra r-open set which contains one, but not the other. (ii) Supra rT1 if for every two distinct points of X, there exists Supra r-open sets U and V such that x U , y U and xV , y V. (iii) Supra rT2 if for every two distinct points of X, there exists supra r-open sets U and V such that x U, y V and U V = . 7. SUPRA R-CLOSED GRAPHS AND STRONGLY SUPRA R-CLOSED GRAPHS 7.1 Definition: A subset A of the product space X × Y is supra r-closed in X × Y if for each (x, y) (X × Y )-A, there exists supra r-open sets U and V containing x and y respectively such that (U × V ) A = . A function f: (X, *) (Y, *) has a supra r-closed graph, if the graph G (f) = {(x, f(x)): x X} is supra r-closed in X ×Y. 7.2 Lemma: A function f: (X, *) (Y, *) has a supra r-closed graph if and only if for each x X, y Y such that y ≠ f(x) , there exists supra r-open sets U and V containing x and y respectively such that f(U) V = . 7.3 Theorem: If a function f: (X; *) (Y,*) is supra* r-continuous and Y is Supra rT2, then f has a supra r-closed graph. Proof: Let (x, y) (X×Y)-G (f). Then y ≠ f(x). Since Y is supra rT2 there exists supra r-open sets U and V such that f(x) U, y V and U V =. Since f is supra* r-continuous , there exists supra open set W of x such that f(W) U. Hence f(W) V = . This implies that f has a supra r-closed graph. 7.4 Theorem: If a function f: (X; *) (Y,*) is supra* r-continuous injection with a supra r-closed graph, then X is Supra rT2. Proof: Let x1, x2 X with x1 ≠ x2. Then f(x1) ≠ f(x2). This implies that (x1, f(x2)) (X×Y)-G (f).Since f has a supra r-closed graph, there exists supra r-open sets U and V of x1 and f(x2) respectively such that f (U) V =. Since f is Supra*r-continuous, there exists supra r-open set W containing x2 such that f (W) V. Hence f (W) f(U) = . So U W =. Hence X is Supra rT2. 418 Anuradha N., et al., Comp. & Math. Sci. Vol.7 (8), 412-419 (2016) 7.5 Definition: A function f: (X; *) (Y,*) has a strongly supra r-closed graph, if for each (x, y) G (f), there exists supra r-open sets U and V containing x and y respectively such that U× Supra rCl (V) G(f) = . 7.6 Lemma: A function f: (X; *) (Y,*) has a strongly supra r-closed graph, if for each (x, y) G(f), there exists supra r-open sets U and V containing x and y respectively such that f(U) Supra rCl(V ) =. 7.7 Theorem: Let f: (X; *) (Y,*) be a surjective function with a strongly supra r-closed graph. Then Y is a Supra rT2 space. Proof: Let y1 and y2 be two distinct points of Y . Then there exists x1 in X such that f(x1) = y1. Thus (x1, y2) G(f). Since f has a strongly supra r-closed graph, there exists supra r-open sets U and V of x1 and y2 respectively such that f(U) Supra rCl (V) =. Consequently y1 V . So Y is a Supra rT2 space. REFERENCES 1. M.E. Abd El-Monsef, S.N. El-Deeb and R.A. Mahmoud, β-open sets and β -continuous mappings, Bull. Fac. Sci. 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