On Supra R-open sets and Supra R-continuity Anuradha N.

On Supra R-open sets and Supra R-continuity

Anuradha N.

and Baby Chacko

1

Assistant Professor in Mathematics,

Government Engineering College, Kozhikode-5, Kerala, INDIA.

email: anuanamica@gmail.com.

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. Mashhour

introduced supra topological space and studied s- continuous maps and s*-continuous maps. In 1996, D. Andrijevic

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 sets

and contains all semi open sets

and all pre open sets. In 2008, R.

Devi

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 Noiri

introduced and studied supra b-open sets and supra b-continuity on topological spaces. S. Sekhar and P.

Jayakumar

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

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 continuous

if inverse image of each open set of Y is clopen in X.

2.2. Definition: A map f: (X,) (Y,) completely continuous

if inverse image of each open set of Y is regular open in X.

2.3. Definition: A map f: (X,) (Y,) almost completely continuous

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 continuous

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 r- open.

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.

(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 V

and V

be supra r-closed. Then (X -V

) ∩ (X -V

) are supra r-open.

Or X-(V

V

) is supra r-open. Hence V

 V

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).

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

(A))  f

(Cl (A)) for every A  Y.

(iv) f(Supra rCl(A))  Cl(f(A)) for every A  X.

(v) f

(Int(B))  Supra rInt(f

(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

(V) is supra r-open. That is f

(V) is supra r-closed.

(ii). (ii)  (iii). Let A Y. Then Cl(A) is closed in Y .

By (ii), f

(Cl(A)) is supra r-closed. So Supra rCl(f

(Cl(A))) = f

(Cl(A)).

Now f

(A)  f

(Cl(A)). So f

(Cl(A)) = Supra rCl(f

(Cl(A)))  Supra rCl(f

(A)). That is

Supra rCl(f

(A))  f

(Cl(A)).

(iii). (iii)  (iv). Let A  X. Then f (A) Y.

By (iii), Supra rCl(f

(f(A)))  f

(Cl(f(A))). That is Supra rCl(A)  f

(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

(Cl(f(A))). Or Supra rInt(X-A)  f

(Int(f(X-A))). Or Supra rInt(f

(B))  f

(Int(B)) where B = f(X - A).

(v) .(v)  (i). Let A be open in Y.

Then by (v), Supra rInt(f

(A))  f

(Int(A)).

This implies Supra rInt (f

(A))  f

(A), since A is open.

But Supra rInt(f

(A))  f

(A). Hence Supra rInt(f

(A)) = f

(A). So f

(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

(Supra rInt(B))  rInt(f

(B)) for every B  Y . (ii) rCl (f

(B))  f

(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

(Supra rInt (V))  rInt (f

(V)). Or f

1

(V)  rInt (f

(V)). But f

(V)  rInt (f

(V)). So f

(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

(V)  f

(supra rCl(V )) for every V Y . Then rInt f

1

(Y -V)  f

(Supra rInt(Y - V)). Then by (i) f is supra r-continuous.

(iii) Let V be open in Y. By (iii), f (rCl(f

(V ))  Supra rCl(V ). So by (ii), f is supra r- continuous.

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.

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 r- continuous.

4.14 Theorem: Every almost completely continuous function into a discrete spaceis supra r- continuous.

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:

(i) f is a supra r-open map.

(ii) f is a supra r-closed map.

(iii) f

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 rT

if for every two distinct points of X, there exists a supra r-open set which contains one, but not the other.

(ii) Supra rT

if for every two distinct points of X, there exists Supra r-open sets U and V such that x  U , y U and xV , y V.

(iii) Supra rT

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 rT

, then f has a supra r-closed graph.

Proof: Let (x, y)  (X×Y)-G (f). Then y ≠ f(x). Since Y is supra rT

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 rT

.

Proof: Let x

, x

 X with x

≠ x

. Then f(x

) ≠ f(x

). This implies that (x

, f(x

))  (X×Y)-G

(f).Since f has a supra r-closed graph, there exists supra r-open sets U and V of x

and f(x

)

respectively such that f (U)  V =. Since f is Supra

r-continuous, there exists supra r-open

set W containing x

such that f (W) V. Hence f (W)  f(U) = . So U W =. Hence X is Supra rT

.

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 rT

space.

Proof: Let y

and y

be two distinct points of Y . Then there exists x

in X such that f(x

) = y

. Thus (x

, y

) G(f). Since f has a strongly supra r-closed graph, there exists supra r-open sets U and V of x

and y

respectively such that f(U)

Supra rCl (V) =. Consequently y

V . So Y is a Supra rT

space.

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