Download View PDF - Journal of Computer and Mathematical Sciences

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 xV , 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. Assiut Univ. 12(1), 77-90 (1983).
2. D. Andrijevic’, On b-open sets, Mat. Vesnik, 48, 59-64 (1996).
3. S.P. Arya and R. Gupta, On strongly continuous mappings, Kyungpook Math. J.14, 131143 (1974).
4. R. Devi, S.Sampath Kumar and M. Caldas, On supra α-open sets and sα-continuous maps,
General Mathematics, 16(2), 77-84 (2008).
5. R. C. Jain, The role of regularly open sets in general topology, Ph.D thesis, Meerut
University, Institute of advanced studies, Meerut -India (1980).
6. J.K. Kohli and D. Singh, Between strong continuity and almost continuity, Applied
General Topology, Vol 11, No.1, 29-42 (2010).
7. N. Levine, Semi open sets and semi continuity in topological spaces, Amer. Math Monthly
70, 36-41 (1963).
8. A.S. Mashhour A, A allam, F.S. Mahmoud and F.H. Khader, On supra topological spaces,
Indian J. Pure. Applied Mathematics 14(4), 502-510 April (1983).
9. O.R. Syed and Takashi Noiri, On supra b-open sets and supra b-continuity on topological
spaces, European Journal of Pure and Applied Mathematics, Vol 3, No.2,295-302 (2010).
10. S. Sekar and J. Jayakumar, On supra I-open sets and Supra I continuous functions,
International journal of Scientific and Engineering Research, Volume 3, Issue 5, May (2012).
11. D. Singh, Almost perfectly continuous functions, Quaest Math 33, 1-11 (2010).
419