 Open Access
 Total Downloads : 449
 Authors : A. Pushpalatha, K. Kavithamani
 Paper ID : IJERTV1IS9469
 Volume & Issue : Volume 01, Issue 09 (November 2012)
 Published (First Online): 29112012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Study On c*gHomeomorphisms In Topological Spaces
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
Vol. 1 Issue 9, November 2012
A. Pushpalatha Professor
Department of Mathematics, Government Arts College, Udumalpet642 126, Tirupur District, Tamil Nadu, India.
Abstract
In this paper, we have introduced the concept of c*gcontinuous, c*g – closed, c*g open, c*g irresolute, c*g homeomorphisms and (c*g)* – homeomorphisms in Topological space.
Key words: c*gcontinuous, c*g – closed, c*g open, c*g irresolute, c*g homeomorphisms and (c*g)* – homeomorphisms
K. Kavithamani Assistant Professor
Department of Mathematics, JCT College of Engineering &
Technology,Coimbatore – 641 105, Coimbatore District
Tamil Nadu, India.

INTRODUCTION
Levine [6], [28] introduced the concept of generalized closed sets and strong continuity in topological space. Dunham and Levine [2] further studied some properties of generalized closed sets. Sundaram [13] introduced the concept of generalized continuous function and proved that the class of generalized continuous function includes the class of continuous function and studied several properties related to it. Pushpalatha [26] introduced the concept of strongly generalized continuous function and studied several properties related to it.
Pallaniappan[23]introduced the concept of regular generalized continuous function and proved that the class of regular generalized continuous function includes the class of continuous function and studied several properties related to it.
Malghan[20]introduced and investigated some properties of generalized closed maps in topological spaces. The concept of generalized open map was introduced by Sundaram[13]. Biswas [01] defined semi open mappings as a generalization of open mappings and studied several of introduced the concepts of strongly generalized closed maps in topological spaces.
Functions and of course irresolute functions stand among the most important and most researched points in the whole of mathematical science. In 1972, Crossely and Hildebrand [30] introduced the notion of irresoluteness. Many different form of irresolute functions have been introduced over the years. Various interesting problems arise when one considers irresoluteness. Its important in significant in various areas of mathematics and related science.
Several mathematicians have generalized homeomorphisms in topological spaces. Biswas[1], Crossely and Hildebrand[30], Gentry and Hoyle[32], Umehara and Maki[33] have introduced and investigated semi homeomorphisms, somewhat homeomorphisms and g – homeomorphisms. Crossely and Hildebrand defined yet another semi homeomorphism which is also a generalization of homeomorphism. Sundaram[13]introduced g
homeomorphism and gc homeomorphism in topological spaces. Puspalatha [26 ] introduced strongly generalized homeomorphism and strongly g*homeomorphism in topological spaces.
In this section, we introduced the concepts of c*g continuous function, c*g closed maps, c*g open maps, c*g irresolute maps and c*g homeomorphisms and (c*g)* homeomorphism in topological spaces and study their properties. It is an extension study of [ 27] for continuous functions.

PRELIMINARIES DEFINITION: 2.1
A map f: xy from a topological space x into a topological space y is called

Continuous if f1(V) is closed in x for each subset V in y.

Strongly continuous if f1(V) is both open and closed in y for each subset V in y(Due to Sundaram).

Strongly generalized if the inverse image of every closed set in y is strongly g closed in x.

Regular generalized continuous if the inverse image of every closed set in y is rg closed in x.
DEFINITION: 2.2
A map f: xy from a topological space x into a topological space y is called

Strongly generalized closed if for each closed set F in x, f (F) is strongly gclosed set in y.

Regular generalized closed if for each closed set F in x, f(F) is Regular generalized closed set in y.
DEFINITION: 2.3
A map f: xy from a topological space x into a topological space y is called
(a) Irresolute if f1(V) is semiopen set in x for every semiopen set V in y[].

Strongly generalized irresolute if f1(V) is strongly gclosed in x for every strongly gclosed set V in y.

Regular generalized irresolute if
f1(V) is rgclosed in x for every rg closed set V in y.
3.1 C*g CONTINUOUS FUNCTIONS IN TOPOLOGICAL
SPACES
DEFINITION: 3.1.1 A map f: xy
from a topological space x into a topological space y is called c*g continuous if the inverse image of every closed set in y is c*g closed in x.
THEOREM: 3.1.2 If a map f: xy is continuous then it is c*g continuous but not conversely.
Proof: – Let f: xy be continuous. Let F be any closed set in y. Then the inverse image f1(F) is closed in x. Since every closed set is c*gclosed, f1(F) is c*g closed in x. Therefore f is c*g continuous.
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE:3.1.3 Let x= y =
{a,b,c};with the topology;
= { ,x,{a, b}} and = { ,y,{a}}. Let f: xy be the identity map. Then f is not continuous, since for the closed set {b,c} in y, f1 ({b,c})={b,c} is not closed in x. But f is c*g continuous.
THEOREM: 3.1.4 If a map f: xy is strongly gcontinuous then it is c*g continuous but not conversely.
Proof: – Let f: xy be a strongly g continuous map. Let F be any closed set in y. Then the inverse image f1(F) is strongly g closed in x. Since every strongly g closed set is c*gclosed set, f1(F) is c*g closed in x. Therefore f is c*g continuous.
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.1.5 Let x = y =
{a,b,c};with the topology;
= { ,X,{a}} and = { ,y,{b}}. Let f: (x, ) (y, ) be the identity map. Then f is c*g continuous, but not strongly g continuous. For {a,c} is closed in y, but f1({a,c})={a,c} is not strongly g closed in x. Therefore f is not strongly g – continuous.
THEOREM: 3.1.6. Let f: xy be a map. Then following statements are equivalent.

f is c*g continuous

The inverse image of each open set in y is c*gopen in x.
Proof:Assume that f: xy is c*g continuous. Let G be open in y. Then Gc is closed in y. Since f is c*g continuous, f1(Gc) is c*g continuous in
x. But f1(Gc) = x f1(G). Thus f1(G) is c*gopen in x.
Conversely assume that the inverse image of each open set in y is c*gopen in x. Let F be any closed set in y. Then Fc is open in y. By assumption f1(Fc) is c*gopen in x. Hence a and b are equivalent.
THEOREM: 3.1.7 If a map f: xy is c*g continuous, then it is regular generalized continuous.
Proof: – Let f: xy be c*gcontinuous. Let F be any closed set in y. Then the inverse image f1(F) is c*g closed in x. Since every c*g closed set is regular generalized closed set, f1(F) is regular
generalized closed in x. Therefore f is regular generalized continuous.
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.1.8 Let x = y =
{a,b,c};with the topology;
= { ,X,{b,c}} and = { ,y,{a,b}}. Let f: xy be the identity map. Then f is not c* continuous. Since for the closed set {c} in y, f1({c}) ={c} is not closed in x. But f is regular generalized continuous.
We illustrate the relations between various generalizations of continuous functions in the following diagram
Continuity Strongly g continuity c*g continuity regular
generalized continuity
In the above diagram none of the implications can be reversed.

c*gCLOSED MAPS AND c*g OPEN MAPS IN TOPOLOGICAL SPACES
DEFINITION: 3.2.1 A map f: xy
from a topological space x into a topological space y is called c*gclosed map if for each closed set F in x, f(F) is a c*g closed set in y.
THEOREM: 3.2.2 If a map f: xy is closed then it is c*g closed but not conversely.
Proof:. Since every closed set is c*g closed, the result follows
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.2.3 consider the topological space x = y = {a,b,c};with the topology; = { ,X,{a}} and
= { ,y,{a,b}}. Here
c (x, ) = { ,X,{b,c }};
c (y, ) = { ,Y,{c}} and c*gc(y, )
= { ,y,{c},{a,b}{b,c},{a,c}}. Let f be the identity map from x onto y. Then f is c*g closed, but not a closed map.
Since for the closed set {b,c} in (x, ),
f{b,c})={b,c} is not closed in y.
DEFINITION: 3.2.4 A map f: xy
from a topological space x into a topological space y is called c*gopen map if f (u) is c*gopen in y for every open set u in x.
THEOREM: 3.2.5 If a map f: xy is open then it is c*g open but not conversely.
Proof:. Let f: xy be an open map. Let U be any open set in x. Then f (U) is an open set in Y. Then f (U) is c*g open, since every open set is c*gopen. Therefore f is c*gopen.
Converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.2.6 consider the topological space x = y = {a,b,c};with the topology; = { ,X,{a, }} and
= { ,y,{a,b}}. Here c*go (y, ) =
{ , y, {a}, {b}, {c}, {a,b}}. Then the identity function f: xy is c*g open, but not open, since for the open set {a}
in (x, ), f{a})={a} is c*g open, but
not an open in (y, ). Therefore f is not an open map in y.
THEOREM: 3.2.7. If f: xy is regular generalized continuous and c*g closed and A is a c*gclosed set of x, then f (A) is c*gclosed in y.
Proof:. Let f (A) O. where O is regular generalized open set of y. Since f is regular generalized continuous, f1(O) is regular generalized open set containing A. Hence cl(A) f1(O) as A is c*gclosed. Since f is c*gclosed, f(cl(A)) is a c*gclosed set contained in the regular generalized open set O, which implies that cl[f(cl(A))] O and hence cl[f(A)] O. So f (A) is c*g closed set in y.
COROLLORY: 3.2.8. If f: xy is
continuous, closed and A is c*gclosed of x, then f(A) is c*gclosed in y.
Proof:.Since every continuous map is regular generalized continuous and every closed map is c*gclosed, by the above theorem the result follows.
THEOREM: 3.2.9. If f : xy is closed and h : yz is c*g closed then
h.f : xz is c*g closed.
Proof: – Let f: xy is a closed map and h : yz is a c*gclosed map. Let v be any closed set in x. Since f: xy is closed, f(v) is closed in y and since h : yz is c*gclosed, h(f(v)) is a c*g closed set in z. Therefore h.f : xz is a c*g closed map.
THEOREM: 3.2.10. If f : xy is
c*g closed and A is closed set in x. Then fA : A y is c*g closed.
Proof: – Let v be closed set in A. Then v is closed in x. Therefore it is c*g closed in x. By Theorem 3.2.9. f (v) is c*g closed. That is fA (v) =f(v) is c*g closed in y. Therefore fA : A y is c*g closed.

c*g IRRESOLUTE MAPS IN TOPOLOGICAL SPACES
DEFINITION: 3.3.1 A map f: xy
from a topological space x into a topological space y is called c*g irresolute map if the inverse of every c*g closed(c*gopen) set in y is c*g closed set(c*gopen) in x.
THEOREM: 3.3.2 If a map f: xy is irresolute then it is c*gcontinuous, but not conversely.
Proof:. Assume that f is c*g irresolute. Let F be any closed set in y. Since every closed set is c*g closed, F is c*g closed in y. Since f is c*g irresolute, f1(F) is c*g closed in x. Therefore f is c*g continuous.
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.3.3 consider the topological space x = y = {a,b,c};with the topology; 1= { ,X,{c}} and
2={ ,y,{a},{a,b}}.
Let f: (x, 1) (y, 2) be the identity map. Then f is c*gcontinuous, but not c*g irresolute. For the c*g closed set
{a}, the inverse image f1({a}) = (a) is not c*g closed in x.
THEOREM: 3.3.4 .Let x, y and z be any topological spaces. For any c*g irresolute map f: xy and any c*g continuous map g: yz the composition g.f : xz is c*g continuous.
Proof: – Let f be any closed set in z. Since g is c*gcontinuous,g1(F) is c*g closed in y. Since f is c*g irresolute, f1 [g1 (F)] is c*g closed in x. But f1 [g1 (F)] = (g.f)1 (F). Therefore g.f is c*gcontinuous.
THEOREM: 3.3.5. If f: xy from a topological space x into a topological space y is bijective, c* set and c*g continuous then f is c*g irresolute.
Proof: – Let A be a c*gclosed set in y. let f1(A) O, where O is c* set in x. Therefore, A f(O) holds. Since f(O) is c* set and A is c*gclosed in y, cl(A) f(O) holds and hence f1() O. Since f is c*gcontinuous and is closed in y, cl[f1()] O and so cl[f1(A)] O. Therefore f1(A) is c*gclosed in x. Hence f is c*g irresolute.
The following examples show that no assumption of the above theorem can be removed.
EXAMPLE: 3.3.6. Let (x, 1) and (y,
2) be the spaces defined in 3.3.3. The identity map f: (x, 1) (y, 2) is c*g continuous, bijective and not c* set. And f is not c*g irresolute. Since for
the c*g closed set G= {a} in y, the inverse image f1(G) =G is not c*g closed in x.
EXAMPLE: 3.3.7. consider the topological space x = y = {a,b,c};with the topology; 1= { ,X,{c}} and
2 = { ,y,{a},{a,b}}. Let f : (x, 1)(y, 2) be the identity map. Then f is c*gcontinuous and c*set, but it is not bijective. And f is not c*g irresolute. Since, for the c*g closed set G= {a}, the inverse image f1(G) =G is not c*g closed in x.
EXAMPLE: 3.3.8. Consider the map f: x y is defined in example 3.3.3. The map f is c*set, bijective but not c*g continuous and f is not c*g irresolute.
The following two examples show that the concepts of irresolute maps and c*g irresolute maps are independent of each other.
EXAMPLE: 3.3.9. consider the topological space x = y = {a,b,c} with the topologies,
1 = { ,X,{a}{b},{a,b}} and
2 = { ,y,{a},{b,c}}. Then the identity map f: (x, 1) (y, 2) is irresolute. But it is not c*g irresolute, since for the c*gclosed set G= {a} in y, f1(G) =G is not c*g closed in x.
EXAMPLE: 3.3.10. Consider the topological space x = y = {a,b,c} with the topologies, 1 =
{ ,X,{a,b}} and 2 = { ,y,{c}}.
Then the identity map f: (x, 1) (y,
2) is c*g irresolute. But it is not irresolute, since for the semiclosed set G= {c} in y, f1(G) =G is not semi closed in x.
THEOREM: 3.3.11. If y is TS and f:
xy from a topological space x into a topological space y is c*g irresolute, then f is regular generalized irresolute. Proof: – Let y be a TS space and f: x y is a c*g irresolute map. Let V
be a c*set in y and since y is TS, V is c*gclosed in y and since f is c*g irresolute, f1(V) is c*g closed in x. But every c*gclosed set is regular generalized closed, f1 (V) is regular generalized closed. Therefore f is a regular generalized irresolute map.
EXAMPLE: 3.3.12. Consider the topological space x = y = {a,b,c} with the topologies,
1 = { ,X,{a},{b},{a,b}} and
2 = { ,y,{a,b}}. Then the idetity map f: (x, 1) (y, 2) is c*g irresolute. But it is not regular generalized irresolute, since for the regular generalized closed set G= {b} in y, f1(G) =G is not regular generalized closed in x.
THEOREM: 3.3.13. If y is TS and f:
xy from a topological space x into a topological space y is regular generalized irresolute, then f is c*g irresolute.
Proof: – Let x be a TS space and f: x y is a regular generalized irresolute map. Let V be a c*gclosed set in y. Since every c*gclosed set is regular generalized closed, V is regular generalized closed in y. Since f is regular generalized irresolute, f1(V) is regular generalized closed set in x. But x is TS and therefore f1 (V) is c*g closed in x. Thus f is c*g irresolute.
EXAMPLE: 3.3.14. Consider the topological space x = y = {a,b,c} with the topologies,
1={ ,X,{a},{c},{a,c}}and
2= { ,y,{a},{a,b}}. Then the identity map f: (x, 1) (y, 2) is regular generalized irresolute. But it is not c*g irresolute, since for the c*gclosed set G= {b} in y, f1(G) =G is not c*g closed in x.
REMARK: 3.3.15. The concepts of regular generalized irresolute and c*g irresolute are independent as seen from example 3.3.12. and example 3.3.14.

c*g HOMEOMORPHISMS IN TOPOLOGICAL SPACES
DEFINITION: 3.4.1. A bijection map f: (x, ) (y, )) from a topological space x into a topological space y is called c*ghomeomorphism if f is both c*g continuous and c*gopen.
THEOREM: 3.4.2. Every
homeomorphism is a c*g homeomorphism, but not conversely.
Proof:. Since every continuous function is c*g continuous and every open map is c*gopen, the proof follows.
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.4.3. Consider the topological space x = y = {a,b,c}; with the topology = { ,X,{a}} and
= { ,y,{a},{b,c}}. Then the identity
map f: (x, ) (y, ) is a c*g homeomorphism but not a homeomorphism.
THEOREM: 3.4.4. Every strongly g homeomorphism is a c*g homeomorphism, but not conversely.
Proof:. Let f: xy be strongly g homeomorphism. Then f is strongly g continuous and strongly gopen. Since every strongly gcontinuous function is c*g continuous and every strongly g open map is c*gopen, f is c*g continuous and c*gopen. Hence f is a c*ghomeomorphism.
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.4.3. Consider the topological space x = y = {a,b,c};with the topology = { ,X,{c},{a,c}} and
= { ,y,{b},{a,b}}. Define the map f: (x, ) (y, ) be the identity. Then
the set {a,c}is a c*ghomeomorphism but not a strongly ghomeomorphism.
Next we shall characterize c*g homeomorphism and c*gopen.
THEOREM: 3.4.6. For any bijection f: xy the following statements are equivalent.

The inverse map f1:yx is c*g continuous.

f is a c*gopen map.

f is a c*gclosed map.
Proof: – (a) (b). Let G be any open set in x. Since f1 is c*gcontinuous, the inverse image of G under f1, namely f(G) is c*gopen in y and so f is a c*gopen map.

(c). Let F be any closed set in x. Then Fc is open in x. Since f is c*g open, f(Fc) is c*gopen in y. But f(Fc) =yf(F) and so f(F) is c*gclosed in y. Therefore f is a c*gclosed map.

(a). Let F be any closed set in x. Then (f1) 1(F) =f (F) is c*gclosed in y. Since f is c*gclosed map. Therefore f1 is c*gcontinuous.
THEOREM: 3.4.7. Let f: (x, ) (y,
) be a bijective and c*gcontinuous map, the following statements are equivalent.

f is a c*gopen map.

f is c*ghomeomorphism.

f is a c*gclosed map.
Proof: .(a) (b). By assumption, f is bijective, c*gcontinuous and c*gopen. Then by definition, f is c*g homeomorphism.

(c). By assumption, f is c*gopen and bijective. By theorem 3.4.4. f is c*gclosed map

(a). By assumption, f is c*g closed and bijective. By theorem 3.4.4. f is c*gopen map.
EXAMPLE: 3.4.8. Let x = y = {a,b,c}
with topologies
1={ ,X,{c},{a,c},{b,c}};
2={ ,y,{b}{b,c}};and
3={ ,z,{b},{c},{b,c}} respectively. Let f and g are identity maps such that f : xy and g: yz. Then f and g are c*ghomeomorphism but their composition g.f : xz is not a c*g homeomorphism. For the open set {a,c} in x, g[f({a,c})]={a,c} is not c*gopen in z.
THEOREM: 3.4.9. Every c*g homeomorphism is a regular generalized homeomorphism, but not conversely.
Proof:. Let f: xy be c*g homeomorphism. Then f is c*g continuous and c*gopen. Since every c*gcontinuous function is regular generalized continuous and every c*g open map is regular generalized open, f is regular generalized continuous and regular generalized open. Hence f is a regular generalized homeomorphism.
The converse of the above theorem need not be true as seen from the following example.
EXAMPLE: 3.4.10. Let x = y =
{a,b,c} with topologies
1={ ,X,{a},{a,b}{a,c}}and
={ ,y,{a},{b,c}}. Define the map f: (x, ) (y, ) be the identity. Then f is regular generalized homeomorphism, but not a c*g homeomorphism.
DEFINITION 3.4.11. A bijection f: (x, ) (y, ) is said to be a (c*g)* homeomorphism if f and its inverse f1 are c*g irresolute maps.
NOTATION: Let family of all (c*g)* homeomorphism from (x, ) onto itself be denoted by (c*g)* h (x, ) and family of all c*g homeomorphism from (x, ) onto itself be denoted by c*g h (x, ). The family of all
homeomorphism from (x, ) onto itself is denoted by h (x, ).
THEOREM: 3.4.12.Let x be a
topological space. Then

The set (c*g)*h(x) is a group under composition of maps.

h(x) is a subgroup of (c*g)*h(x).

(c*g)*h(x) c*gh(x).
Proof: i). Let f,g(c*g)*h(x). Then g.f
(c*g)*h(x) and so (c*g)*h(x) is closed under the composition maps. The composition of maps is associative. The identity map I :xx is a (c*g)* homeomorphism and so I(c*g)*h(x). Also f.I = I.f =f for everyf(c*g)*h(x). If f(c*g)*h(x), the f1(c*g)*h(x) and f.f1= f1.f=I. Hence (c*g)*h(x) is a group under the composition of maps.

Let f: xy be a homeomorphism. Then by theorem 3.3.5. both of f and f1are (c*g)* irresolute and so f is a (c*g)* – homeomorphism. Therefore every homeomorphism is a (c*g)* – homeomorphism and so h (x) is a subset of (c*g)* h (x). Also h(x) is a group under the composition of maps. Therefore h(x) is a sub group of the group (c*g)*h(x).

Since every (c*g)* – irresolute map is c*g continuous, (c*g)*h(x) is a subset of (c*g)h(x).
From the above observation we get the following diagram.
Homeomorphism strongly g Homeomorphism c*g Homeomorphism regular generalized Homeomorphism.
In the above diagram none of the implications can be reversed.
REFERENCES
[1].N.Biswas,On some mapping in topological spaces, Bull.Call.Math.Soc: 61 (1969), 127135. [2]. W.Dunham and N.Levine, Further results on generalized closed sets in topology,Kyungpook.Math.J,20 (1980),169175.
[3]. R. Engelking, Outlines of general topology, North Holland publishing company, Amsterdam, 1968. [4]. S.Fomin, Extension of topological spaces, Ann.Math ;44(1943),471480. [5]. N.Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly; 68(1961),4446. [6]. N.Levine, Generalized closed sets in topology, Rend.Circ.Mat.Palermo, 19(1970),8996. [7].T,Noiri, A Generalization of closed mappings, Atti. Acad. Lincei. Rend. Cl. Sci. Fis. Mat. Nature; 54(1973), 412415.
[8]. P.Sundaram and M.Rajamani, On decomposition of Generalized continuous maps in topological spaces, Acta Ciencia Indica, Vol.26(2000), 101104. []. P.Sundaram and M.Rajamani, On decomposition of Regular generalized continuous maps in topological spaces, Far East.J.Math Sci.,III (2000),179188. [10].I.A.Stean and J.A.Seebach, Jr., Counter Examples in Topology, Springer Verlag, Newyork, 1978. [11].D.D.Smith, An alternate characterization of continuity, Proc. Amer. Math. Soc; 39(1973), 318320. [12]. I.Arockiarani, Studies on generalization of generalized closed sets and maps in topological spaces,Ph.D. Thesis, Bharathiar University, Coimbatore,(1997).
[13].K.Balachandran, P.Sundaram and H.Maki, On Generalized continuous maps in Topological spaces, Mem. Fac. Sci. Kochi.Univ. Math., 12(1991), 513. [14].N.Bourbaki,General topology, AddisonWesley, Reading, Mass., 1966. [15]. J.Chaber, Remarks on open closed mappings. Fund. Math., 74(1972),197208. [16].R.Devi, Studies On generalizations of closed maps and homeomorphisms in topological spaces, Ph.D. Thesis , Bharathiar University, Coimbatore (1994).[17].W.Dunham,T1/2Spaces, Kyungpook. Math. J., 17(1977), 161
169.
[18]. G.L.Faro, On strongly – irresolute mappings, Indian J.Pure Appl. Math., 18(1987), 146151. [19].S.N.Maheswari and R.Prasad., Onirresolute mappings, Tankang J. Math., 11(1980), 209214.
[20].S.R.Malghan, Generalized closed maps,J. Karnataka Univ.Sci., 27(1982), 8288. [21].J.R.Munkers,TopologyA First course, Prentice Hall, Inc, Englewood cliffs S.N.J., 1975. [21]. M.G. Murdeshwar, Genreal Topology, Wiley Eastern limited, New Delhi 1986. [22]. A. Neubrunnova, On certain Generalization of the notion of continuity, Mat, Casopsis., 23[1973], 374380. [23]. N. Palainyappan and K.C. Rao, Regular generalized closed sets,Kyungpook,Math.J., 33[1933], 211
219.
[24]. S.F. Tadeos and A.S. AbdAllah, On some mappings of Topological spaces, Ain shams. sci. Bull., 28A [1990], 8198. [25]. P.Sundaram, On ccontinuous maps in Topological spaces, proc.of 82nd session of the Indian Science Congress, Calcutta [1997], 4849. [26]. A. Pushpalatha, Studies on Generalized of mappings in Topological spaces, PhD Thesis, Bharathiar University, Coimbatore 2000.[27]. K. Kavithamani ,A Study of c*g closed and c(s)closed sets in Topological spaces., Vol 10(2013); Antartica Journal of Mathematics.
[28]. N.Levine, Strong Continuity in topological spaces, Amer.Math. Monthly; 67(1960),269. [30].S.G.Crossely and S.K.Hildebrand, Semi topological properties, Fund. Math., 74(1972), 233254. [31]. P.Sundaram and N.Nagaveni, On weekly generalized continuous maps , Weekly generalized closed maps and Weekly generalized irresolute maps, Far East.J.Math Sci., Vol.6(1998),903 912. [32]. K.R.Gentry and H.G.Hoyle,Some what continuous functions, Czechoslovak math.J.,21(1971)5 12. [33].J. Umehara and H.Maki, A note onthe homeomorphic image of a T
space, Mem.Fac.Sci.Cochi Univ. (Math).,10 (1989), 3945.