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 Authors : A. Pushpalatha, A.Kavitha
 Paper ID : IJERTV3IS10833
 Volume & Issue : Volume 03, Issue 01 (January 2014)
 Published (First Online): 25012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Some Mapping on c*gOpen & Closed Maps in Topological Spaces
A. Pushpalatha A. Kavitha
Department of Mathematics, Department of Mathematics,
Government Arts College Dr.Mahalingam College of
Udumalpet642 126, Tirupur District Engineering and Technology,
Tamil Nadu, India Pollachi642003,Coimbatore District
TamilNadu, India
Abstract
In this paper we have introduced the concept of Closed maps ,Open maps , Irresolute and Homeomorphism on the c*gclosed set and study some properties on them.

Introduction
Malghan [1] introduced and investigated some properties of generalized closed maps in topological spaces. The concept of generalized open map was introduced by Sundaram[2]. In this paper we introduced the concepts of c*gclosed maps and c*gopen maps in topological spaces.

Premilinaries
Definition: 2.1: A subset A of a topological space (X, ) is called

Generalized closed set (gclosed)[3] if cl(A) U whenever A U, and U is open .

generalized closed set gclosed[4] if cl(A) U whenever A U, and U is open in .

cg closed set[5] if cl(A) U whenever A U and U is Cset. The complement of
cg closed set is cg open set[5].

c*gclosed set[5] if cl(A) U whenever A U and U is C*set. The complement of c*g – closed set is c*g – open set[5].

c(s)g closed set[5] if cl(A) U whenever A U and U is C(s) set. The complement of c(s)g closed set is c(s)g open set[5].
Definition: 2.3:For a subset A of X is called

a Cset(Due to Sundaram)[2] if A= GF where G is gopen and F is a tset in .

a Cset (Due to Hatir, Noiri and Yuksel)[9] if
A = GF where G is open and F is an *set in .

a C*set[11] if A= GF where G is gopen and F is an *set in .
Definition 2.4: A function is said to be

gclosed[3] in for each closed set F in .

generalized continuous (gcontinuous)[15] if is gclosed in for each closed set F in .

closed map[1] if for each closed set F in , is closed in .

open map[1] if for each open set F in ,
is open in .
3.c*gClosed maps & c*gOpen maps in topological spaces
Definition3.1: A map from a topological space into a topological space is
called c*gclosed map if for each closed set in , is a c*gclosed set in
Theorem 3.2: If a map is closed map then it is c*gclosed map but not conversely.
Proof: Since every closed set is c*gclosed set then it is c*gclosed map.
The converse of the above theorem need not be true as seen from the following example.
Example 3.3: Let . Let be a identity map such that
,
.
Here
Then is c*gclosed map but not closed map. Since for the closed set in ,
is not closed in
Theorem 3.4: If a map is gclosed map then it is c*gclosed map but not conversely.
Proof: Let be a gclosed map. Then for
each closed set in , is gclosed set in Since every gclosed set is c*gclosed set. Therefore is c*gclosed set. Hence is c*gclosed map.
The converse of the above theorem need not be true as seen from the following example.
Example 3.9: Let . Let be a identity map such that
.Then is
The converse of the above theorem need not be true as seen from the following example.
Example 3.5: Let . Let be a identity map such that
.Then is
c*gclosed but not gclosed because for the closed set in , is not g closed in Therefore is not gclosed map.
Theorem 3.6: If a map is closed map then it is c*gclosed map but not conversely.
Proof: Let be a closed map. Then for
each closed set in , is closed set in Since every closed set is c*gclosed set. Therefore is c*gclosed set. Hence is c*gclosed map.
c*gclosed but not
gclosed because for the closed set in ,
is not gclosed in Therefore is not gclosed map.
Theorem 3.10: If a map is gsclosed map then it is c*gclosed map but not conversely.
Proof: Let be a gsclosed map. Then
for each closed set in , is gsclosed set in Since every gsclosed set is c*gclosed set. Therefore is c*gclosed set. Hence is c*gclosed map.
The converse of the above theorem need not be true as seen from the following example.
Example 3.11: Let . Let be a identity map such that
The converse of the above theorem need not be true as seen from the following example.
c*gclosed but not
.Then is
Example 3.7: Let . Let be a identity map such that
.Then is
c*gclosed but not
closed because for the closed set in ,
is not closed in Therefore is not closed map.
Theorem 3.8: If a map is gclosed map then it is c*gclosed map but not conversely.
Proof: Let be a gclosed map. Then for each closed set in , is gclosed set
in Since every gclosed set is c*gclosed set. Therefore is c*gclosed set. Hence is c*gclosed map.
gsclosed because for the closed set in ,
is not gsclosed in Therefore is not gsclosed map.
Definition3.12: A map from a topological space into a topological space is called c*gopen map if is a c*gopen set
in for every open set in .
Theorem 3.13: If a map is open map then it is c*gopen map but not conversely.
Proof: Let be a open map. Let be any open set in , is open set in Then is c*gopen set. Since every open set is c*gopen set. Hence is c*gopen map.
The converse of the above theorem need not be true as seen from the following example.
Example 3.14: Let . Let be a identity map such that
.Then is
c*gopen map but not open map because for the open set in , is not open in Therefore is not open map.
Theorem 3.15: If a map is gopen map then it is c*gopen map but not conversely.
Proof: Let be a gopen map. Let be any open set in , is gopen set in Since every gopen set is c*gopen set. Then is c*gopen set. Hence is c*gopen map.
gopen map because for the open set in ,
is not gopen in Therefore is not gopen map.
Theorem 3.19: If a map is open map then it is c*gopen map but not conversely.
Proof: Let be a open map. Let be any open set in , is open set in Since every open set is c*gopen set. Then is
c*gopen set. Hence is c*gopen map.
The converse of the above theorem need not be true as seen from the following example.
Example 3.20: Let . Let be a identity map such that
.Then is
The converse of the above theorem need not be true as seen from the following example.
Example 3.16: Let . Let be a identity map such that
c*gopen map but not
open map because for the open set in ,
is not open in Therefore is not open map.
c*gopen map but not
.Then is
Theorem 3.21: If a map is gsopen map then it is c*gopen map but not conversely.
gopen map because for the open set in ,
is not gopen in Therefore is not gopen map.
Theorem 3.17: If a map is gopen map then it is c*gopen map but not conversely.
Proof: Let be a gopen map. Let be any open set in , is gopen set in Since every gopen set is c*gopen set. Then is c*gopen set. Hence is c*gopen map.
Proof: Let be a gsopen map. Let be any open set in , is gsopen set in Since every gsopen set is c*gopen set. Then is
<>c*gopen set. Hence is c*gopen map.
The converse of the above theorem need not be true as seen from the following example.
Example 3.22: Let . Let be a identity map such that
.Then is
The converse of the above theorem need not be true as seen from the following example.
Example 3.18: Let . Let be a identity map such that
c*gopen map but not
gsopen map because for the open set in ,
is not gsopen in Therefore is not gsopen map.
c*gopen map but not
.Then is
Theorem 3.23: If is c*gcontinuous and c*gclosed and A is a c*gclosed set of
then is c*gclosed in
Proof: Let where is c*set of Since is c*gcontinuous, is c*set
containing A. Hence as is c*gclosed. Since is c*gclosed, is c*gclosed set contained in c*set , which implies that and hence
So is c*gclosed in Y.
Corollary 3.24: If is continuous and closed map and if is c*gclosed set in then
is c*gclosed in
Proof: Since every continuous map is c*g continuous and every closed map is c*gclosed, by the above theorem the result follows.
Theorem 3.25: If is closed and
is c*gclosed then is c*gclosed.
Proof: Let is a closed map and is c*gclosed map. Let be any closed
set in Since is closed, is
closed in and since is c*gclosed
, is c*gclosed set in Therefore is c*gclosed map.
Theorem 3.26: If is c*gclosed and is closed set in Then is
c*gclosed.
Proof: Let be closed set in A. Then V is closed
in X. Therefore is c*gclosed set in Y. By theorem 1.24 is c*gclosed. That is is c*gclosed set in Y. Therefore
is c*gclosed.

c*g irresolute map in Topological Spaces
Crossely and Hildebrand[9] introduced and investigated the concept of irresolute function in topological spaces. Sundaram[2] , Maheshwari and Prasad[10], Jankovic[11] have defined gc irresolute maps,
irresolute maps and popen maps in topological spaces.
In this section, we have introduced a new class of map called
c*g irresolute map and study some of their properties.
Definition 4.1: A map from topological space X into a topological space Y is called
c*g irresolute map in the inverse of every c*g – closed(c*g open) set in Y is c*g closed
(c*g open) in X.
Theorem 4.2: If a map is
c*g irresolute, then it is c*g continuous, but not conversely.
Proof: Assume that 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 is c*g irresolute, irresolute, is c*g closed in X. Therefore is c*g continuous.
The converse of the above theorem need not be true as seen from the following example.
Example 4.3: Consider the topological space
with topology
,
Let be
the identity map then is c*g continuous ,
because for the inverse image of every closed in Y is c*g closed in X, but not c*g irresolute. Because for the inverse image of every c*g closed in Y is not c*g closed in X. (ie) for the
c*g closed set {b} in Y the inverse image
is not c*g closed in X.
Theorem 4.4: Let X,Y,and Z be any topological spaces.For any c*g irresolute map
and any c*g continuous map the composition is c*g continuous.
Proof: Let F be any closed set in Z. Since is c*g continuous, is c*g closed in Y.
Since is c*g irresolute is
c*g closed .
Therefore is c*g continuous.
Theorem 4.5: If from topological space X into a topological space Y is bijective,
c*g open set and c*g continuous then is c*g irresolute.
Proof: Let A be a c*g closd set in Y. Let
,Where O is C*set in X. Therefore
holds. is c*g open set and A is c*g closed in Y,
Since is c*g continuous and cl(A) is closed
Remark 4.9: The following two examples show that the concepts of irresolute maps and
c*g irresolute maps are independent of each other.
Example 4.10: Consider the topological space
with topology
,
in Y. cl( f 1 ( cl(A)) O
2 {,Y,{a},{a,b}}
Then the defined
and so cl( f 1 (A)) O .Therefore is c*g closed in X. Hence is c*g irresolute.
The following examples show that no assumption of the above theorem can be removed.
Example 4.6:Consider the topological space
with topology
,
Then the defined identity
map is c*g continuous,
bijective and not cgopen. So is not cg irresolute. Since for the c*g closed set {a} in Y the inverse image is not
c*g closed in X.
Example 4.7:Consider the topological space
with topology
,
Then the map
be defined by
Then is
c*g continuous, c*g open and not bijective. So is not c*g irresolute. Since for the c*g closed set {b} in Y the inverse image is not c*g closed in X.
Example 4.8: Consider the topological space
with topology
,
identity map is irresolute
but not c*g irresolute. Since {b} is c*g closed set in Y has its inverse image is not c*g closed in X.
Example 4.11: Consider the topological space
with topology
1 {, X ,{a},{a,b}} ,
2 {,Y,{a},{b},{a,b}} . Then the defined identity map is c*g –
irresolute but not irresolute. Since the closed set
{a,c} in Y has its inverse image
f 1 ({a, c}) {a, c} is not closed in X.
Remark 4.12:From the following diagram we can conclude that c*g irresolute map is independent with irresolute map.
c*g irresolute irresolute map

c*g homeomorphism maps in Topological Spaces
Several mathematicians have generalized homeomorphism in topological spaces. Biswas[14],Crossely and Hildebrand[9], Gentry and Hoyle[13] and Umehara and Maki[12] have introduced and investigated semihomeomorphism, which also a generalization of homeomorphism. Sundaram[2] introduced ghomeomorphism and gchomeomorphism is topological spaces.
In this section we introduce the concept of c*g homeomorphism and study some of their properties.
Definition 5.1: A bijection
2 {,Y,{a},{a,b},{a, c}} Then the
is called c*g – homeomorphism if f is both
defined identity map is
bijective, c*g open and not c*g continuous,. So is not c*g irresolute. Since for the c*g closed set {b} in Y the inverse image f 1 ({b}) {b} is not c*g closed in X.
c*g open and c*g continuous.
Theorem 5.2: 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*g open the proof follows.
The converse of the above theorem need not be true as seen from the following example.
Example 5.3: Let with
then is c*g –
homeomorphism but not homeomorphism.
Theorem 5.4: For any bijection the following statements are equivalent.

is c*g continuous.

is a c*g open map.

is a c*g closed map.

Proof: Let G be any open set in X.Since is c*g continuous, the inverse
image of G under namely is c*g open
in Y.So is c*g open map.
Let F be any closed set in X.Then Fc is open in X. Since is c*g open map
is c*g open map in Y.But
and so is c*g open map in Y.Therefore is a c*g closed map.
Let F be any closed set in X. Then is c*g closed map in

herefore is c*g continuous.
Theorem 5.5: Let be a bijective and c*g continuous map the following statement are equivalent.

is a c*g open map.

is a c*g homeomorphism.

is a c*g closed map.

Proof: The proof easily follows from definitions and assumptions.
The following examples shows that the composition of wo c*g homeomorphism need not be c*g homeomorphism.
Example 5.6: Let with
topologies
Let and be identity maps such that and then and are
c*g homeomorphism, but their composition
is not c*g homeomorphism.
Theorem 5.7: Every homeomorphism is a c*g homeomorphism.
Proof: Let be a homeomorphism then is continuous and closed. Since every
continuous is c*g continuous and every
closed is c*g closed, is c*g continuous
and c*g closed. Therefore is c*g homeomorphism.
The converse of the above theorem need not to be true as seen from the following example.
Example 5.8:
Consider the topological space with topology
,
Then the defined
identity map is
c*g homeomorphism but not
homeomorphism.Since for the open set {a} in X the inverse image f1({a})={a} is not open in Y.
From the above observations we get the following diagram:
homeomorphism homeomorphism
c*g homeomorphism
Definition 5.9 : A bijection is said to be (c*g)* homeomorphism if and its inverse are c*g irresolute map.
Notation 5.10: Let the family of all (c*g)* homeomorphism from onto itself be
denoted by and the family of all c*g homeomorphism from onto itself be denoted by The family of all
homeomorphism from onto itself be denoted by
Theorem 5.11: Let X be a topological space. Then
i) The set is group under composition of maps. (ii) h(x) is a subgroup of
(iii) .
Proof for (i): Let then
and so
is closed under the composition of maps.The composition of maps is associative. The identity map I:XX is a homeomorphism and
so Also
for every

Kavitha.A., cg, c*g, c(s)gclosed sets in Topological spaces, In the 99th Indian
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Balachandran, K. Sundaram, P. and Maki, H., On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. Math., 12(1991), 513.

Mashhour, A. S., Hasanein, I. A. and ElDeeb,
S. N., continuous and – open mappings, Acta. Math. Hunger., 41(1983), 213218.

Pushpalatha .A., Kavitha.A., A New class
cgset weaker form of closed sets in Topological Spaces, International Journal of computer Application Vol 54No 26 September 2012.
. If ,
then and
Hence
is a group under the composition of maps.
Proof for (ii): Let be a homeomorphism. Then by theorem 4.5.Both of and are )* irresolute and so is a
homeomorphism.Therefore every homeomorphism is a homeomorphism and so is a subset of .
Also is a group under composition of maps.Therefore is a subgroup of group
.
Proof for (iii): Since every irresolute map is continuous, is a subset of .
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