Some Mapping on αc*g-Open & Closed Maps in Topological Spaces

Some Mapping on c*g-Open & Closed Maps in Topological Spaces

A. Pushpalatha A. Kavitha

Department of Mathematics, Department of Mathematics,

Government Arts College Dr.Mahalingam College of

Udumalpet-642 126, Tirupur District Engineering and Technology,

Tamil Nadu, India Pollachi-642003,Coimbatore District

TamilNadu, India

Abstract

In this paper we have introduced the concept of Closed maps ,Open maps , Irresolute and Homeomorphism on the c*g-closed set and study some properties on them.

1. 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*g-closed maps and c*g-open maps in topological spaces.

2. Premilinaries

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

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

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

3. cg- closed set[5] if cl(A) U whenever A U and U is C-set. The complement of

   cg- closed set is cg- open set[5].

4. c*g-closed 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].

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

1. a C-set(Due to Sundaram)[2] if A= GF where G is g-open and F is a t-set in .

2. a C-set (Due to Hatir, Noiri and Yuksel)[9] if

   A = GF where G is open and F is an *-set in .

3. a C*set[11] if A= GF where G is g-open and F is an *-set in .

Definition 2.4: A function is said to be

1. g-closed[3] in for each closed set F in .

2. -generalized continuous (g-continuous)[15] if is g-closed in for each closed set F in .

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

4. open map[1] if for each open set F in ,

is open in .

3.c*g-Closed maps & c*g-Open maps in topological spaces

Definition3.1: A map from a topological space into a topological space is

called c*g-closed map if for each closed set in , is a c*g-closed set in

Theorem 3.2: If a map is closed map then it is c*g-closed map but not conversely.

Proof: Since every closed set is c*g-closed set then it is c*g-closed 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*g-closed map but not closed map. Since for the closed set in ,

is not closed in

Theorem 3.4: If a map is g-closed map then it is c*g-closed map but not conversely.

Proof: Let be a g-closed map. Then for

each closed set in , is g-closed set in Since every g-closed set is c*g-closed set. Therefore is c*g-closed set. Hence is c*g-closed 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*g-closed but not g-closed because for the closed set in , is not g- closed in Therefore is not g-closed map.

Theorem 3.6: If a map is -closed map then it is c*g-closed 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*g-closed set. Therefore is c*g-closed set. Hence is c*g-closed map.

c*g-closed but not

g-closed because for the closed set in ,

is not g-closed in Therefore is not g-closed map.

Theorem 3.10: If a map is gs-closed map then it is c*g-closed map but not conversely.

Proof: Let be a gs-closed map. Then

for each closed set in , is gs-closed set in Since every gs-closed set is c*g-closed set. Therefore is c*g-closed set. Hence is c*g-closed 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*g-closed but not

.Then is

Example 3.7: Let . Let be a identity map such that

.Then is

c*g-closed 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 g-closed map then it is c*g-closed map but not conversely.

Proof: Let be a g-closed map. Then for each closed set in , is g-closed set

in Since every g-closed set is c*g-closed set. Therefore is c*g-closed set. Hence is c*g-closed map.

gs-closed because for the closed set in ,

is not gs-closed in Therefore is not gs-closed map.

Definition3.12: A map from a topological space into a topological space is called c*g-open map if is a c*g-open set

in for every open set in .

Theorem 3.13: If a map is open map then it is c*g-open map but not conversely.

Proof: Let be a open map. Let be any open set in , is open set in Then is c*g-open set. Since every open set is c*g-open set. Hence is c*g-open 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*g-open 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 g-open map then it is c*g-open map but not conversely.

Proof: Let be a g-open map. Let be any open set in , is g-open set in Since every g-open set is c*g-open set. Then is c*g-open set. Hence is c*g-open map.

g-open map because for the open set in ,

is not g-open in Therefore is not g-open map.

Theorem 3.19: If a map is -open map then it is c*g-open 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*g-open set. Then is

c*g-open set. Hence is c*g-open 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*g-open map but not

-open map because for the open set in ,

is not -open in Therefore is not -open map.

c*g-open map but not

.Then is

Theorem 3.21: If a map is gs-open map then it is c*g-open map but not conversely.

g-open map because for the open set in ,

is not g-open in Therefore is not g-open map.

Theorem 3.17: If a map is g-open map then it is c*g-open map but not conversely.

Proof: Let be a g-open map. Let be any open set in , is g-open set in Since every g-open set is c*g-open set. Then is c*g-open set. Hence is c*g-open map.

Proof: Let be a gs-open map. Let be any open set in , is gs-open set in Since every gs-open set is c*g-open set. Then is

<>c*g-open set. Hence is c*g-open 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*g-open map but not

gs-open map because for the open set in ,

is not gs-open in Therefore is not gs-open map.

c*g-open map but not

.Then is

Theorem 3.23: If is c*g-continuous and c*g-closed and A is a c*g-closed set of

then is c*g-closed in

Proof: Let where is c*-set of Since is c*g-continuous, is c*-set

containing A. Hence as is c*g-closed. Since is c*g-closed, is c*g-closed set contained in c*-set , which implies that and hence

So is c*g-closed in Y.

Corollary 3.24: If is continuous and closed map and if is c*g-closed set in then

is c*g-closed in

Proof: Since every continuous map is c*g- continuous and every closed map is c*g-closed, by the above theorem the result follows.

Theorem 3.25: If is closed and

is c*g-closed then is c*g-closed.

Proof: Let is a closed map and is c*g-closed map. Let be any closed

set in Since is closed, is

closed in and since is c*g-closed

, is c*g-closed set in Therefore is c*g-closed map.

Theorem 3.26: If is c*g-closed and is closed set in Then is

c*g-closed.

Proof: Let be closed set in A. Then V is closed

in X. Therefore is c*g-closed set in Y. By theorem 1.24 is c*g-closed. That is is c*g-closed set in Y. Therefore

is c*g-closed.

1. 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 p-open 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 cg-open. 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

2. 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 semi-homeomorphism, which also a generalization of homeomorphism. Sundaram[2] introduced g-homeomorphism and gc-homeomorphism 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.

   1. is c*g -continuous.

   2. is a c*g -open map.

   3. 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

1. herefore is c*g -continuous.

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

   1. is a c*g -open map.

   2. is a c*g -homeomorphism.

   3. 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 f-1({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

1. Kavitha.A., cg, c*g, c(s)g-closed sets in Topological spaces, In the 99th Indian

   Science Congress, Bhubaneswar(2012),.

2. Balachandran, K. Sundaram, P. and Maki, H., On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. Math., 12(1991), 5-13.

3. Mashhour, A. S., Hasanein, I. A. and El-Deeb,

   S. N., -continuous and – open mappings, Acta. Math. Hunger., 41(1983), 213-218.

4. Pushpalatha .A., Kavitha.A., A New class

cg-set weaker form of closed sets in Topological Spaces, International Journal of computer Application Vol 54-No 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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