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REGULAR WEAKLY HOMEOMORPHISM IN IDEAL TOPOLOGICAL SPACES

Mohanarao Navuluri1 and A. Vadivel2* _

1 Department of Mathematics, Govt. College of Engg. Bodinayakkanur-2; Mathematics Section (FEAT), Annamalai University, Annamalainagar-608 002, Tamilnadu.

mohanaraonavuluri@gmail.com 2 Post Graduate and Research Department of Mathematics, Government Arts

College(Autonomous), Karur - 639 005, Tamilnadu; Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, TamilNadu.

Abstract

In this paper we introduce and study two new homeomorphisms namely rwI

-homeomorphism and *

rwI -homeomorphism and study some of their properties in ideal

topological spaces.

Key words and phrases: rwI -closed map,

rwI -irresolute, *

rwI -open, rwI -homeomorphism,

*

rwI -homeomorphism.

AMS (2000) subject classification: 54C10, 54A05

1. Introduction The notion of ideal topological spaces was first studied by Kuratowski [4] and Vaidyanathaswamy [8]. In 1990, Jankovic and Hamlett [3] investigated further properties of ideal topological spaces. R. S. Wali [9] introduce rw-continuous, rw-homeomorphism using

rw-closed sets in topologcal spaces. A. Vadivel et. al. [6, 7] introduce rwI -continuous using

rwI

-open sets in Ideal topological spaces. In this paper, we introduce the concepts of

rwI -homeomorphism and study the relationship

with I -homeomorphisms. Also we introduce new class of maps *

rwI -homeomorphism which

forms a subclass of rwI -homeomorphism. This class of maps is closed under compositions of

maps. We prove that the set of all rwI -homeomorphisms form a group under the composition

of maps.

2. Preliminaries Let ( , )X be a topological space with no separation properties assumed. For a subset A of

a topological space ( , )X , ( )cl A and ( )int A denote the closure and interior of A in

( , )X , respectively. An ideal I on a topological space ( , )X is a non-empty collection of

subsets of X which satisfies the following properties: (1) A I and B A implies B I ,

(2) A I and B I implies A B I . An ideal topological space is a topological space ( , )X with an ideal I on X and is denoted by ( , , )X I . For a subset A X ,

*( , ) ={ |A I x X A U I for every ( , )}U X x is called the local function of A with

respect to I and [3, 4]. We simply write *A instead of *( )A I in case there is no chance

for confusion. For every ideal topological space ( , , )X I , there exists a topology *( )I , finer

than , generated by the base ( , ) ={ |I U J U and }J I . It is known in [3] that

( , )I is not always a topology. When there is no ambiguity, *( )I is denoted by * . For a

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subset A X , *( )cl A and *( )int A will, respectively, denote the closure and interior of A

in *( , )X .

Definition 2.1 (i) A subset A of a space ( , )X is said to be regular open [5] if

= ( ( ))A int cl A and A is said to be regular closed [5] if = ( ( )).A cl int A

(ii) A subset A of a space ( , )X is said to be regular semiopen [2] if there is a regular open

set U such that ( )U A cl U . The complement of a regular semiopen set is said to be

regular semiclosed. (iii) A subset A of a space ( , )X is said to be rw -closed [9] if ( )cl A U whenever

A U and U is regular semiopen. A is said to be rw-open if X A is rw-closed.

Definition 2.2 (1) A subset A of an ideal space ( , , )X I is said to be I -open [1] if *( )A int A . The complement of an I -open set is said to be I -closed.

(2) A subset A of an ideal space ( , , )X I is said to be a regular weakly closed set with

respect to the ideal I (rwI -closed) [6] if *A U whenever A U and U is regular

semiopen. A is called a regular weakly open set (rwI -open) if X A is an

rwI -closed set.

3. rwI -homeomorphism in ideal topological space

We introduce the following definitions

Definition 3.1 A map : ( , , ) ( , , )f X I Y J is said to be

(i) rwI -closed if the image ( )f A is

rwI -closed set in ( , )Y for each closed set A in

( , , )X I .

(ii) rwI -continuous [7] if 1( )f A is

rwI -closed set in ( , , )X I for each closed set A in

( , )Y

Definition 3.2 A bijective function : ( , , ) ( , , )f X I Y J is called

(i) rwI -irresolute if the inverse image 1( )f A is

rwI -closed set in ( , , )X I for each rwI

-closed set A in ( , , )Y I .

(ii) *

rwI -homeomorphism if both f and 1f are rwI -irresolute.

(iii) rwI -homeomorphism if both f and 1f are

rwI -continuous.

We say the spaces ( , , )X I and ( , , )Y J are rwI -homeomorphic if there exists a

rwI

-homeomorphism from ( , , )X I onto ( , , )Y J .

We denote the family of all rwI -homeomorphisms (resp. *

rwI - homeomorphisms) of an ideal

topological space ( , , )X I onto itself by rwI -h ( , , )X I (resp. *

rwI -h ( , , )X I ).

Theorem 3.1 Every I -homeomorphism is a rwI -homeomorphism but not conversely.

Proof. Let : ( , , ) ( , , )f X I Y J be a I -homeomorphism. Then f and 1f are I

-continuous and f is bijection. As every I -continuous function is rwI -continuous, we have

f and 1f are rwI -continuous. Therefore, f is

rwI -homeomorphism.

The converse of the above theorem need not be true, as seen from the following example.

Example 3.1 Let = ={ , , }X Y a b c , ={ , ,{ },{ },{ , }}X a c a c , ={ , ,{ },{ , }}X c a b and

={ ,{ }}I c . Then the identity map on X is a rwI -homeomorphism, but it is not I

-homeomorphism. Since the inverse image of the open set { }c in ( , )Y is { }c which is not

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I -open set in ( , , )X I .

Theorem 3.2 For any bijection : ( , , ) ( , , )f X I Y J , the following statements are

equivalent:

(i) 1 : ( , , ) ( , , )f Y J X I is rwI -continuous.

(ii) f is a rwI -open map.

(iii) f is a rwI -closed map.

Proof.(i) (ii): Let U be a I -open set of ( , , )X I . By assumption 1 1( ) ( ) = ( )f U f U

is a rwI -open set in ( , , )Y J and so f is a

rwI -open map.

(ii) (iii): Let F be a I -closed set of ( , , )X I . Then cF is I -open in ( , , )X I . Since

f is rwI -open, ( )cf F is

rwI -open in Y . But ( ) = ( ( ))c cf F f F , we have ( )f F is rwI

-closed in Y and so f is a rwI -closed map.

(iii) (i): Suppose F is a I -closed set in ( , , )X I . By assumption 1 1( ) = ( ) ( )f F f F is

rwI -closed set in ( , , )Y J and so 1f is rwI -continuous.

Theorem 3.3 Let : ( , , ) ( , , )f X I Y J be a bijective and rwI -continuous, then the

following statements are equivalent: (i) f is a

rwI -open map.

(ii) f is a rwI -homeomorphism.

(ii) f is a rwI -closed map.

Proof. Proof follows from the Definitions 3.1,3.2and Theorem 3.2.

Remark 3.1 The composition of two rwI -homeomorphism need not be a

rwI -homeomorphism

as seen from the following example.

Example 3.2 Let = = ={ , , }X Y Z a b c , ={ ,{ },{ },{ , }, }a c a c X , ={ ,{ }}I c ,

={ ,{ },{ , }, }c a b Y and ={ ,{ },{ },{ , }, }a b a b Z respectively. Let : ( , , ) ( , , )f X I Y I

and : ( , , ) ( , )g Y I Z be identity map respectively. Then both f and g are rwI

-homeomorphisms but their composition : ( , , ) ( , )g f X I Z , is not a rwI

-homeomorphism, because for the open sets { }b of ( , , )X I ,

({ }) = ( ({ })) = ({ }) ={ }g f b g f b g b b which is not a rwI -open in ( , )Z . Therefore, g f is

not a rwI -open and not a

rwI -homeomorphism.

Definition 3.3 A map : ( , , ) ( , , )f X I Y J is called *

rwI -open if ( )f U is rwI -open in

( , , )Y J for every rwI -open set U of ( , , )X I .

Theorem 3.4 For any bijection : ( , , ) ( , , )f X I Y J , the following statements are

equivalent:

(i) 1 : ( , , ) ( , , )f Y J X I is rwI -irresolute.

(ii) f is a *

rwI -open map.

(iii) f is a *

rwI -closed map.

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Proof. (i) (ii) Let U be a rwI -open in ( , , )X I . By (i) 1 1( ) ( ) = ( )f U f U is

rwI -open in

( , , )Y J . Hence (ii) holds.

(ii) (iii) Let V be rwI -closed in ( , , )X I . Then X V is

rwI -open and by (ii)

( ) = ( )f X V Y f V is rwI -open in ( , , )Y J . That is ( )f V is

rwI -closed in Y and so f

is *

rwI -closed map.

(iii) (i) Let W be rwI -closed in ( , , )X I . By (iii), ( )f W is

rwI -closed in ( , , )Y J . But 1 1( ) = ( ) ( )f W f W . Thus (i) holds.

Theorem 3.5 For any spaces *

rwI -h ( , , ) rwX I I -h ( , , )X I .

Proof. The result follows from the fact that every

rwI -irresolute function is rwI -continuous and

every *

rwI -open map is rwI -open.

Theorem 3.6 Let : ( , , ) ( , , )f X I Y J and : ( , , ) ( , , )g Y J Z k are *

rwI

-homeomorphism, then their composition : ( , , ) ( , , )gof X I Z k is also *

rwI

-homeomorphism.

Proof. Let U be a rwI -open set in ( , , )Z k . Since g is

rwI -irresolute, 1( )g U is rwI

-open in ( , , )Y J . Since f is rwI -irresolute, 1 1 1( ( )) = ( ) ( )f g U gof U is

rwI -open set in

( , , )X I . Therefore, 1( )gof is rwI -irresolute. Also, for a

rwI -open set G in ( , , )X I , we

have ( )( ) = ( ( )) = ( )gof G g f G g W , where = ( )W f G . By hypothesis ( )f G is rwI -open in

( , , )Y J and so again by hypothesis ( ( ))g f G is a rwI -open set in ( , , )Z k . That is

( )( )gof G is a rwI -open set in ( , , )Z k and therefore, gof is

rwI -irresolute. Also, gof is

a bijection. Hence gof is *

rwI -homeomorphism.

Theorem 3.7 The set *

rwI -h ( , , )X I is a group under the composition of maps.

Proof. Define a binary operation **: rwI -h *( , , ) rwX I I -h *( , , ) rwX I I -h ( , , )X I by

* =f g gof for all *, rwf g I -h ( , , )X I and is the usual operation of composition of

maps. Then by theorem 3.6. *

rwgof I -h ( , , )X I . We know that the composition of maps is

associative and the identity map : ( , , ) ( , , )I X I X I belonging to *

rwI -h ( , , )X I serves

as the identity element. If *

rwf I -h ( , , )X I , then 1 *

rwf I -h ( , , )X I such that 1 1= =fof f of I and so inverse exists for each element of *

rwI -h ( , , )X I . Thus *

rwI -h

( , , )X I forms a group under the operation of composition of maps.

Theorem 3.8 Let : ( , , ) ( , , )f X I Y J be a *

rwI -homeomorphism. Then f induces an

isomorphism from the group *

rwI -h ( , , )X I onto the group *

rwI -h ( , , )Y J .

Proof. Using the map f , we define a map *:f rwI -h *( , , ) rwX I I -h ( , , )Y J by 1( ) =f h fohof for every *

rwh I -h ( , , )X I . Then f is a bijection. Further, for all *

1 2, rwh h I -h ( , , )X I , 1 1 1

1 2 1 2 1 2 1 2( ) = ( ) = ( ) ( ) = ( ) ( )f f fh oh fo h oh of foh of o foh of h o h .

Therefore f is a homeomorphism and so it is an isomorphism induced by f .

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