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© 2019 JETIR March 2019, Volume 6, Issue 3 www.jetir.org (ISSN-2349-5162)
JETIRAW06017 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 130
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
© 2019 JETIR March 2019, Volume 6, Issue 3 www.jetir.org (ISSN-2349-5162)
JETIRAW06017 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 131
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
© 2019 JETIR March 2019, Volume 6, Issue 3 www.jetir.org (ISSN-2349-5162)
JETIRAW06017 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 132
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.
© 2019 JETIR March 2019, Volume 6, Issue 3 www.jetir.org (ISSN-2349-5162)
JETIRAW06017 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 133
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 .
© 2019 JETIR March 2019, Volume 6, Issue 3 www.jetir.org (ISSN-2349-5162)
JETIRAW06017 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 134
References
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[4] K. Kuratowski, Topology I , Warszawa, 1933.
[5] M. H. Stone, Application of the theory of boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 374-481.
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