T1T2#RG – Homeomorphisms in Bitopological Spaces

T1T2#RG – Homeomorphisms in Bitopological Spaces

S.Thilaga Leevathi

Associate Professor of Mathematics, Popes College(Autonomous), Sawyerpuram, Tamil Nadu – 627 251, India.

S. Sivanthi

Assistant Professor of Mathematics, Popes College(Autonomous), Sawyerpuram, Tamil Nadu – 627 251, India

Abstract – A bijection : (, 1, 2) (, 1, 2 ) is called 12#regular generalized homeomorphism if

and 1 are 12#rg-continuous. Also we introduce new class of maps, namely 12#rgc-homeomorphisms which form a subclass of 12#rg-homeomorphisms. This class of maps is closed under composition of maps. We prove that the set of all 12#rgchomeomorphisms forms a group under the operation composition of maps.

Mathematical Subject Classification: 54C10, 54C08, 54C05

Keywords: 12#rg-homeomorphism, 12#rgc homeomorphism

1. INTRODUCTION

   The notion homeomorphism plays a very important role in topology. By definition, a homeomorphism between two topological spaces X and Y is a bijective map : when both and 1 are continuous. It is well known that as Jnich [[9], p.13] says correctly: homeomorphisms play the same role in topology that linear isomorphism play in linear algebra, or that biholomorphic maps play in function theory, or group isomorphism in group theory, or isometries in Riemannian geometry. In the course of generalizations of the notion of homeomorphism, Maki et al. [12] introduced g -homeomorphisms and gc -homeomorphisms in topological spaces.

   In this paper, we introduce the concept of 12#rg-homeomorphism and study the relationship between homeomorphisms, 12ghomeomorphism, g12s- homeomorphism and 12rghomeomorphism.

   Also we introduce new class of maps 12#rgc-homeomorphism which form a subclass of 12#rg-

   homeomorphism. This class of maps is closed under composition of maps. We prove that the set of all

   12#rgchomeomorphisms forms a group under the operation composition of maps.

   Let us recall the following definition which we shall require later.

   Definition 1.1. A subset A of a bitopological space (X, 1, 2) is called:

   1. 12 preopen set if A 1int2cl (A) and a 12preclosed set if 2cl1int (A) A.

   2. 12 semiopen set[1] if A 2cl1int (A) and a 12semiclosed set if 1int2cl (A) A.

   3. 12 regular open set if A = 1int2cl (A) and a 2regular closed set if A = 2cl1int (A).

   4. 12 – open set if A is a finite union of regular open sets.

   5. 12 regular semi open if there is a 1regular open U such U A 2cl(U).

   Definition 1.2. A subset A of (X,1, 2) is called

   1. 12 generalized closed set (briefly, 12g-closed) if 2cl (A) U whenever AU and U is open in X.

   2. 12 regular generalized closed set (briefly, 12 rg-closed) if 2 cl (A)U whenever AU and U is 1 – regular open in X.

   3. 12 generalized preregular closed set (briefly,12gpr-closed) if 2pcl (A)U whenever A U and U is

      1-regular open in X.

   4. 12 regular weakely generalized closed set (briefly,12wg-closed) if 2cl1int (A) U whenever A U and U is 1- regular open in X.

   5. 12 rw-closed if 2cl(A)U whenever AU and U is 1regular semi open.

   6. 12#-closed if 2cl(A)U whenever AU and U is 1rw-open.

      The complements of the above mentioned closed sets are their respective open sets.

      Definition: 1.3. A function : (, 1, 2) (, 1, 2 ) is called #rg-continuous if f -1 (V ) is 12#rg-closed in (X, 1, 2) for every closed subset V of (Y, 1, 2).

      Definition: 1.4. A function : (, 1, 2) (, 1, 2 ) is called 12 #rg-irresolute if f -1 (V ) is 12 #rg-closed in X for every 12#rg-closed subset V of Y.

      Definition 1.5. A function : (, 1, 2) (, 1, 2 ) is said to be 12#rg-closed (resp. 12#rg-open) if for every #rg-closed (resp. 12#rg-open) set U of X the set f(U) is 12#rg-closed (resp. 12#rg-open) in Y.

      Definition 1.6. A map : (, 1, 2) (, 1, 2 ) is said to be

      1. 12 g homeomorphism[12] if both and 1are 12 g-continuous,

      2. 12 gs- homeomorphism [6] if both and 1are 12 gs-continuous,

      3. 12 rwg- homeomorphism[14] if both and 1are 12 rwg-continuous,

      4. 12 gc- homeomorphism[12] if both and 1are 12 gc-irresolute.

2. #RG-homeomorphism in Bitopological Spaces

Definition 2.1. A bijection : (, 1, 2) (, 1, 2 ) is called 12 #regular generalized homeomorphism (briefly, 12 #rg-homeomorphism) if and 1 are 12 #rg-continuous. We denote the family of all 12 #rg homeomorphisms of a topological space (, 1, 2) onto itself by 12#rg-h(, 1, 2).

Example 2.2. Consider = = {, , , } with topologies 1={ X, , {a}, {b}, {a,b},{a,b,c}} and 2 ={X, ,

{a}, {a, b}}, 1= {Y, , {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}} and 2 = {Y, , {a}, {b}, {c}, {a,b}, {a,c},

{b,c}, {a,b,c}} Let : (, 1, 2) (, 1, 2 ) be the identity map. Then f is bijective, 12#rgcontinuous and 1 is 12#rg-continuous. Hence f is 12#rg-homeomorphism.

Theorem 2.3. Every homeomorphism is 12 #rg homeomorphism, but not conversely.

Proof. Let : (, 1, 2) (, 1, 2 ) be a homeomorphism. Then and 1 are continuous and f is bijection. Since every continuous function is 12 #rg-continuous, and 1is 12#rg-continuous. Hence f is

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

Example 2.4.Consider = = {, , , } with topologies 1= {X, , {a}, {b}, {c}, {a,b}, {a,c}, {b,c},

{a,b,c}} and 2 ={X, , {a}, {a, b}}, 1 = {X, , {a}, {b}, {a,b},{a,b,c}} and 2 = {Y, , {a}, {b}, {c}, {a,b},

{a,c}, {b,c}, {a,b,c}}. Let : (, 1, 2) (, 1, 2 ) be the identity map. Then f is 12#rg-homeomorphism it is not homeomorphism, since the inverse image of closed set of {a,d} in X is {a,d} which is not closed in Y.

Theorem 2.5. Every 12 #rg-homeomorphism is g-homeomorphism.

Proof. Let : (, 1, 2) (, 1, 2 ) be a 12 #rg homeomorphism. Then f and f-1 are 12 #rgcontinuous and f is bijection. Since every 12 #rgcontinuous function is g-continuous, and 1 are g-continuous. Hence f is 12 ghomeomorphism.

.

Corollary 2.6. Every 12 #rg-homeomorphism is 12 gs-homeomorphism.

Proof. By the fact that every 12 g homeomorphism is 12 gs-homeomorphism and by theorem 2.5.

Corollary 2.7. Every 12 #rg-homeomorphism is 12 gsp-homeomorphism.

Proof. By the fact that every gshomeomorphism is 12 gsp-homeomorphism and by corollary 2.6.

Theorem 2.8. Every 12 #rg-homeomorphism is 12 rg-homeomorphism.

Proof. Let : (, 1, 2) (, 1, 2 ) be a 12 #rg homeomorphism. Then f and f-1 are 12 #rgcontinuous and f is bijection. Since every #rgcontinuous function is 12 rg-continuous, and 1 are 12 rg-continuous. Hence f is12 rg homeomorphism.

Corollary 2.9. Every 12 #rg-homeomorphism is 12 rwg-homeomorphism and 12 gp rhomeomorphism.

Proof. By the fact that every 12 rg homeomorphism is 12 rwg-homeomorphism and 12 gpr-homeomorphism, and by theorem 2.8.

Theorem 2.10. Let : (, 1, 2) (, 1, 2 ) be a bijective 12 #rg- continuous map. Then the following are equivalent.

1. f is a 12 #rg- open map

2. f is 12 #rg-homeomorphism,

3. f is a 1 2 #rg -closed map.

Proof. Suppose (i) holds. Let V be open in (, 1, 2). Then by (i), f (V) is 12 #rg-open in (, 1, 2 ) . But

() = (1)1 (V) and so (1)1 (V) is 12 #rgopen in (, 1, 2 ) . This shows that 1 is

12 #rgcontinuous and it proves (ii).

Suppose (ii) holds. Let F be a closed set in (, 1, 2). By (ii), 1 is 12 #rg-continuous and so

( 1)1 (V) (F) = () is 12 #rg-closed in (, 1, 2 ) . This proves (iii).

Suppose (iii) holds. Let V be open in (, 1, 2). Then c is closed in (, 1, 2). By (iii), f(c ) is

12 #rg-closed in (, 1, 2 ). But (c)= (())c . This implies that (())c is 12 #rg-closed in (, 1, 2 )

and so () is 12 #rg-open in (, 1, 2 ) . This proves (i).

Definition 2.11. A bijection : (, 1, 2) (, 1, 2 ) is said to be 12 #rgc-homeomorphism if both and

1are 12 #rg-irresolute. We say that spaces (, 1, 2) and (, 1, 2 ) are 12 #rgchomeomorphic if there exists a 12 #rgchomeomorphism form (, 1, 2) onto (, 1, 2 )We denote the family of all 12 #rgchomeomorphisms of a topological space (, 1, 2) onto itself by 12 #rgc-h(, 1, 2).

Theorem 2.12. Every 12 #rgc-homeomorphism is a 12 #rg homeomorphism but not conversely.

Proof. Let : (, 1, 2) (, 1, 2 ) be an 12 #rgchomeomorphism. Then and 1 are 12 #rg irresolute and f is bijection. By Theorem 4.2 in [22], and 1 are 12 #rg- continuous. Hence f is 12 #rg-homeomorphism. The converse of the above theorem is not true in general as seen from the following example.

Example 2.13. Consider X = Y = {a, b, c, d } with 1= {X, , {a}, {b}, {c}, {a,b}, {a,b,c}} and 2 ={X, , {a},

{a, b}, {a,b,c}}, 1 = {X, , {c},{a,b}, {a,b,c}} and 2 = {Y, , {c},{a,b}, {a,b,c}}. Let : (, 1, 2)

(, 1, 2 ) be defined by () = , () = , () = and () = . Then f is 12 #rg-homeomorphism but it is not 12 #rgc-homeomorphism, since f is not 12 #rg-irresolute.

Theorem 2.14. Every 12 #rgc-homeomorphism is 12 g-homeomorphism but not conversely.

Proof. Proof follows from Theorems 2.5 and Theorem 2.12.

Remark 2.15 12 #rgc-homeomorphism and 12 gc – homeomorphism are independent as seen from the following example.

Example 2.16 Let X = Y = {a,b,c,d} with 1= {X, , {a},{b},{a,b},{a,b,c}} and 2 ={X, ,

{a},{b},{a,b},{a,b,c}}, 1 = {X, , {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}} and 2 = {Y, ,{a}, {b}, {c},

{a,b}, {a,c}, {b,c}, {a,b,c}}. Let : (, 1, 2) (, 1, 2 ) be the identity map. Then f is 12 #rgc homeomorphism but it is not 12 gc-homeomorphism, since f is not 12 gc-irresolute.

Theorem 2.17. Let : (, 1, 2) (, 1, 2 ) and g : ( , 1, 2 ) (, 1, 2) are 12 #rgc-homeomorphisms, then their composition : (, 1, 2) ( , 1, 2) is also 12 #rgc-homeomorphism.

Proof. Let U be a 12 #rg-closed set in ( , 1, 2 ). Since g is 12 #rg-homeomorphism, g-1 (U) is

12 #rgclosed in ( , 1, 2 ).Since f is 12 #rghomeomorphism, 1 ( 1 (U)) = ()1 (U) is 12 #rg closed in (, 1, 2 ). Therefore is 12 #rgirresolute. Also for a 12 #rg-closed set G in (, 1, 2), We have ()() = (() ) = (), where = ( ) . By hypothesis, () is 12 #rg-closed in (, 1, 2) and so again by hypothesis, (()) is a 12 #rg-closed set in (, 1, 2). That is ()() is a

12 #rg-closed set in (, 1, 2) and therefore ()1 is 12 #rg-irresolute. Also is a bijection. Hence

is 12 #rg-homeomorphism.

Theorem 2.18. The set 12 #rgc-h(, 1, 2 ) is a group under the composition of maps.

Proof. Define a binary operation * : 12 #rgch(, 1, 2) × 12 #rgc-h(, 1, 2) 12 #rgc-h(, 1, 2) by

= for all , 12 #rgc-h(, 1, 2) and is the usual operation of composition of maps. Then by theorem 2.17, 12 #rgc-h(, 1, 2 ). We know that the composition of maps is associative and the identity map I: (, 1, 2) (, 1, 2) belonging to 12 #rgc-h(, 1, 2) serves as the identity element. If

12 #rgc-h(, 1, 2), then 1 12#rgc-h(, 1, 2) such that 1 = 1 = and so inverse exists for

each element of 12 #rgc-h(, 1, 2). Therefore (12 #rgch(, 1, 2), ) is a group under the operation of composition of maps.

Theorem 2.19. Let : (, 1, 2) (, 1, 2 ) be a 12 #rgchomeomorphism. Then f induces an isomorphism from the group 12 #rgc-h(, 1, 2)onto the group 12 #rgc-h(, 1, 2 ) .

Proof. Using the map f, we define a map : 12 #rgc-h ( , 1, 2 ) 12 #rgc-h( , 1, 2 ) by

1 2

1 2 1 2 1 2

()=1 for every 12 #rgc-h(, 1, 2). Then is a bijection. Further, for all , 12 #rgc-h(, 1, 2); ( ) = ( ) 1 = ( 1) ( 1) = (1) (2) . Hence is a homomorphism and so it is an isomorphism induced by f.

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