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 Authors : S. Syed Ali Fathima, M. Mariasingam
 Paper ID : IJERTV1IS5040
 Volume & Issue : Volume 01, Issue 05 (July 2012)
 Published (First Online): 02082012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
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#RG – Homeomorphisms in Topological Spaces
S. Syed Ali Fathima Department of Mathematics Sadakathullah Appa College Tirunelveli 627 011, India.
M. Mariasingam
Post Graduate and Research Department of Mathematics,
V.O. Chidambaram College Thoothukudi628 008 (T.N.), India.
Abstract
A bijection f: (X,) (Y,) is called #regular generalized homeomorphism if f and f 1 are
#rgcontinuous. Also we introduce new class of maps, namely #rgchomeomorphisms which form a subclass of #rghomeomorphisms. This class of maps is closed under composition of maps. We prove that the set of all #rgc homeomorphisms forms a group under the operation composition of maps.
Mathematical Subject Classification: 54C10, 54C08, 54C05
Keywords: #rghomeomorphism, #rgc homeomorphism

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 f: X Y when both f and f1 are continuous. It is well known that as J nich [[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 #rghomeomorphism and study the relationship between homeomorphisms, g homeomorphism, gs homeomorphism and rg homeomorphism. Also we introduce new class of maps #rgchomeomorphism which form a subclass of #rg homeomorphism. This class of maps is closed under composition of maps. We prove that the set of all #rgc
homeomorphisms 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 space X is called

a preopen set[13] if A intcl(A) and a preclosed set if clint (A) A.

a semiopen set[10] if A clint(A) and a semi closed set if intcl (A) A.

a regular open set[16]if A = intcl(A) and a regular closed set if A = clint(A).

a – open set[20] if A is a finite union of regular open sets.

regular semi open[4]if there is a regular open U such U Acl(U).
Definition 1.2. A subset A of (X,) is called

generalized closed set (briefly, g closed)[11] if cl (A) U whenever AU and U is open in X.

regular generalized closed set (briefly, rg closed)[15] if cl (A)U whenever AU and U is regular open in X.

generalized preregular closed set (briefly, gprclosed)[8] if pcl (A)U whenever A U and U is regular open in X.

regular weakly generalized closed set (briefly, rwgclosed)[14] if clint(A)U whenever A U and U is regular open in X.

rwclosed [3] if cl(A)U whenever AU and U is regular semi open.
6)#rgclosed[17] if cl(A)U whenever AU and U is rwopen.
The complements of the above mentioned closed sets are their respective open sets.
Definition: 1.3. A map f:(X,)(Y,) is called gcontinuous[2](resp.rgcontinuous[15] resp. gscontinuous[6], resp. gspcontinuous [7], resp. gprcontinuous[8], resp. regular continuous[15], resp. rwgcontinuous[14],
resp. #rgcontinuous[19]) if f1(V) is gclosed( resp. rgclosed, resp. gsclosed, resp. gsp closed, resp. gprclosed, resp. regular closed, resp. rwgclosed, resp. #rgclosed) in X for every closed subset V of Y.
Definition: 1.4. A map f:(X,)(Y,) is said to be
(i) Irresolute[5] if f1 (V) is semi open in (X,) for each semi open set V of (Y,).
(iii) #rgirresolute [19 ] if f1 (V) is #rgopen in (X,) for each #rgopen set V of (Y,).
Definition 1.5. A map f:(X,)(Y,) is said to be

#rgclosed[19] if f(F) is #rgclosed in (Y,) for every #rgclosed set F of (X,)

#rgopen[19] if f(F) is #rgopen in (Y,) for every #rgopen set F of (X,).
Definition 1.6. A map f:(X,)(Y,) is said to be

g homeomorphism[12] if both f and f1 are gcontinuous,

gs homeomorphism [6] if both f and f1 are gscontinuous,

rwg homeomorphism[14] if both f and f1 are rwgcontinuous,

gc homeomorphism[12] if both f and f1 are gcirresolute.


#RGhomeomorphism in Topological Spaces
Definition 2.1. A bijection f:(X,) (Y,) is called #regular generalized homeomorphism (briefly, #rghomeomorphism) if f and f1 are
#rgcontinuous.
We denote the family of all #rg homeomorphisms of a topological space (X,) onto itself by #rgh(X,).
Example 2.1. Consider X=Y={a,b,c,d} with topologies ={X,, {a}, {b}, {a,b},{a,b,c}}
and = {Y, , {a}, {b}, {c}, {a,b}, {a,c},
{b,c}, {a,b,c}}. Let f:(X,) (Y,) be the identity map. Then f is bijective, #rg continuous and f1 is #rgcontinuous. Hence f is #rghomeomorphism.
Theorem 2.1. Every homeomorphism is #rg homeomorphism, but not conversely.
Proof. Let f:(X,) (Y,) be a homeomorphism. Then f and f1 are continuous and f is bijection. Since every continuous function is #rgcontinuous, f and f
1 is #rgcontinuous. Hence f is #rg homeomorphism.
The converse of the above theorem need not be true, as seen from the following example.
Example 2.2.Consider X=Y={a,b,c,d} with topologies = {X, , {a}, {b}, {c}, {a,b},
{a,c}, {b,c}, {a,b,c}} and = {X,, {a}, {b},
{a,b},{a,b,c}}. Let f : (X,) (Y,) be the identity map. Then f is #rghomeomorphism 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 2.2. Every #rghomeomorphism is ghomeomorphism, but not conversely.
Proof. Let f:(X,) (Y,) be a #rg homeomorphism. Then f and f1 are #rg continuous and f is bijection. Since every #rg continuous function is gcontinuous, f and f1 are gcontinuous. Hence f is g homeomorphism.
The converse of the above theorem need not be true, as seen from the following example.
Example 2.3. Consider X=Y={a,b,c,d} with topologies = {X, , {a}, {b}, {a ,b}} and
= {X,, {a}, {b}, {a ,b},{a ,b ,c}}. Define a map f:(X,) (Y,) be the identity map. Then f is bijection, gcontinuous and f1 is g continuous. Hence f is ghomeomorphism. But f is not #rghomeomorphism, since the inverse image of closed set of {d} in Y is {d} which is not #rgclosed in X.
.
Corollary 2.1. Every #rghomeomorphism is gshomeomorphism, but not conversely.
Proof. By the fact that every g homeomorphism is gshomeomorphism and by theorem 2.2.
Corollary 2.2. Every #rghomeomorphism is gsphomeomorphism, but not conversely.
Proof. By the fact that every gs homeomorphism is gsphomeomorphism and by corollary 2.1.
Theorem 2.3. Every #rghomeomorphism is rghomeomorphism, but not conversely.
Proof. Let f:(X,) (Y,) be a #rg homeomorphism. Then f and f1 are #rg continuous and f is bijection. Since every #rg continuous function is rgcontinuous, f and f1 are rgcontinuous. Hence f is rg homeomorphism.
The converse of the above theorem need not be true, as seen from the following example.
Example 2.4. Consider X=Y={a,b,c,d} with topologies = {X, , {a}, {b}, {a,b}} and =
{X,, {a}, {b}, {a,b},{a,b,c}}. Define a map f:(X,) (Y,) be the identity map. Thenf is bijection, rgcontinuous and f1 is rg continuous. Hence f is rghomeomorphism. But f is not #rghomeomorphism, since the inverse image of closed set of {d} in Y is {d} which is not # rgclosed in X.
Corollary 2.3. Every #rghomeomorphism is rwghomeomorphism and gpr homeomorphism, but not conversely.
Proof. By the fact that every rg homeomorphism is rwghomeomorphism and gprhomeomorphism, and by theorem 2.3.
Theorem 2.4. Let f : ( X, ) (Y, ) be a bijective rg continuous map. Then the following are equivalent.

f is a rg open map

f is rghomeomorphism,

f is a rg closed map.

Proof. Suppose (i) holds. Let V be open in (X, ). Then by (i), f (V) is #rgopen in (Y,).
But f(V) = ( f1)1(V) and so ( f1)1(V) is #rg open in (Y,). This shows that f1 is #rg continuous and it proves (ii).
Suppose (ii) holds. Let F be a closed set in (X,) . By (ii), f1 is #rgcontinuous and so ( f1)1(V) (F) = f(F) is #rgclosed in (Y,). This proves (iii).
Suppose (iii) holds. Let V be open in (X,). Then Vc is closed in (X,). By (iii),
f(Vc) is #rgclosed in (Y,). But f(Vc)= (f(V))c
. This implies that (f(V))c is #rgclosed in (Y,) and so f (V) is is #rgopen in (Y,). This proves (i).
Remark 2.1. The composition of two #rg homeomorphism need not be a #rg
homeomorphism in general as seen from the following example.
Example 2.5. Consider X=Y=Z={a,b,c,d},=
{X, , {a}, {b}, {c}, {a,b}, {a,c}, {b,c},
{a,b,c}} and = {X,, {a}, {b},
{a,b},{a,b,c}} and ={Z,,{c},{a,b},{a,b,c}}. Define a map f:(X,)(Y,) be the identity map. Then clearly f is #rghomeomorphism. g: (Y,) (Z, ) defined by g(a)=b, g(b)=c, g(c)=a and g(d)=d. Then g is also #rg homeomorphism, but their composition gof
:(X,)(Z,) is not #rghomeomorphism, because for the closed set {a,d } of ( X, ), gof({a,d})=g(f({a,d}))=g({a,d})={b,d}, which is not #rgclosed in (Z,). Therefore gf is not
#rgclosed and so gof is not #rg homeomorphism.
Definition 2.2. A bijection f: (X,) (Y,) is said to be #rgchomeomorphism if both f and f1 are #rgirresolute.
We say that spaces (X,) and (Y,) are #rgc homeomorphic if there exists a #rgc homeomorphism form (X,) onto (Y,).We denote the family of all #rgc homeomorphisms of a topological space (X,) onto itself by #rgch(X,).
Theorem 2.5. Every #rgchomeomorphism is a #rg homeomorphism but not conversely.
Proof. Let f : ( X, ) ( Y, ) be an #rgc homeomorphism. Then f and f1 are #rg irresolute and f is bijection. By Theorem 4.2 in [19], f and f1 are #rg continuous. Hence f is #rghomeomorphism.
The converse of the above theorem is not true in general as seen from the following example.
Example 2.6. Consider X = Y = {a, b, c,d } with = { X,,{a},{b},{a,b} ,{a,b,c}} and
= {Y, , {c},{a,b}, {a,b,c}}. Let f : (X, ) ( Y, ) be defined by f(a)=b, f(b)=c, f(c)=a and f(d)=d. Then f is #rghomeomorphism but it is not #rgchomeomorphism, since f is not
#rgirresolute.
Theorem 2.6. Every #rgchomeomorphism is ghomeomorphism but not conversely.
Proof. Proof follows from Theorems 2.5 and Theorem 2.2.
Remark 2.2 #rgchomeomorphism and gc – homeomorphism are independent as seen from the following example.
Example 2.7 Let X = Y = {a,b,c,d} with =
{X, , {a},{b},{a,b},{a,b,c}} and =
{Y,,{a}, {b}, {c}, {a,b}, {a,c}, {b,c},
{a,b,c}}. Let f : (X, ) ( Y, ) be the identity map. Then f is #rgc – homeomorphism but it is not gchomeomorphism, since f is not gcirresolute.
Example 2.8 Consider X = Y={a,b,c,d} with topologies ={X,,{a},{b},{a,b},{a,b,c} } and
= {Y,,{c},{a,b},{a,b,c}}. Let f : ( X, ) (Y,) be the identity map. Then f is gc homeomorphism but it is not #rgc homeomorphism, since f is not #rg irresolute.
Remarks 2.3. From the above discussions and known results we have the following implications. In the following figure, by A B we mean A implies B but not conversely and A B means A and B are independent of each other.
rgchomeomorphismgchomeomorphism homeomorphism ghomeomorphism
#rghomeomorphism gshomeomorphism rghomeomorphism gsphomeomorphism
rwghomeomorphism gprhomeomorphism
Figure 1.
Theorem 2.7. Let f : (X, ) (Y, ) and g : (Y, ) ( Z, ) are #rgc homeomorphisms, then their composition gof: ( X, ) ( Z, ) is also #rgchomeomorphism.
Proof. Let U be a #rgclosed set in (Z, ). Since g is #rghomeomorphism, g1 (U) is #rg closed in (Y,).Since f is #rg homeomorphism, f 1 (g1(U)) = (gof) 1 (U) is
#rg closed in (X, ). Therefore gof is #rg irresolute. Also for a #rgclosed set G in (X,), We have (gof)(G) = g(f(G) ) = g(W), where W= f(G) . By hypothesis, f(G) is #rgclosed in (Y,) and so again by hypothesis, g(f(G)) is a
#rgclosed set in (Z,). That is (gof)(G) is a
#rgclosed set in (Z, ) and therefore (gof )1 is #rgirresolute. Also gof is a bijection. Hence gof is #rghomeomorphism.
Theorem 2.8. The set #rgch(X, ) is a group under the composition of maps.
Proof. Define a binary operation: #rgc h(X,) #rgch(X,) #rgch(X,) by f*g = gf for all f,g#rgch(X,) and is the usual operation of composition of maps. Then by theorem 2.7, gf #rgch(X,). We know that the composition of maps is associative and the identity map I: (X,) (X,) belonging to
#rgch(X,) serves as the identity element. If f
#rgch(X,), then f1 #rgch(X,) such that ff1 = f1f =I and so inverse exists for each element of #rgch(X,). Therefore (#rgc h(X,),) is a group under the operation of composition of maps.
Theorem 2.9. Let f: (X, ) (Y,) be a #rgc homeomorphism. Then f induces an isomorphism from the group #rgch(X,) onto the group #rgch(Y,).
Proof. Using the map f, we define a map f:
#rgch (X,) #rgch(Y,) by f(h)=fhf1 for every h #rgch(X,). Then f is a bijection. Further, for all p,p #rgch(X,);
f(pp) = f(pp)f1
= (fpf1)(fpf1)
= f(p) f(p) .
Hence f is a homomorphism and so it is an isomorphism induced by f.
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