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On rgw-Homeomorphism in Topological Space

Sanjay Mishra

Department of Mathematics, Lovely Professional University, Phagwara, (Punjab)- 144411, India

Email:drsanjaymishra@rediifmail.com

Abstract
In this paper, we first introduce new class of homeomorphism called regular generalized weakly homeomorphism (briefly rgw-homeomorphism) in topological space and after that we compare this new class of homeomorphism with recently introduced generalized form of homeomorphism. Moreover, we study algebraic properties of the family of rgw - homeomorphism in topological space.
Keywords: rgw-homeomorphism; topological space.
1.Introduction
The homeomorphism in topological space play a significant role in the study of topology. The role of the homeomorphism in topology is same as like group isomorphisms in group theory, linear isomorphisms in linear algebra, biholomorphic maps in function theory, and isometries in Riemannian geometry. First we provide basic requirement for better understanding of rgw-Homeomorphism in topological space. In this paper, we de- fine new class of homeomorphism called rgw-Homeomorphism and study the properties of this map and also compare this homeomorphism with well-known homeomorphism in order to have a better understanding of rgw-Homeomorphism. In last, we investigate algebraic properties of family of rgw-Homeomorphisms like equivalence relation, group under binary operation etc.
2. PRELIMINARIES
First, we recall some definition and results which will help to understand the new class of homomorphism. Closed set as like open set is one of important tool to define the topology on any set and generalization of closed set always give better conditions for finding weak or strong topologies. In this direction, first step was taken by L. Nevine [6] in 1970, who defined generalized closed in topological space and studied their properties. In past years, many researchers introduced various form of generalized closed which is defined under different conditions. In this section, we are presenting those generalized closed set which are directly related to our core discussion.

A non-empty  subset A of topological space X is said to be

(1) Generalized closed (briefly g-closed) [6] set if and only if Cl(A)  G whenever A G and G is open.

(2) Semi closed (briefly s-closed) [5] set if Int(Cl(A))  A and complement  of A is called semi open (briefly s-open) set.

(3) Regular closed (briefly r-closed) [12] set if A =Cl(Int(A)).

(4) Regular  generalized  closed (briefly  rg-closed)  [10] set  if Cl(A)    G  whenever A G and G is regular open in X. (5) Weakly closed (briefly w-closed) [11] if Cl(A) G whenever A G and  G is semi-open in X .

(6) Regular semi-open (briefly rs-closed) set [2] if there  is a regular open set G such that  G A Cl(G).

(7) Regular generalized weakly closed (briefly rgw-closed) set [8] if Cl(Int(A)) G whenever A G and G is regular semi-open in X .

Complements of above mentioned closed sets are said to be their respective open sets. Now, we are giving following generalized continuous functions in term of above closed sets.

Definition 2.1.  A function f : (X, τ1 ) → (Y, τ2) is said to be

(1) g-continuous [1] if f−1(H ) is g−τ1-closed set  of topological  space X  for every τ2-closed set H of Y .

(2) rg-continuous [10] if f−1(H ) is rg−τ1-closed set of topological space X for every τ2-closed set H of Y .

(3) w-continuous [4] if f−1(H ) is w−τ1-closed set of topological space X  for every τ2-closed set H of Y .

(4) rgw-continuous [9] if f−1(H ) is rgw−τ1-closed in topological space X  for every τ2-closed set H in Y .

Recently, introduced following generalized form of homeomorphism as follows.

Definition 2.2.  A bijection function f :(X, τ1) → (Y, τ2 ) is called

(1) Generalized homeomorphism [7] (briefly, g-homeomorphism) if both f and f−1  are g-continuous.

(2) Weakly homeomorphism [4] (briefly w-homeomorphism)  if  both  f and  f−1   are w-continuous.

(3) Semi-generalized homeomorphism [3] (briefly, sg-homeomorphism) if f is sg-continuous and sg-open.

(4) Generalized semi-homeomorphism [3] (briefly, gs-homeomorphism) if f is gs-continuous and gs-open.

3.The rgw-Homeomorphism in Topological Space

Now, in this section we are going to define new class of homeomorphism called regular generalized weakly homeomorphism on topological space and further we will discuss some its interesting properties.

Definition 3.1.  A bijection map f : (X, τ1) → (Y, τ2 ) is said to be rgw-homeomorphism if f and f−1 both are rgw-continuous map.

The family of all homeomorphism and  rgw-homeomorphism  of a  topological  space

(X, τ1) onto itself is denoted by h(X, τ1) and rgw−h(X, τ1) respectively.

Definition 3.2.  A bijection f : (X, τ1) → (Y, τ2) is said to be rgw-homeomorphism  if both f and f−1  are rgw-irresolute.

The family of all rgw-homeomorphism of a topological space (X, τ1) onto  itself is denoted  by rgw−h(X, τ1) respectively.

Now we compare new defined rgw-homeomorphism with recently introduced well-known generalized form of homeomorphism.

Theorem 3.3.  Every homeomorphism is a rgw-homeomorphism.

Proof.  Let f : (X, τ1) → (Y, τ2 ) is homeomorphism, then f and f−1  are continuous and also bijections.  As every continuous map is rgw-continuous so f and f−1 are rgw-continuous. Therefore, f is rgw-homeomorphism. But the converse of this result is not  true.   Let X  = Y   = {p, q, r},  τ  = {φ, X, {p}} and  ρ = {φ, Y, {q}}.   Let us consider a map f : (X, τ1 ) → (Y, τ2) defined by f (p) = r, f (q) = p and f (r) = q. Then we can verify that f is a rgw-homeomorphism.  But, f is not a homeomorphism.

Theorem 3.4.  Every rgw-homeomorphism is a g-homeomorphism.

Proof.  Since every rgw-continuous map is g-continuous map.  So easily can proof this.    

Theorem 3.5.  Every rgw-homeomorphism is a rgw-homeomorphism but not conversely. Proof.  As we know that  if f : (X, τ1 ) → (Y, τ2) is rgw-irresolute, then it is rgw-continuous and the fact that every rgw-open map is rgw-open.                                                          

Theorem 3.6.  Let the function f : (X, τ1 ) → (Y, τ2 ) be a one-one onto rgw-continuous map.  Then the following statements are equivalent:

(1) The function f is a rgw-open map.

(2) The function f is a rgw-homeomorphism.

(3) The function f is a rgw-closed map.

Theorem 3.7.  For  any topological space (X, τ ), h(X, τ ) rgw-h(X, τ ).

Proof.  Let us consider f h(X, τ ),  then  by  the  definition  of homeomorphism  f and f−1   are continuous.   Since every continuous function is rgw-continuous so f and f−1   is rgw-continuous map.   Now by the definition of rgw-homeomorphism we can say that f  rgw-h(X, τ ).                                                                                                                        

Theorem 3.8.  Let the function f : (X, τ1) → (Y, τ2) and g : (Y, τ2) → (Z, τ3 ) are rgw-homeomorphism, then composition of these two function as gof : (X, τ1) → (Z, τ3) is also rgw-homeomorphism.

Proof.  Let G be rgw-open in Z. Now (gof )−1(G)  = f−1 (g−1(G))  = f−1(G),  where V  = g−1(G).  By given, if G us rgw-open in Z and g is rgw-homeomorphism, then g−1(G) = H is rgw-open  in Y .  Again, as given g−1(G)  is open in Y   and f is rgw-homeomorphism, then f−1(g−1(G)) id rgw-open in X . Therefore, gof is continuous.  Similarly we can show that  (gof )−1  is also continuous.  Hence, by the definition gof  is rgw-homeomorphism.  By the above result we can proof that

Theorem 3.9.  The rgw-homeomorphism is an equivalence relation in the family of all topological spaces.

Theorem 3.10.  Let (X, τ ) be a topological space, then the collection rgw-h(X, τ ) forms a group under the composition of functions.

Proof.  Let a binary  operation  o' : rgw-h(X, τ ) × rgw-h (X, τ ) → rgw-h(X, τ ) defined by f o' g  = gof  for all f, g rgw-h(X, τ ), where gof : X  → X  is composite maps of f and g such that  (gof )(x)  = g(f (x))  for all x in X .By the theorem  3.8, gof  rgw-h(X, τ ). The following properties hold by the collection rgw-h(X, τ ).

(1) Associativity:  Since the composition of maps is associative, so easily prove

(f  o' g)o' h = f o'(g o' h)    for all    f, g, h rgw-h(X, τ )

(2) Existence of identity:  Since the identity map iX  : X → X is also rgw-homeomorphism, then easily we can say that  for all element f rgw-h(X, τ ), there exists an element iX   such that

f o' iX =  iX o' f = iX

(3) Existence of inverse: We know that the composition of maps is associative and the identity map iX : (X, τ ) → (X, τ ) belonging to rgw-h(X, τ ) servers as the identity element.  If f rgw-h(X, τ ), then f −1  rgw-h(X, τ ) such that  f of −1  = f−1 of = i and so inverse exists for each element of  rgw-h(X, τ ).

Therefore, (rgw-h(X, τ ), o) is a group under the operation of composition of maps.       

Theorem 3.11.  The homeomorphism group h(X, τ ) is a subgroup of the group rgw-h(X, τ ).

Proof.  It is obvious that 1X : (X, τ ) → (X, τ ) is a homeomorphism  and so h(X, τ ) ≠ φ.It follows from that  h(X, τ ) rgw-h(X, τ ).   Let   a, b h(X, τ ).   Then  we have  that o'(a, b−1)  = b−1oa  h(X, τ ), here o' : rgw-h(X, τ ) × rgw-h(X, τ ) → rgw-h(X, τ ) is the binary operation.  Therefore, the group h(X, τ ) is a subgroup of rgw-h(X, τ ).                 

Next important result give the relation between homeomorphism of space with group isomorphism of their family of rgw-homeomorphism.

Theorem 3.12.  If f : (X, τ1 ) → (Y, τ2) is homeomorphism, then there exists isomorphism between rgw− h(X, τ1 ) and rgw−h(Y, τ2) i.e.  rgw−h(X, τ1 ) rgw − h(Y, τ2).

Proof.  Using the map f , we define a map ψf : rgw − h(X, τ1) → rgw − h(Y, ρ) by ψf (h) = f ohof1  for every h rgw − h(X, τ1).  Then  ψf   is bijection.  Further, for all h1, h2  ψf , ψf (h1oh2) = f o(h1 oh2)of−1  = (f oh1of −1)o(f oh2 of−1) = ψf (h1)oψf (h2).  Therefore ψf   is a homeomorphism and so it is an isomorphism induced by f .                                            

Definition 3.13.  A function f : (X, τ1) → (Y, τ2 ) is said to contra  rgw-irresolute if f −1 is rgw-closed in (X, τ1) for every rgw-open set H of (Y, ρ).

Lemma 3.14.  Let two function  f : (X, τ1 ) → (Y, τ2) and g : (Y, τ2) → (Z, τ3 ) defined on topological spaces (X, τ1) and (Y, τ2 ) respectively, then

(1) If functions  f and g are  contra  rgw-irresolute, then the composition  gof  is also rgw-irresolute.

(2) If function  f is  rgw-irresolute (resp. contra  rgw-irresolute) and  g are  contra rgw-irresolute (resp.  rgw-irresolute), then the composition function gof  is contra rgw-irresolute.

Definition 3.15.   For  a topological  space (X, τ ),  we define the  collection of functions contra-rgw-h(X, τ ) as follows.

con − rgw-h(X, τ ) = {f : f (X, τ ) → (X, τ ) is a contra  rgw-irresolute bijection and f −1  is rgw-irresolute}

For a topological space (X, τ), we construct  alternative groups, say rgw-h(X, τ ) con −rgw-h(X, τ ).

Theorem 3.16.  If (X, τ) be a topological space, then union of two collections, rgw-h(X, τ )con−rgw-h(X, τ ), forms a group under the composition of functions.

Proof.  Let us BX   = rgw-h(X, τ )con−rgw-h(X, τ ). A binary operation wX : BX ×BX → BX   is well defined by wX (a, b) = boa, where boa : X → X is the composite function of the functions a and b. Indeed, let (a, b) BX ; if a rgw-h(X, τ ) and b con−rgw-h(X, τ ), then  boa : (X, τ )→  (X, τ ) a contra  rgw-irresolute  bijection  and  (boa)−1   is also contra rgw-irresolute  and  so wX (a, b)  = boa rgw-h(X, τ )    BX ,  if a  rgw-h(X, τ )  and b rgw-h(X, τ ) then boa : (X, τ ) → (X, τ ) is a rgw-irresolute bijection and so wX (a, b) = boa rgw-h(X, τ ) BX , if a  con − rgw-h(X, τ ) and  b  con − rgw-h(X, τ ),  then boa : (X, τ ) → (X, τ ) is a rgw-irresolute bijection  and (boa)−1   is also rgw-irresolute and so wX (a, b) = boa rgw-h(X, τ ) BX   is a con−rgw-h(X, τ ) and b rgw-h(X, τ ) then boa : (X, τ ) → (X, τ ) is a contra reg-irresolute bijection and (boa)−1  is also rgw-irresolute and and so wX (a, b) = boa con − rgw-h(X, τ ) BX .  By the similar arguments,  it is claimed that  the binary  operation  wX : BX  × BX   → BX   satisfies the axiom of group; for the  identity  element  e of BX , e = 1X : (X, τ ) → (X, τ ).  Thus  the  pair (BX , wX ) forms a group under  the  composition  of functions,  i.e., rgw-h(X, τ ) con − rgw-h(X, τ ) is a group.   

Theorem 3.17.  The homeomorphism group h(X, τ ) is a subgroup of rgw-h(X, τ ) con −rgw-h(X, τ ).

Proof.  By Theorem 3.11, it can be show that h(X, τ ) is a subgroup of rgw-h(X, τ ) con −rgw-h(X, τ ).                                                                                                                               

Theorem 3.18.  Let f : (X, τ1) → (Y, τ2) is rgw-continuous and collection of functions f is defined as Gg (f ) = {(x, y) X × Y : y = f (x)},  where X × Y   is product  topological space and Gg (f ) is called rgw-graph f . Then the following properties are satisfied:

(1) Gg (f ), as a subspace of X × Y , rgw-homeomorphism to X .

(2) If Y  is rgw-Housdrofff space, then Gg (f ) is rgw-closed in X × Y .

Proof.

(1) Consider  the  function  g : X    Gg (f )  is defined  by  g(x)  = (x, f (x))  for each x  X  is rgw-continuous  and  g1  is also rgw-continuous.   It  is obvious  that  g is an  injective  function.   Let  P  and  Q are  an  arbitrary neighbourhood  x  X and  (x, f (x)) in Gg(f ) respectively.   So, there  exists  two  rgw-open  sets U and  V in X  and Y  respectively containing  x and f (x) foe which (U × V ) ∩ Gg (f ) E and  U  P  and  f (U )  V .   Let  N  = (U × V ) ∩ Gg (f ),  then  (x, f (x))  N and  x g−1 (N ) U P .  This  shows that  g−1   is rgw-continuous.   Therefore, g(U ) (U × V ) ∩ Gg (f ) Q.  Hence g is rgw-continuous which means that  g is a rgw-homeomorphism.

(2) Let (x, y) Gg (f ).  Then y1 = f (x) = y. By hypothesis,  there exist disjoint rgw-open sets V1  and V  in Y  such that  y1 V1, y V . Since f is rgw-continuous, there exists an open set U in X containing x such that  f (U ) V1. Then g(U ) U × V1. It follows from this and the fact that  V1 ∩ V  = φ that  (U × V ) ∩ Gg (f ) = φ.

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