**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, τ_{1} )
→ (Y, τ_{2}) is said to be

(1) g-continuous [1] if f^{−1}(H ) is g−τ_{1}-closed
set of topological space X
for every τ_{2}-closed set H of Y .

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

(3) w-continuous [4] if f^{−1}(H ) is w−τ_{1}-closed
set of topological space X for every τ_{2}-closed
set H of Y .

(4) rgw-continuous [9] if f^{−1}(H ) is
rgw−τ_{1}-closed in topological space X for every τ_{2}-closed set H in
Y .

Recently, introduced following generalized form of homeomorphism
as follows.

**Definition 2.2.** A bijection function f :(X, τ_{1})
→ (Y, τ_{2} ) 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, τ_{1})
→ (Y, τ_{2} ) 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, τ_{1}) onto itself is denoted by h(X,
τ_{1}) and rgw−h(X, τ_{1}) respectively.

**Definition 3.2.** A bijection f : (X, τ_{1})
→ (Y, τ_{2}) 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, τ_{1})
onto itself is denoted by rgw^{∗}−h(X, τ_{1}) 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,
τ_{1}) → (Y, τ_{2} ) 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, τ_{1}
) → (Y, τ_{2}) 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, τ_{1} ) → (Y,
τ_{2}) 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, τ_{1} ) → (Y, τ_{2} ) 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, τ_{1}) → (Y, τ_{2}) and g :
(Y, τ_{2}) → (Z, τ_{3} ) are rgw-homeomorphism,
then composition of these two function as gof : (X, τ_{1}) →
(Z, τ_{3}) 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 i_{X} : X → X is also rgw-homeomorphism, then
easily we can say that for all element f
∈ rgw-h(X, τ ), there
exists an element i_{X} such
that

f o' i_{X }= i_{X}
o' f = i_{X}

(3) Existence of inverse: We know that the composition
of maps is associative and the identity map i_{X} : (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 1_{X} : (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^{−1}oa ∈ 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,
τ_{1} ) → (Y, τ_{2}) is homeomorphism, then
there exists isomorphism between rgw− h(X, τ_{1} ) and rgw−h(Y,
τ_{2}) i.e. rgw−h(X,
τ_{1} ) ≅ rgw
− h(Y, τ_{2}).

**Proof.** Using the map
f , we define a map ψ_{f} : rgw − h(X, τ_{1})
→ rgw − h(Y, ρ) by ψ_{f} (h) = f ohof^{1 } for every h ∈ rgw − h(X, τ_{1}). Then
ψ_{f} is
bijection. Further, for all h_{1},
h_{2} ∈ ψ_{f }, ψ_{f }(h_{1}oh_{2})
= f o(h_{1} oh_{2})of^{−1} = (f oh_{1}of ^{−1})o(f
oh_{2} of^{−1}) = ψ_{f} (h_{1})oψ_{f}
(h_{2}). Therefore ψ_{f} is a homeomorphism and so it is an
isomorphism induced by f .

**Definition 3.13.** A function f : (X, τ_{1})
→ (Y, τ_{2} ) is said to contra rgw-irresolute if f ^{−1} is
rgw-closed in (X, τ_{1}) for every rgw-open set H of (Y, ρ).

**Lemma 3.14.** Let two
function f : (X, τ_{1} )
→ (Y, τ_{2}) and g : (Y, τ_{2}) → (Z,
τ_{3} ) defined on topological spaces (X, τ_{1}) and
(Y, τ_{2} ) 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 w_{X}
: B_{X} ×B_{X} → B_{X} is well defined by w_{X} (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
w_{X} (a, b) = boa ∈ rgw-h(X, τ )
⊂ B_{X} , 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 w_{X}
(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 w_{X}
(a, b) = boa ∈ rgw-h(X, τ ) ⊂ B_{X}
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 w_{X} (a, b) = boa ∈ con − rgw-h(X, τ ) ⊆ B_{X} .
By the similar arguments, it is
claimed that the binary operation
w_{X} : B_{X} × B_{X} → B_{X} satisfies the axiom of group; for the identity
element e of B_{X} , e =
1_{X} : (X, τ ) → (X, τ ). Thus
the pair (B_{X} , w_{X}
) 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,
τ_{1}) → (Y, τ_{2}) 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
→ G_{g} (f ) is defined
by g(x) = (x, f (x))
for each x ∈ X is
rgw-continuous and g^{1}
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 G_{g}(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
) ∩ G_{g} (f ) ⊂ E
and U
⊂ P and f
(U ) ⊂ V . Let N = (U
× V ) ∩ G_{g} (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 ) ∩ G_{g} (f ) ⊂ Q. Hence g is
rgw-continuous which means that g is a
rgw-homeomorphism.

(2) Let (x, y) ∉ Gg (f
). Then y_{1} = f (x) = y. By
hypothesis, there exist disjoint rgw-open
sets V_{1} and V in Y
such that y_{1} ∈ V_{1}, y ∈ V . Since f is rgw-continuous, there exists an open
set U in X containing x such that f (U )
⊂ V_{1}. Then g(U ) ⊂ U × V_{1}. It follows from this and the fact
that V_{1} ∩ V = φ that
(U × V ) ∩ G_{g} (f ) = φ.

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