Download RW - Homeomorphism in Topological Spaces

International Journal of Computer Trends and Technology (IJCTT) – volume 6 Number 5– Dec 2013
RW - Homeomorphism in Topological Spaces
M.Karpagadevi 1, A.Pushpalatha 2
1
Assistant Professor, Mathematics, S.V.S. College of Engineering, Coimbatore, India
2
Assistant Professor, Mathematics, Government Arts College, Udumalpet, India
Abstract : A bijection f: (X,) → (Y, σ) is called
-1
regular weakly homeomorphism if f and f are
rw-continuous. Also we introduce new class of maps
namely rw-homeomorphism. This class of maps is
closed under composition of maps.
Definition 2.4 : A Subset A of (X, ) is called
gpr-closed [5] if pcl(A)  U whenever A  U and
U is regular open in X.
Definition 2.5: A map f: (X,) → (Y, σ) is called
rg-continuous [3] if f -1(V) is rg – closed.
Mathematical Subject Classification: 54C10,
54C08, 54C05.
Definition 2.6: A map f: (X,) → (Y, σ) is called
rwg-continuous [4] if f -1(V) is rwg – closed.
Keywords: rw-homeomorphism
Definition 2.7: A map f: (X,) → (Y, σ) is called
rw-continuous [2] if f -1(V) is rw – closed.
I. Introduction
Definition 2.8: A map f: (X,) → (Y, σ) is called
gpr-continuous [5] if f -1(V) is gpr – closed.
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 f -1 are
continuous.
Benchalli S.S and Wali R.S [1] introduced
RW – closed set in topological spaces.
Karpagadevi.M. and Pushpalatha.A [2] introduced
RW – continuous and RW– irresolute maps in
topological spaces.
In this paper we introduce the concept of
RW – homeomorphism and study the relationship
between homeomorphism, rw – homeomorphism, gpr
– homeomorphism and rwg – homeomorphism.
II. Preliminaries
Definition 2.1 : A Subset A of (X, ) is called
rw-closed [1] if cl(A)  U whenever A  U and U
is regular semi open in X.
Definition 2.2 : A Subset A of (X, ) is called
rg-closed [3] if cl(A)  U whenever A  U and U
is regular open in X.
Definition 2.3 : A Subset A of (X, ) is called
rwg-closed [4] if cl(int(A))  U whenever A  U
and U is regular open in X.
ISSN: 2231-2803
Definition 2.9: A map f: (X,) → (Y, σ) is called
rg-homeomorphism [3] if f and f -1 are rg –
continuous.
Definition 2.10: A map f: (X,) → (Y, σ) is called
rwg-homeomorphism [4] if both f and f -1 are
rwg – continuous.
Definition 2.11: A map f: (X,) → (Y, σ) is called
gpr-homeomorphism [5] if both f and f -1 are
gpr – continuous.
III. RW-homeomorphism in topological
spaces
Definition 3.1 : A bijection f : (X, ) → (Y, σ) is
called regular weakly homeomorphism if f and f -1
are rw-continuous .
We denote the family of all rw-homeomorphisms of a
topological space (X,) onto itself by
rw-(X,).
Example 3.2 : Consider X=Y={a,b,c} with
topologies  = { X, , {a}, {b}, {a,b} } and
σ = {Y, , {c} }. Let f: X → Y be a map defined by
f (a) = a, f (b) = b and f(c) = c. Then f is bijective,
rw-continuous and f -1 is rw –continuous. Hence f is
rw – homeomorphism.
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International Journal of Computer Trends and Technology (IJCTT) – volume 6 Number 5– Dec 2013
Theorem 3.3: Every homeomorphism is rwhomeomorphism.
Proof: Let f: (X, ) → (Y, σ) be a
homeomorphism. Then f and f -1 are continuous and f
is bijection . Since every continuous function is rwcontinuous, f and f -1 are rw-continuous .Hence f is
rw-homeomorphism.
Remark 3.4: The Converse of the above theorem
need not be true as seen from the following example.
Example 3.5 : Consider X=Y={a,b,c} with
topologies  = { X, , {a}, {b}, {a,b} } and
σ = {Y, , {c} }. Let f: X → Y be the identity map.
Then f is rw-homeomorphism. But it is not
homeomorphism since the inverse image of the
closed set {c} in X is {c} is not closed in Y.
Theorem 3.6: Every regular homeomorphism is rwhomeomorphism.
Proof: The proof follows from the theorem 3.3
Remark 3.7: The Converse of the above theorem
need not be true as seen from the following example.
Example 3.8 : Consider X=Y={a,b,c} with
topologies  = { X, , {a}, {b}, {a,b} } and
σ = {Y, , {c} }. Let f: X → Y be the identity map.
Then f is rw-homeomorphism. But it is not regular
homeomorphism since the inverse image of the
closed set {c} in X is {c} is not regular closed in Y.
Theorem 3.9: Every rw-homeomorphism is gprhomeomorphism.
Proof: Let f: (X,) → (Y, σ) be a rwhomeomorphism. Then f and f -1 are rw- continuous
and f is bijection. Since every rw-continuous function
is gpr-continuous, f and f -1 are gpr-continuous.
Hence f is gpr-homeomorphism.
Remark 3.10: The Converse of the above theorem
need not be true as seen from the following example.
Example 3.11 : Consider X=Y={a,b,c,d} with
topologies  = { X, , {a}, {b}, {a,b}, {a,b,c} } and
σ = {Y, , {c} }. Let f: X → Y be the identity map.
Then f is gpr-homeomorphism. But it is not
rw-homeomorphism since the inverse image of the
closed set {c} in X is {c} is not rw - closed in Y.
Theorem 3.12: Every rw-homeomorphism is rwghomeomorphism.
Proof: The proof follows from the definition and fact
that every rw-closed set is rwg-closed.
Remark 3.13: The Converse of the above theorem
need not be true as seen from the following example.
ISSN: 2231-2803
Example 3.14 : Consider X=Y={a,b,c,d} with
topologies  = { X, , {a}, {b}, {a,b}, {a,b,c} } and
σ = {Y, , {c} }. Let f: X → Y be the identity map.
Then f is rwg-homeomorphism. But it is not
rw-homeomorphism since the inverse image of the
closed set {c} in X is {c} is not rw-closed in Y.
Theorem 3.15: Every rw-homeomorphism is rghomeomorphism.
Proof: The proof follows from the definition and fact
that every rw-closed set is rg-closed.
Remark 3.16: The Converse of the above theorem
need not be true as seen from the following example.
Example 3.17 : Consider X=Y={a,b,c,d} with
topologies  = { X, , {a}, {b}, {a,b}, {a,b,c} } and
σ = {Y, , {c} }. Let f: X → Y be the identity map.
Then f is rg-homeomorphism. But it is not
rw-homeomorphism since the inverse image of the
closed set {c} in X is {c} is not rw - closed in Y.
Theorem 3.18: Let f: (X,) → (Y, σ) be a bijective
rw-continuous map. Then the following statements
are equivalent.
(i)
(ii)
(iii)
f is a rw-open map.
f is rw-homeomorphism.
f is a rw-closed map.
Proof: Suppose (i) holds. Let V be open in (X,).
Then by (i), f (V) is rw-open in (Y, σ). But
f (V) = (f -1)-1(V) and so (f -1)-1(V) is rw-open in
(Y, σ). This shows that f -1 is rw-continuous and it
proves (ii).
Suppose (ii) holds. Let F be a closed set in
(X,). By (ii), f -1 is rw-continuous and so
(f -1)-1(V) (F) = f (F) is rw-closed in (Y, σ).This
proves (iii).
Suppose (iii) holds. Let V be open in (X,).
Then Vc is closed in (X,). By (iii), f (Vc) is rwclosed in (Y, σ). But f (Vc) = (f (V)) c. This implies
that (f (V)) c is rw-closed in (Y, σ) and so f (V) is
rw-open in (Y, σ). This proves (i).
Remark 3.18: The Composition of two rwhomeomorphism need not be a rw-homeomorphism
in general as seen from the following example.
Example 3.19 : Let X=Y=Z={a,b,c} with topologies
 = { X, , {a}, {b}, {a,b} },
σ = {Y, ,{b}, {a,b} },  = {Z, ,{a}, {c}, {a,c} }.
Let g : (X,) → (Y, σ) be a map defined by g(a) = a,
g(b) = b and g(c) = c, let f : (Z, ) → (X,) be a map
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International Journal of Computer Trends and Technology (IJCTT) – volume 6 Number 5– Dec 2013
defined by f(a)=b, f(b)=a and f(c)=c. Both f and g are
rw-homeomorphism. Define g  f: (Z,) → (Y, σ).
Here {c} is closed set of (Y, σ). Therefore
(g  f)-1{c} = {c} is not a rw-closed set of (Z,).
Hence g  f is not rw-homeomorphism.
IV. CONCLUSION
In this paper we have introduced and studied the
properties of rw-homeomorphism. Our future
extension is rw-homeomorphism in bitopological
spaces and in Intuitionistic Fuzzy topological spaces.
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