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Applied Mathematical Sciences, Vol. 8, 2014, no. 9, 415 - 422
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http://dx.doi.org/10.12988/ams.2014.311639
On Generalized gp*- Closed Map
in Topological Spaces
S. Sekar
Department of Mathematics
Government Arts College (Autonomous)
Salem – 636 007, Tamil Nadu, India
P. Jayakumar
Department of Mathematics
Paavai Engineering College,
Pachal, Namakkal – 637 018, Tamil Nadu, India
Copyright © 2014 S. Sekar and P. Jayakumar. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, the authors introduce a new class of generalized gp*-closed map in
topological spaces and study some of their properties as well as inter relationship
with other closed map
Mathematics Subject Classification: 54C05, 54C08, 54C10
Keywords: gp*-closed map and gp*-open map
1 Introduction
Different types of Closed and open mappings were studied by various
researchers. In 1998, T.Noiri introduced new type of set called gp-closed set.
A.A.Omari and M.S.M. Noorani introduced and studied b-closed map.
The aim of this paper is to introduce gp* closed map and to continue the
study of its relationship with various generalized closed maps. Throughout this
paper (X, τ ) and (Y, σ ) represent the non-empty topological spaces on which no
separation axioms are assumed, unless otherwise mentioned.
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S. Sekar and P. Jayakumar
2 Preliminaries
Definition 2.1[4]: Let A subset A of a topological space (X, τ ), is called a
pre-open set if A ⊆ Int(cl (A)).
Definition 2.2 [1]: Let A subset A of a topological space (X, τ ), is called a
generalized closed set (briefly g-closed) if cl (A) ⊆ U whenever A ⊆ U and U is
open in X.
Definition 2.3 [5]: Let A subset A of a topological space (X, τ ), is called a
generalized pre- closed set (briefly gp- closed) if pcl (A) ⊆ U whenever A ⊆ U
and U is open in X.
Definition 2.4 [3]: Let A subset A of a topological space (X, τ ), is called a
generalized pre-closed set (briefly pg-closed) if pcl(A) ⊆ U whenever A ⊆ U and U
is pre-open in X.
Definition 2.5 [8]: Let A subset A of a topological space (X, τ ), is called a
generalized pre- closed set (briefly g*- closed) if cl (A) ⊆ U whenever A ⊆ U and
U is g-open in X.
Definition 2.6 [10]: Let A subset A of a topological space (X,τ ), is called a
generalized pre- closed set (briefly g*p-closed) if pcl (A) ⊆ U whenever A ⊆ U
and U is g-open in X.
Definition 2.7 [7]: Let A subset A of a topological space (X, τ ), is called a
generalized pre- closed set (briefly strongly g- closed) if cl (A) ⊆ U whenever
A ⊆ U and U is g-open in X.
Definition 2.8 [6]: Let A subset A of a topological space (X, τ ), is called a
generalized pre- closed set (briefly α g- closed) if α cl(A) ⊆ U whenever A ⊆ U
and U is open in X.
Definition 2.9 [9]: Let A subset A of a topological space (X, τ ), is called a
generalized pre- closed set (briefly g# closed) if cl(A) ⊆ U whenever A ⊆ U and
U is α g-open in X.
Definition 2.10 [15]: A subset A of a topological space (X, τ ), is called
gp*-closed set if cl (A) ⊆ U whenever A ⊆ U and U is gp open in X.
3 On regular generalized b-closed map
In this section we introduce gp*-closed map(gp*-closed),gp*-open map
(gp*-open) and study the some of their properties.
Definition 3.1: Let X and Y be two topological spaces. A map f: (X, τ ) → (Y, δ )
is called gp*-closed map(briefly gp*-closed), if the image of every closed set in X
is gp*-closed in Y.
Theorem 3.2. Every closed map is gp*-closed but not conversely.
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417
Proof: Let f: (X, τ ) → (Y, δ ) is closed map and V be an closed set in X then f (V)
is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed.
The converse of above theorem need not be true as seen from the following
example.
Example 3.3 Consider X = Y = {a,b,c}, τ ={ X, ϕ , {b,c}} and
σ ={Y, ϕ ,{a},{c},{a,c},{b,c}}. Let f: (X, τ ) → (Y, σ ) be defined by f(a) = a,
f(b) = b, f(c) = c. The map is gp*-closed but not closed as the image of and {a,b}
in X is {a,b} is not closed in Y.
Theorem 3.4. Every g*-closed map is gp*-closed but not conversely.
Proof: Let f: (X, τ ) → (Y, δ ) is g*-closed map and V be an closed set in X then f
(V) is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed.
Example 3.4 Consider X = Y = {a,b,c}, τ ={ X, ϕ ,{b}} and σ ={Y, ϕ }. Let f:
(X, τ ) → (Y, σ ) be defined by f(a) = c, f(b) = a, f(c) = b. The map is gp*-closed
but not g*closed as the image of and {a,c} in X is {b,c} is not g*-closed in Y.
Theorem 3.5 Every pg-closed map is gp*-closed but not conversely.
Proof: Let f: (X, τ ) → (Y, δ ) be pg-closed map and V be an closed set in X then
f (V) is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed.
Example 3.6 Consider X = Y = {a,b,c}, τ ={ X, ϕ , {a},{a,c}} and
σ ={Y, ϕ ,{a},{c},{a,c},{b,c}}. Let f: (X, τ ) → (Y, σ ) be defined by f(a) = b,
f(b) = a, f(c) = c. The map is gp*-closed but not pg closed as the image of {b,c} in
X is {a,c} is not pg closed in Y.
Theorem 3.7 Every g*p map is gp*-closed but not conversely.
Proof: Let f: (X, τ ) → (Y, δ ) be g*p map and V be an closed set in X then f (V)
is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed.
Example 3.8 Consider X = Y = {a,b,c}, τ ={ X, ϕ , {a},{a,b}} and
σ ={Y, ϕ ,{b}}.
Let f: (X, τ ) → (Y, σ ) be defined by f(a) = b,
f(b) = c, f(c) = a. The map is gp*-closed but not g*p-closed as the image of and
{c,b} in X is {a,c} is not g*p-closed in Y.
Theorem 3.9 Every gp*closed map is strongly g closed map but not
conversely.
Proof: Let f: (X, τ ) → (Y, δ ) gp*- closed map and V be an closed set in X then f
(V) is closed in Y. Hence strongly g closed in Y. Then f is strongly g closed.
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Example 3.10 Consider X = Y = {a,b,c}, τ ={ X, ϕ ,{b},{a,b}} and
σ ={Y, ϕ ,{c},{b,c}}.
Let f: (X, τ ) → (Y, σ ) be defined by f(a) = c, f(b)
= b, f(c) = a. The map is strongly g closed but not gp*closed as the image of and
{a,c} in X is {a,c} is not gp*closed in Y.
Theorem 3.11 Every gp*-closed map is g#-closed but not conversely.
Proof: Let f: (X, τ ) → (Y, δ ) be gp*-closed map and V be an closed set in X
then f (V) is closed in Y. Hence g# closed in Y. Then f is g# closed in Y.
Example 3.12 Consider X = Y = {a,b,c}, τ ={ X, ϕ ,{a},{a,b}} and
σ ={Y, ϕ ,{a},{b},{a,b},{b,c}}. Let f: (X, τ ) → (Y, σ ) be defined by f(a) = b,
f(b) = c, f(c) = a. The map is gp*-closed but not g α -closed as the image of and
{b,c} in X is {a,c} is not
closed in Y.
Theorem 3.13 Every gp*closed map is α g*closed but not conversely.
Proof: Let f: (X, τ ) → (Y, δ ) be gp* closed map and V be an closed set in X then
f (V) is closed in Y. Hence α g*-closed in Y. Then f is α g*-closed.
Example 3.14 Consider X = Y = {a,b,c}, τ ={ X, ϕ ,{a}} and σ ={Y, ϕ ,{b}}.
Let f: (X, τ ) → (Y, σ ) be defined by f(a) = b, f(b) = a, f(c) = c. The map is
α g*closed but not gp* closed as the image of {a,c} in X is {b,c} is not closed in
Y.
4 Application
Theorem 4.1 A map f: (X, τ ) → (Y, σ ) is continuous and gp*-closed set A is
gp*-closed set of X then f(A) is gp* closed in Y.
Proof: Let f(A) ⊆ U where U is gp open set in Y. Since f is continuous, f-1 (U)
is open set containing A. Hence cl(A) ⊆ f-1 (U) (as A is gp*-closed). Since
f is gp*-closed f (cl(A)) ⊆ U is gp* closed set ⇒ cl(f(cl(A)) ⊆ U, Hence
cl(A) ⊆ U. So that f(A) is gp*-closed set in Y.
Theorem 4.2. If a map f: (X, τ ) → (Y, σ ) is continuous and closed set and A is
gp*-closed then f(A) is gp*-closed in Y.
Proof: Let F be a closed set of A then F is gp*-closed set. By theorem 4.8 f(A) is
gp*-closed. Hence fA(F) = f(F) is gp*-closed set of Y. Here fA is gp*-closed
and also continuous.
Theorem 4.3 If f: (X, τ ) → (Y, σ ) is closed map and g: (Y, σ ) → (Z, η ) is
gp*-closed map , then the composition g • f : (X, τ ) → (Z, η ) is gp*-closed map.
Proof: Let F be any closed set in (X, τ ). Since f is closed map, f(F) is closed set
in (Y, σ ). Since g is gp*-closed map, g(f(F)) is gp*-closed set in (Z, η ).
That is g.f(F) = g(f(F)) is gp* closed . Hence g • f is gp* closed map.
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Remark: If f: (X, τ ) → (Y, σ ) is gp*-closed map and g: (Y, σ ) → (Z, η ) is
closed map, then the composition need not gp*-closed map .
Theorem 4.4 A map f: (X, τ ) → (Y, σ ) is gp*-closed if and only if for each
subset S of (Y, σ ) and each open set U containing f-1(S) ⊂ U, there is a
gp*-open set V of (Y, σ ) such that S ⊂ V and f-1(V) ⊂ U.
Proof: Suppose f is gp*-closed. Let S ⊂ Y and U be an open set of (X, τ ) such
that f-1(S) ⊂ U. Now X-U is closed set in (X, τ ). Since f is gp*-closed, f(X-U)
is gp*-closed set in (Y, σ ). There fore V = Y – f(X-U) is an gp*-open set in
(Y, σ ). Now f-1(S) ⊂ U implies S ⊂ V and f-1(V) = X - f-1(f(X-U)) ⊂ X-(X-V)
-1
= U. (ie) f (V) ⊂ U.
Conversely, Let F be a closed set of (X, τ ). Then f-1(f(Fc)) ⊂ Fc. and Fc. is an
open in(X, τ ).
By hypothesis, ∃ A gp*-open set V in (Y, σ ) such that f(Fc)
⊂V
and f-1(V) ⊂ Fc ⇒
F ⊂ f-1(V)c.
Hence Vc ⊂ f(F) ⊂
-1
c c
c
c
c
f(((f (V)) ) ⊂ V ⇒ f(V) ⊂ V . Since V is gp*-closed, f(F) is gp*-closed. (ie)
f(F) is gp*-closed in Y and . Therefore f is gp*-closed.
Theorem 4.5 If f: X1 × X2 → Y1 × Y2 is defined as f(x1,x2) = (f1(x1),f2(x2)),
Y1 × Y2 is gp* closed map.
then f: X1 × X2 →
Proof: Let U1 × U2 ⊂ X1 × X2 where Ui ∈ gp*cl(Xi), for i = 1,2. Then f(U1 ×
U2) = f1(U1) × f2(U2) ∈ gp*cl(X1 × Y2). Hence f is gp*-closed set.
Theorem 4.6 Let h: X → X1 × X2 be gp*-closed map and Let fi : X → Xi be
define as h(x) = (x1,x2) and fi(x) = xi, then fi : X → Xi is gp*-closed map for i =
1,2.
Proof: Let U1 × U2 ∈ X1 × X2, then f1(U1) = h1(U1 × X2) ∈ gp*cl(X), there fore
f1 is gp*-closed. Similarly we have f2 is gp*-closed. Thus fi is gp*-closed map
for i=1, 2.
Theorem 4.7 For any bijection map f: (X, τ ) → (Y, σ ), the following statements
are equivalent:
(i) f-1 : (Y, σ ) → (X, τ ) is gp*-continuous.
(ii) f is gp*-open map
(iii) f is gp*-closed map.
Proof: (i) ⇒ (ii) Let U be an open set of (X, τ ). Bu assumption, (f-1)-1(U) =
f(U) is gp*-open in (Y, σ ) and so f is gp*-open.
(ii) ⇒ (iii) Let F be a closed set of (X, τ ). Then Fc is open set in (X, τ ). By
assumption f(Fc) is gp*-open in (Y, σ ). Therefore f (Fc) = f (F)c is gp*-open
in (Y, σ ). That is f(F) is gp*-closed in (Y, σ ).
Hence f is gp*-closed.
(iii) ⇒ (i) Let F be a closed set of (X, τ ). By assumption, f (F) is
gp*-closed in (Y, σ ). But
f (F) = (f-1)-1(F) ⇒ (f-1) is continuous.
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5. On regular generalized b-open map
Definition 5.1: Let X and Y be two topological spaces. A map f: (X, τ ) → (Y, δ )
is called regular generalized b-open(briefly gp*-open) if the image of every open
set in X is gp*-open in Y.
Theorem 5.2 Every open map is gp*-open but not conversely.
Proof: Let f: (X, τ ) → (Y, δ ) is open map and V be an open set in X then f(V) is
open in Y. Hence gp*-open in Y. Then f is gp*-open.
The converse of above theorem need not be true as seen from the following
example.
Example 5.3 Consider X = Y = {a,b,c}, τ ={X, ϕ ,{a}} and σ ={Y, ϕ ,{b}}.
Let f: (X, τ ) → (Y, σ ) be defined by f(a) = b, f(b) = c, f(c) = a. The map is
gp*-open but not open as the image of and {b,c} in X is {a,c} is not open in Y.
Theorem 5.4 A map f: (X, τ ) → (Y, σ ) is gp*-closed set if and only if for each
subset S of Y and for each open set U containing f-1(S) ⊂ U there is a gp*-open
set V of Y such that S ⊂ U and
f-1(V) ⊂ U.
Proof: Suppose f is gp*-closed set. Let S ⊂ Y and U be an open set of (X, τ )
such that f-1(S) ⊂ U. Now X-U is closed set in (X, τ ). Since f is gp* closed,
f(X-U) is gp* closed set in(Y, σ ). Then V = Y- f(X-U) is gp* open set in
(Y, σ ). Therefore f-1(S) ⊂ U implies S ⊂ V and f-1(V) = X- f-1(f(X-U))
⊂ X-(X-V) = U. (ie) f-1(V) ⊂ U.
Conversely, Let F be a closed set of (X, τ ). Then f-1(f (Fc)) ⊂ Fc and Fc is an
open in (X, τ ). By hypothesis, ∃ a gp* –open set V in (Y, σ ) such that f(Fc)
⊂ V and f-1(V) ⊂ Fc ⇒ F ⊂ (f-1(V)c. Hence Vc ⊂ f(F) ⊂ f(((f-1 (V))c )c )
⊂ Vc ⇒ f(V) ⊂ Vc . Since Vc gp*-closed, f(F) is gp*-closed. (ie) f(F) is
gp*-closed in (Y, σ ) and therefore f is gp*-closed.
Theorem 5.5 For any bijection map f: (X, τ ) → (Y, σ ), the following statements
are equivalent.
(i)
f-1: (X, τ ) → (Y, σ ) is gp*-continuous.
(ii) f is gp* open map.
(iii) f is gp*-closed map.
Proof: (i) ⇒ (ii) Let U be an open set of (X, τ ). By assumption (f-1)-1(U) = f(U) is
gp*-open in (Y, σ ). Therefore f is gp*-open map.
(ii) ⇒ (iii) Let F be closed set of (X, τ ), Then Fc is open set in (X, τ ). By
assumption, f(Fc) is gp*-open in (Y, σ ). Therefore f(F) is gp*-closed in (Y, σ ).
Hence f is gp*-closed.
(iii) ⇒ (i) Let F be a closed set of (X, τ ), By assumption f(F) is gp*-closed in
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421
(Y, σ ). But f(F) = (f-1)-1(F). Hence f-1: (X, τ ) → (Y, σ ) is gp*-continuous.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the Dr. A. R. Rajamani, Principal,
Government Arts College (Autonomous), Salem-636 007, for encouragement and
support. The authors also thank Mr. K. Mariappa, Assistant Professor, Department
of Mathematics, King College of Technology, Namakkal – 637 020, Tamil Nadu,
India, for his kind help in basic ideas of this research paper and encouragement.
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Received: November 16, 2013