Download Generalized star semi regular closed sets in topological spaces

Malaya J. Mat. S(1)(2015) 42-56
Generalized star semi regular closed sets in topological spaces
D. Sreejaa,∗ and S. Sasikalab
a Department
of Mathematics,CMS College of Science and Commerce, Coimbatore, Tamil Nadu, India.
b Department
of Mathematics, Pioneer College of Arts and Science, Coimbatore, Tamil Nadu, India.
Abstract
In this paper a new class of sets called generalized star semi regular closed sets is introduced and
its various properties are discussed. g ∗ sr continuous function is defined and its results are studied.
Also comparative study has been done with the existing sets.
Keywords:
Topological spaces, generalized closed sets, g-closed, gs-closed, g ∗ sr-closed sets, g ∗ sr
-closed map, g ∗ sr -continuous functions.
2010 MSC: 54D10.
1
c 2012 MJM. All rights reserved.
Introduction
The notion of closed set is fundamental in the study of topological spaces. In 1970 Levine [12]
introduced the class of generalized closed sets in the topological space by comparing the closure of
subset with its open supersets. The investigation on generalized of closed set has lead to significant
contribution to the separation axiom, covering properties and generalization of continuity. T.K.
Kong, R. Kopperman and P. Meyer [12] shown some of the properties of generalized closed set have
been found to be useful in computer science and digital topology . Maki et al. [15] defined αg-closed
sets in 1994. S.P. Arya and N. Tour [2] defined gs-closed sets in 1990. Dontchev [6], Gnanambal and
Palaniappan and Rao [18] introduced gsp-closed sets, gpr closed sets and rg-closed sets respectively.
M.K.R.S. Veerakumar [11] introduced g ∗ - closed sets in 1991. J. Dontchev [6] introduced gsp-closed
sets in 1995. In 1993, N.Palaniappan and Chandra Sekran Rao [18] introduced rg-closed sets.
∗
Corresponding author.
E-mail address: [email protected] (D. Sreeja), [email protected]( S. Sasikala).
D. Sreeja et al. / Generalized star semi...
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Preliminaries
Throughout this paper (X, τ ) and (Y, σ) represent nonempty topological spaces are mentioned.
For a subset A of a space (X, τ ) cl A and int A denote the closure of A and the interior of A
respectively. (X, τ ) will be replaced by X if there is no chance of confusion. Let us recall the
following definitions which we shall require later.
Definition 2.1. A subset A of a space (X, τ ) is called
(1) The closure of A is defined as the intersection of all closed sets containing A.
(2) a pre-open set [13] if A ⊆ int(cl(A)) and a pre-closed set if cl(int(A)) ⊆ A.
(3) a semi-open set [11] if A ⊆ cl(int(A)) and semiclosed set if int(cl(A)) ⊆ A.
(4) a α-open set [16] if A ⊆ int(cl(int(A))) and α-closed set if cl(int(cl(A))) ⊆ A.
(5) a semi pre-open set [1] if A ⊆ cl(int(cl(A)) and a semi pre-closed set if int(cl(int(A))) ⊆ A.
(6) a regular-open set if A = int(cl(A)) and regular closed set A = cl(int(A)).
The family of all pre-open (resp.preclosed, semi preopen, semi preclosed, regularopen, regularcosed) subsets
of space (X, τ ) wil be denoted by po(τ )(resp, pc(τ ), spo(X), spc(X), ro(τ ), rc(τ )). The intersection of all
semi-closed subsets of x containing A is called semiclosure of A is denoted by scl(A).
Definition 2.2. A subset A of a space (X, τ ) is called a
(1) generalized closed (briefly g-closed)[12] if cl(A) ⊆ U . whenever A ⊆ U and U is open in X.
(2) semi-generalized closed set (briefly sg-closed) [3] if scl(A) ⊆ U , whenever A ⊆ U and U is semi open
in X.
(3) generalized semi-closed set (briefly gs-closed) [2] if scl(A) ⊆ U , whenever A ⊆ U and U is open in X.
(4) α - generalized -closed set (briefly αg-closed) [15] if αcl(A) ⊆ U , whenever A ⊆ U and U is open in
X.
(5) a generalized α-closed set (briefly gα -closed) [14]. if αcl(A) ⊆ U , whenever A ⊆ U and U is open in
X.
(6) generalized semi-pre closed set (briefly gsp-closed) [6] if spcl(A) ⊆ U , whenever A ⊆ U and U is open
in X.
(7) regular generalized closed set (briefly rg-closed) [18] if cl(A) ⊆ U , whenever A ⊆ U and U is regular
open in X.
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(8) generalized pre-closed set (briefly gp-closed) [4] if pcl(A) ⊆ U , whenever A ⊆ U and U is open in X.
(9) generalized pre-regular closed set (briefly gpr-closed) if pcl(A) ⊆ U , whenever A ⊆ U and U is regular
open in X.
(10) sg ∗ r closed if rcl(A) ⊆ U whenever A ⊆ U and U is sg open.
Definition 2.3. A map f : (x, τ ) → (y, σ) is called
(1) generalized closed (briefly g-closed ) if[12] f (U ) is g-open in (y, σ) for every open set of U of (x, τ )
(2) regular closed (briefly r-closed) if f (U ) is r-open in (y, σ) for every open set of U of (x, τ )
(3) α generalized closed (briefly αg-closed) [15] if f (U ) is αg-open in (y, σ) for every open set of U of (x, τ )
(4) generalized pre-closed (briefly gp-closed)[4] if f (U ) is rg-open in (y, σ) for every open set of U of (x, τ )
(5) regular generalized closed (briefly rg-closed)[18] if f (U ) is rg-open in (y, σ) for every open set of U of
(x, τ )
(6) generalized semi-pre closed (briefly gsp-closed) [6] if f (U ) is rg-open in (y, σ) for every open set of U of
(x, τ ).
Definition 2.4. A function f : (x, τ ) → (y, σ) is called
(1) r continuous if f −1 (V ) is r closed set of (X, τ ) → (Y, σ) for every closed set v of (Y, σ).
(2) αg continuous if f −1 (V ) is αg closed set of (X, τ ) → (Y, σ) for every closed set v of (Y, σ).
(3) gpr continuous if f −1 (V ) is gpr closed set of (X, τ ) → (Y, σ) for every closed set v of (Y, σ).
(4) gp continuous if f −1 (V ) is gp closed set of (X, τ ) → (Y, σ) for every closed set v of (Y, σ)
(5) rg continuous if f −1 (V ) is r closed set of (X, τ ) → (Y, σ) for every closed set v of (Y, σ).
3
Basic properties of g ∗ sr closed sets
Definition 3.1. A subset A of a topological space (X, τ ) is said to be g ∗ sr closed set if rcl(A) ⊆ U whenever
A ⊆ U and U is gs open in X.
Example 3.1. Let X = {a, b, c, d} with topology τ = {ϕ, {a}, {a, b}, {c, d}, {a, c, d}, X}. The closed sets
are
τ c = {ϕ, {b, c, d}, {c, d}, {a, b}, {b}, X}.
gs open sets are {ϕ, {a}, {c}, {d}, {a, c}, {a, b}, {a, d}, {c, d}, {a, b, d}, {a, c, d}, {a, b, c}, X} g ∗ sr closed
sets are {ϕ, {b}, {a, b}, {c, d}, {b, c, d}, X}.
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Theorem 3.1. Every regular closed set is g ∗ sr closed
Proof. Let A be a regular closed set in X. Then A = rcl(A). Let us prove that A is g ∗ sr closed in
X. Let A ⊆ U and U is gs open. Then rcl(A) = A ⊆ U whenever A ⊆ U and U is gs open. Hence
rcl(A) ⊆ U . Hence A is g ∗ sr closed set in X.
Remark 3.1. The converse of the above theorem need not be true in general, as shown in the following example.
Example 3.2. Let X = {a, b, c, d} with topology τ = {ϕ, {a}, {b}, {a, b}, X}, Regular closed sets are
{ϕ, {a, c, d}, {b, c, d}, X}. g ∗ sr closed sets are {ϕ, {c, d}, {b, c, d}, {a, c, d}, X}. Hence A = {c, d} is g ∗ sr
closed but not regular closed.
Theorem 3.2. Every g ∗ sr closed set is g closed
Proof. Let A be g ∗ sr closed set in X. Then rcl(A) ⊆ U whenever A ⊆ U and U is gs open. Let
U be open set such that A ⊆ U . Since every open set is g open and A is g ∗ sr closed . We have
rcl(A) ⊆ cl(A) ⊆ U . Hence cl(A) ⊆ U . Therefore A is g closed set in X.
Remark 3.2. The converse of the above theorem need not be true in general, as shown in the following example.
Example 3.3. Let X = {a, b, c, d} with topology τ = {ϕ, {a}, {b}, {a, b}, X}.
g closed sets are
{ϕ, {a, c, d}, {b, c, d}, X}.
g ∗ sr closed sets are {ϕ, {c, d}, {b, c, d}, {a, c, d}, X}. Hence A = {c, d} is g ∗ sr closed but not g closed.
Theorem 3.3. Every g ∗ sr closed set is gs closed.
Proof. Let A be a g ∗ sr closed set in X. (i.e) rcl(A) ⊆ U whenever A ⊆ U and U is gs open. To prove
that A is gs closed. Let U be open set such that A ⊆ U . Since every open set is gs-open and A is g ∗ sr
closed, We have scl(A) ⊆ rcl(A) ⊆ U . Therefore A is gs- closed set in X.
Remark 3.3. The converse of the above theorem need not be true in general, as shown in the following example.
Example
τ
=
3.4.
Let
X
=
{a, b, c, d}
{ϕ, {b}, {d}, {b, d}, {a, b}, {b, c}, {a, b, c}, {b, c, d}, {a, b, d}, X}
with
gs
closed
topology
sets
are
{ϕ, {a}, {c}, {d}, {c, d}, {a, c}, {a, d}, {a, c, d}, {a, b, c}, X} g ∗ sr closed sets are {ϕ, {d}, {a, c, d}, X}
Hence A = {a} is gs closed but not g ∗ sr closed.
Theorem 3.4. Every g ∗ sr closed set is rg closed.
Proof. Let A be g ∗ sr closed set in X. Then rcl(A) ⊆ U and U is gs open. To prove A is rg closed in
X. Let U be r open set such that A ⊆ U . Since every r open set is gs open and A is g ∗ sr closed. We
have cl(A) ⊆ rcl(A) ⊆ U . Hence cl(A) ⊆ U . Therefore A is rg closed set in X.
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Remark 3.4. The converse of the above theorem need not be true in general,as shown in the following example.
Example 3.5. Let X = {a, b, c, d} with topology τ = {ϕ, {a}, {b}, {a, b}, X} rg closed are sets
{ϕ, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, d}, {c, d}, {a, c, d}, {a, b, d}, {a, b, c}, {b, c, d}, X} g ∗ sr closed sets
are {ϕ, {a, c, d}, {b, c, d}, {c, d}, X} Hence A = {c} rg closed but not g ∗ sr closed.
Theorem 3.5. Every g ∗ sr closed set is αg closed
Proof. Let A be g ∗ sr closed set in X. Let us prove that A is αg closed. Let U be open set such that
A ⊆ U . Then A ⊆ U and U is gs open and rcl(A) ⊆ A since A is g ∗ sr closed. But αcl(A) ⊆ rcl(A) ⊆ U
and U is open. Therefore A is αg closed.
Remark 3.5. The converse of the above theorem need not be true in general, as shown in the following example.
Example 3.6. Let X = {a, b, c, d} with topology τ = {ϕ, {a}, {a, b}, {c, d}, {a, c, d}, X} αg closed sets are
{ϕ, {b}, {c}, {d}, {a, b}, {b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d}, {a, b, d}, {a, c, d}, X} g ∗ sr closed sets are
{ϕ, {b}, {a, b}, {c, d}, {b, c, d}}
Hence A = {c} is αg closed but not g ∗ sr closed.
Theorem 3.6. Every g ∗ sr closed set is gp closed.
Proof. Let A be g ∗ srclosed set in X. Then rcl(A) ⊆ U whenever A ⊆ U and U is gs open. To prove
that A is gp closed. Let U be open set such that A ⊆ U . Since every open set is gs open and A is g ∗ sr
closed. We have P cl(A) ⊆ U . Hence pcl(A) ⊆ U . Therefore A is gp closed set in X.
Remark 3.6. The converse of the above theorem need not be true in general, as shown in the following example.
Example 3.7. Let X = {a, b, c, d} with topology τ = {ϕ, {a}, {c}, {d}, {a, c}, {c, d}, {a, c, d}, {a, d}, X}
gp closed sets are {ϕ, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}, X} g ∗ sr closed sets are
{ϕ, {b}, {a, b}, {b, d}, {b, c, d}, X} Hence A = {a, b, c} is gp closed but not g ∗ sr closed.
Theorem 3.7. Every g ∗ sr closed set is gsp closed
Proof. Let A be g ∗ sr closed set in X. Then rcl(A)U and U is gs open. To prove A is gsp closed
in X. Let U be open set such that A ⊆ U . Since every open set is gs open and A is g ∗ sr closed.
Spcl(A) ⊆ scl(A) ⊆ rcl(A) ⊆ U . Hence spcl(A)U . Therefore A is gsp closed.
Remark 3.7. The converse of the above theorem need not be true in general, as shown in the following example.
Example 3.8. Let X = {a, b, c, d} wih topology τ = {ϕ, {a}, {c}, {d}, {a, c}, {ad}, {c, d}, {a, c, d}, X} gsp
closed sets are {ϕ, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, X} g ∗ sr
closed sets are {ϕ, {b}, {a, b}, {b, c, d}, {b, d}, X}
Hence A = {{a}, {c}, {a, d}, {c, d}} are gsp closed but not g ∗ sr closed.
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Theorem 3.8. Every g ∗ sr closed set is gpr closed
Proof. Let A be g ∗ sr closed set in X. Then A ⊆ U and U is gs open. To prove A is gpr closed. Let
U be open set such that A ⊆ U . Since every r-open set is gs open and A is g ∗ sr closed. We have
P cl(A) ⊆ rcl(A) ⊆ U . Therefore A is gpr closed set in X.
Remark 3.8. The converse of the above theorem need not be true in general, as shown in the following example.
Example 3.9. Let X = {a, b, c, d} with topology
τ = {ϕ, {a}, {a, b}, {c, d}, {a, c, d}, X} g ∗ sr closed sets are {ϕ, {b}, {a, b}, {c, d}, {b, c, d}} gpr closed sets
are
{ϕ, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, c, d}, {a, b, d}, {a, b, c}, {b, c, d}, X}.
Hence
A = {{a}, {c}, {d}, {b, c}, {a, c}, {a, d}, {b, d}, {a, b, c}, {a, c, d}, {a, b, d}} are gpr closed but not g ∗ sr
closed.
Theorem 3.9. The finite union of the g ∗ sr closed set is g ∗ sr closed.
Proof. Let A and B be g ∗ sr closed sets in X. Let U be a gs open in X such that AU B ⊆ U . Then
A ⊆ U and B ⊆ U . Since A and B are g ∗ sr closed set rcl(A) ⊆ U and rcl(B) ⊆ U . Hence
rcl(AU B) = rcl(A)U rcl(B) ⊆ U . Therefore AU B is g ∗ sr closed.
Theorem 3.10. The finite intersection of two g ∗ sr closed sets is closed.
Proof. Let A and B be g ∗ sr closed sets in X. Let U be a gs open in X such that A ∩ B ⊆ U . Then
A ⊆ U and B ⊆ U . Since A and B are g ∗ sr closed set rcl(A) ⊆ U and rcl(B) ⊆ U . Hence
rcl(A ∩ B) = rcl(A) ∩ rcl(B) ⊆ U . Therefore A ∩ B is g ∗ sr closed.
Theorem 3.11. Let A and B be subsets such that A ⊆ B ⊆ rcl(A). If A is g ∗ sr closed then B is g ∗ sr closed.
Proof. Let A and B be subsets such that A ⊆ B ⊆ rcl(A) . Suppose that A is g ∗ sr closed. To show
that B is g ∗ sr closed set B ⊆ U and U is gs open in X. Since A ⊆ B and B ⊆ U , we have A ⊆ U .
Hence A ⊆ U and U is gs open in X. Since A is g ∗ sr closed we have rcl(A) ⊆ U . Since B ⊆ rcl(A),
we have rcl(B) ⊆ rcl(rcl(A)) = rcl(A) ⊆ U . Hence rcl(B) ⊆ U . Hence B is g ∗ sr closed.
Theorem 3.12. If x ∈ rcl(A)if f U ∩ A 6= φ for every regular open set U containing x.
Proof. Suppose that x ∈ rcl(A). To show that U ∩ A 6= ϕ for every gs open set U containing x such
that U ∩ A = φ. Then A ⊆ U and U is gs closed set. Since A ⊆ U, rcl(A) ⊆ rcl(U )c . Since x ∈ rcl(A)
We have x ∈ rcl(U )c . Since U is gs closed set. We have x ∈ U c . Hence x ∈
/ U which is contradiction
that x ∈ U . Therefore U ∩ A 6= φ Hence U ∩ A 6= φ for every gs open set U containing x.
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Conversely , suppose that U ∩ A 6= φ. for every open set containing x. To show that x ∈ rcl(A).
Suppose that x ∈
/ rcl(A). Then there exists a gs open set U containing x such that U ∩ A = φ. This is
contradiction to U ∩ A 6= φ. Hence x ∈ rcl(A).
Theorem 3.13. A subset A of X is g ∗ sr-closed set in X iff scl(A) − A contains no non empty gs-closed set
in X.
Proof. Suppose that F is a non empty gs-closed subset of scl(A)-A. Now F ⊆ scl(A) − A. Then
F ⊆ scl(A) ∩ Ac . Therefore F ⊆ scl(A) and F ⊆ Ac . Since F c is gs-open set and A is g ∗ sr-closed ,
scl(A) ⊆ F c . That is F ⊆ scl(A)c . Since F c is gs-open set and A is g ∗ sr-closed , scl(A) ⊆ F c . That
is F ⊆ scl(A)c . Hence F ⊆ scl(A) ∩ [scl(A)]c . That is F = Thus scl(A) − A contains no non empty
gs-closed set.
Conversely, assume that scl(A) − A contains no non empty gs-closed set. Let A ⊆ U , U is gsopen. Suppose that scl(A) is not contained in U . Then scl(A) ∩ U c is a non empty gs closed set and
contained in scl(A) − A Which is a contradiction. Therefore scl(A) ⊆ U and hence A is g ∗ sr-closed
set.
Definition 3.2. An topological space (X, τ ) is called a g ∗ sr compact space if every g ∗ sr covering has a finite
subcover.
Definition 3.3. Let (X, τ ) be a g ∗ sr compact space. If A is g ∗ sr closed subset of X, then A is g ∗ sr compact.
Theorem 3.14. Every closed subset of a g ∗ sr compact. space is g ∗ sr compact.
Proof. Let Y be a closed set in a g ∗ s-I compact space (X, τ ). Therefore, Y is g ∗ s-I compact, since
every closed set is g ∗ sr closed. Given, a covering A of Y by g ∗ sr open sets in X, we can form an
open covering B of X by adjoining to A the single g ∗ sr open set X − Y . ie, B = A ∪ (X − Y ). Since
X is g ∗ sr compact, some finite subcollection of B covers X. If this subcollection contains the set
X − Y , discard X − Y , otherwise leave the subcollection alone. The resulting collection is a finite
subcollection A that covers Y . Hence Y is g ∗ sr compact.
Theorem 3.15. Let (X, τ ) be a compact topological space. If A is g ∗ sr-closed subset of X, then A iscompact.
Proof. Let {Ui } be a open cover of A. Since every open set is gs-open and A is g ∗ sr-closed. We get
scl(A) ⊆ Ui Since a closed subset of a compact space is compact, scl(A) is compact. Therefore there
exist a finite subover say {U1 ∪ U2 ... ∪ Un } of {Ui } for A scl(A). So,A ⊆ scl(A) ⊆ U1 ∪ U2 ∪ ... ∪ Un .
Therefore A is compact
Theorem 3.16. For each x ∈ X, either {x} is gs-closed or {x}c is g ∗ s-closed in X.
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Proof. If {x} is not gs-closed, then the only gs-open set containing {x}c in X. Thus semi closure of
{x}c is contained in X and hence {x}c is g ∗ s-closed in X.
4
g ∗ sr closed maps
In this section,we introduce the concepts of g ∗ sr- closed maps in topological Spaces and study
their properties.
Definition 4.1. A map f : (X, τ ) → (Y, σ) is called g ∗ sr- closed map if the image of each closed set in X is
g ∗ sr- closed in Y .
Theorem 4.1. Every regular closed map is g ∗ sr closed.
Proof. Let F be a closed set in (X, τ ) and f : (X, τ ) → (Y, σ) is regular closed map. Hence f (F ) is
regular closed in Y . As every regular closed set is g ∗ sr closed set in Y . Therefore f (F ) is g ∗ sr closed
in Y . Hence f is g ∗ sr closed map.
Remark 4.1. The converse of the above theorem need not be true in general .
Example 4.1. Every g ∗ sr closed map is not regular closed. Let X = {a, b, c, d}
τ = {ϕ, {a}, {b}, {a, b}, X}
τ c = {X, {b, c, d}, {a, c, d}, {c, d}, ϕ}
σ = {ϕ, {a}, {b}, {a, b}, X}
Y = {a, b, c, d}
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σ c = {X, {b, c, d}, {a, c, d}, {c, d}, ϕ} Let f : (X, τ ) → (Y, σ) be the identity mapping. g ∗ sr-closed sets
of (Y, σ) are {ϕ, X, {c, d}, {a, c, d}, {b, c, d}}. Regular closed sets of (Y, σ) are {ϕ, X, {a, c, d}, {b, c, d}}
consider the closed sets A = {ϕ, X, {c, d}, {a, c, d}, {b, c, d}}. f (A) = f ({c, d}) = {c, d} is g ∗ sr closed in
(Y, σ)f (A) = f ({a, c, d}) = {a, c, d} is g ∗ sr closed in (Y, σ)f (A) = f ({c, d}) = {b, c, d}isg ∗ sr closed in
(Y, σ) therefore f is g ∗ sr closed map. f (A) = f ({c, d}) = {c, d} is g ∗ sr closed, but not regular closed.
Theorem 4.2. Every g ∗ sr closed map is αg closed.
Proof. Let F be a closed set in (X, τ ) and f : (X, τ ) → (Y, σ) be g ∗ sr closed map in (Y, σ). Hence f (F )
is g ∗ sr closed in Y . As every g ∗ sr closed set is αg closed set in Y . Therefore f (F ) is αg closed in Y .
Hence f is αg closed map.
Remark 4.2. The converse of the above theorem need not be true in general.
Example 4.2. Every αg closed map is not g ∗ sr closed. Let X = {a, b, c}
Y = {a, b, c}
τ = {ϕ, {a}, {a, b}, X}
τ c = {X, {b, c}, {c}, ϕ}
σ = {ϕ, {a}, {a, b}, X}
σ c = {X, {b, c}, {c}, ϕ} Let f : (X, τ ) → (Y, σ) be the identity mapping. g ∗ sr -closed sets of (Y, σ)
are {ϕ, X, {b, c}}. αg closed sets of (Y, σ) are {ϕ, X, {b}, {c}, {a, c}, {b, c}} consider the closed sets A =
{ϕ, X, {c}, {b, c}}. f (A) = f ({b, c}) = {b, c} is g ∗ sr closed in (Y, σ) But f (A) = f {a, c} is αg closed but
not g ∗ sr closed set.
Theorem 4.3. Every g ∗ sr closed map is gp closed.
Proof. Let F be a closed set in (X, τ ) and f : (X, τ ) → (Y, σ) be g ∗ sr closed map in (Y, σ). Hence f(F)
is g ∗ sr closed in Y . As every g ∗ sr closed set is gp closed set in Y . Therefore f(F) is gp closed in Y .
Hence f is gp closed map.
Remark 4.3. The converse of the above theorem need not be true in general.
Example 4.3. Every gp closed map is not g ∗ sr closed. Let X = {a, b, c, d}
Y = {a, b, c, d}
τ = {ϕ, {a}, {c}, {d}, {a, d}, {a, c}, {c, d}, {a, c, d}X}
τ c = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ}
σ = {ϕ, {a}, {c}, {d}, {a, d}, {a, c}, {c, d}, {a, c, d}X}
σ c = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ} Let f : (X, τ ) → (Y, σ) be the identity
mapping. g ∗ sr - closed sets of (Y, σ) are {ϕ, X, {b}, {a, b}, {b, d}, {b, c, d}}. gp closed sets of (Y, σ) are
{ϕ, X, {b, c, d}, {a, b, d}, {b, d}, {b, c}, {a, b}, {b}, ϕ} consider the closed sets
A = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ}f (A) = f {b, c} is gp closed but not g ∗ sr
closed set.
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Theorem 4.4. Every g ∗ sr closed map is rg closed.
Proof. Let F be a closed set in (X, τ ) and f : (X, τ ) → (Y, σ) be g ∗ sr closed map in (Y, σ). Hence f (F )
is g ∗ sr closed in Y . As every g ∗ sr closed set is rg closed set in Y . Therefore f(F) is rg closed in Y .
Hence f is rg closed map
Remark 4.4. The converse of the above theorem need not be true in general.
Example 4.4. Every rg closed map is not g ∗ sr closed. Let X = {a, b, c} Y = {a, b, c}
τ = {ϕ, {a}, {a, b}, X}
τ c = {X, {b, c}, {c}, ϕ}
σ = {ϕ, {a}, {a, b}, X}
σ c = {X, {b, c}, {c}, ϕ} Let f : (X, τ ) → (Y, σ) be the identity mapping. g ∗ sr- closed sets of (Y, σ) are
{ϕ, X, {b, c}}. rg closed sets of (Y, σ) are {ϕ, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}. Consider the closed sets
A = {ϕ, X, {c}, {b, c}}. f (A) = f ({b, c}) = {b, c} is g ∗ sr closed in (Y, σ)f (A) = f {a, b} is rg closed but
not g ∗ sr closed set.
Theorem 4.5. Every g ∗ sr closed map is gsp closed.
Proof. Let F be a closed set in (X, τ ) and f : (X, τ ) → (Y, σ) be g ∗ sr closed map in (Y, σ). Hence f (F )
is g ∗ sr closed in Y . As every g ∗ sr closed set is gsp closed set in Y . Therefore f (F ) is gsp closed in Y .
Hence f is gsp closed map.
Remark 4.5. The converse of the above theorem need not be true in general.
Example 4.5. Every gsp closed map is not g ∗ sr closed. Let X = {a, b, c, d}
Y = {a, b, c, d}
τ = {ϕ, {a}, {c}, {d}, {a, d}, {a, c}, {c, d}, {a, c, d}X}
τ c = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, a, b}, {b}, ϕ}
σ = {ϕ, {a}, {c}, {d}, {a, d}, {a, c}, {c, d}, {a, c, d}X}
σ c = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ}
Let f
:
(X, τ )
→
(Y, σ) be the identity mapping.
{ϕ, X, {b}, {a, b}, {b, d}, {b, c, d}}.
gsp
closed
g ∗ sr - closed sets of (Y, σ) are
sets
of
(Y, σ)
{ϕ, X, {a}, {b}, {z}, {d}, {a, c}, {a, d}, {c, d}, {b, c, d}, {a, b, d}, {b, d}, {b, c}, {a, b}, {a, b, c}}
are
consider
the closed sets A = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ} f (A) = f {a, b, c} is gsp
closed but not g ∗ sr closed set.
Theorem 4.6. A map f : x → y is g ∗ sr -closed if and only if for each subset S of y and for each open set U
Containing f −1 (s) there is a g ∗ s -open set V of y such that S ⊆ V and ⊆ (V )
Proof. Suppose f is g ∗ s -closed. Let S be a sub set of Y and is an open set of X such that f −1 (V ) ⊆ (U )
Conversely suppose that F is a closed set in X. Then f −1 (y − f (F )) = X − f and X − F is open.
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By hypothesis, there is a g ∗ s -open set V of Y such that f −1 (F ) ⊆ U V and f −1 (v) ⊆ X − F is open.
Therefore y − v ⊆ f (F )f (x− ⊆ y − v) which implies f (F ) = y − v. Since y − v is g ∗ s -closed, f(F) is
g ∗ s -closed and thus f is g ∗ s -closed map.
Theorem 4.7. If f : X → Y is closed and h : Y → Z is g ∗ sr closed then h ◦ f : X → Z is g ∗ sr closed
Proof. Let f : X → Y is closed map and h : Y → Z is a g ∗ sr closed map. Let f be any closed set in X.
Since f : X → Y is a closed f(F) is closed in Y . Since h : Y → Z is a g ∗ sr closed map. Then h(f (F ))
is g ∗ sr closed set in Z. Hence hf : X → Z is a g ∗ sr closed map.
5
g ∗ srontinuous functions
In this section,we introduce the concept of g ∗ sr- continuous functions in topological Spaces and
study their properties.
Definition 5.1. A map f : X → Y from a topological space X into a topological space Y is called g ∗ sr
continuous if the inverse image of every closed set in Y is g ∗ sr closed in X.
Theorem 5.1. If a map f : X → Y is regular continuous then it is g ∗ sr continuous.
Proof. Let f : X → Y be regular continuous. Let F be any closed set in Y . The inverse image of
f −1 (F ) is regular closed in X. Since every regular closed set is g ∗ sr closed . Hence f −1 (F ) is g ∗ sr
closed in X. Hence f is g ∗ sr continuous.
Remark 5.1. The converse of the above theorem need not be true in general as shown in the following example.
D. Sreeja et al. / Generalized star semi...
53
Example 5.1. If a map f : X → Y is g ∗ sr continuous but not regular continuous. Let X = {a, b, c, d}
Y = {a, b, c, d}
τ = {ϕ, {a}, {b}, {a, b}, X}
τ c = {X, {b, c, d}, {a, c, d}, {c, d}, ϕ}
σ = {ϕ, {a}, {b}, {a, b}, X}
σ c = {X, {b, c, d}, {a, c, d}, {c, d}, ϕ} Let f : (X, τ ) → (Y, σ) be the inverse mapping. g ∗ sr- closed sets
of (X, τ ) are {ϕ, X, {c, d}, {a, c, d}, {b, c, d}}. Regular closed sets of (X, τ ) are {ϕ, X, {a, c, d}, {b, c, d}}.
Let F be a closed set in (Y, σ). Let F = {c, d}, f −1 (F ) = f −1 (c, d) = {c, d} is g ∗ sr closed set in (X, τ ).
Therefore f is g ∗ sr continuous then f −1 (F ) = f −1 (c, d) is not regular continuous. Hence here f is g ∗ sr
continuous but not regular continuous.
Theorem 5.2. If a map f : X → Y is g ∗ sr continuous then it is gpr continuous.
Proof. Let f : X → Y be g ∗ sr continuous. Let F be any closed set in Y . The inverse image of f −1 (F )
is g ∗ sr closed in X. Since every g ∗ sr closed set is gpr closed . Hence f −1 (F ) is gpr closed in X.
Hence f is gpr continuous.
Remark 5.2. The converse of the above theorem need not be true in general as shown in the following example.
Example 5.2. If a map f : X → Y is gpr continuous but not g ∗ sr continuous. Let X = {a, b, c}
Y = {a, b, c}
τ = {ϕ, {a}, {a, b}, X}
τ c = {X, {b, c}, {c}, ϕ}
σ = {ϕ, {a}, {a, b}, X}
σ c = {X, {b, c}, {c}, ϕ} Let f : (X, τ ) → (Y, σ) be the inverse mapping. g ∗ sr- closed sets of (X, τ ) are
{ϕ, X, {b, c}} gpr closed sets of (X, τ ) are {ϕ, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}} Let F be a closed set in
(Y, σ). Let F = {a, b}, f −1 (F ) = f −1 (a, b) = {a, b} is gpr closed set in (X, τ ). Therefore f is gpr continuous
then f −1 (F ) = f −1 (a, b) is not g ∗ sr continuous. Hence here f is gpr continuous but not g ∗ sr continuous.
Theorem 5.3. If a map f : X → Y is g ∗ sr continuous then it is αg continuous.
Proof. Let f : X → Y be g ∗ sr continuous. Let F be any closed set in Y . The inverse image of f −1 (F )
is g ∗ sr closed in X. Since every g ∗ sr closed set is αg closed . Hence f −1 (F ) is αg closed in X. Hence
f is αg continuous.
Remark 5.3. The converse of the above theorem need not be true in general as shown in the following example.
Example 5.3. If a map f : X → Y is αg continuous but not g ∗ sr continuous. Let X = {a, b, c}
Y = {a, b, c}
τ = {ϕ, {a}, {a, b}, X}
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τ c = {X, {b, c}, {c}, ϕ}
σ = {ϕ, {a}, {a, b}, X}
σ c = {X, {b, c}, {c}, ϕ} Let f : (X, τ ) → (Y, σ) be the inverse mapping. g ∗ sr -closed sets of (X, τ ) are
{ϕ, X, {b, c}. αg closed sets of (X, τ ) are {ϕ, X, {a}, {b}, {c}, {a, b}, {a, c, }, {b, c}} Let F be a closed set in
(Y, σ). Let F = {a, b}, f −1 (F ) = f −1 (a, b) = {a, b} is αg closed set in (X, τ ). Therefore f is αg continuous
then f −1 (F ) = f −1 (a, b) is not g ∗ sr continuous. Hence here f is αg continuous but not g ∗ sr continuous.
Theorem 5.4. If a map f : X → Y is g ∗ sr continuous then it is rg continuous.
Proof. Let f : X → Y be g ∗ sr continuous. Let F be any closed set in Y . The inverse image of f −1 (F )
is g ∗ sr closed in X. Since every g ∗ sr closed set is rg closed . Hence f −1 (F ) is rg closed in X. Hence
f is rg continuous.
Remark 5.4. The converse of the above theorem need not be true in general as shown in the following example.
Example 5.4. If a map f : X → Y is rg continuous but not g ∗ sr continuous. Let X = {a, b, c}
Y = {a, b, c}
τ = {ϕ, {a}, {a, b}, X}
τ c = {X, {b, c}, {c}, ϕ}
σ = {ϕ, {a}, {a, b}, X}
σ c = {X, {b, c}, {c}, ϕ} Let f : (X, τ ) → (Y, σ) be the inverse mapping. g ∗ sr- closed sets of (X, τ ) are
{ϕ, X, {b, c}}. rg closed sets of (X, τ ) are {ϕ, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}} Let F be a closed set in
(Y, σ) Let F = {a, b}, f −1 (F ) = f −1 (a, b) = {a, b} is rg closed set in (X, τ ). Therefore f is rg continuous
then f −1 (F ) = f −1 (a, b) is not g ∗ sr continuous. Hence here f is rg continuous but not g ∗ sr continuous.
Theorem 5.5. If a map f : X → Y is g ∗ sr continuous then it is gp continuous.
Proof. Let f : X → Y be g ∗ sr continuous. Let F be any closed set in Y . The inverse image of f −1 (F )
is g ∗ sr closed in X. Since every g ∗ sr closed set is gp closed . Hence f −1 (F ) is gp closed in X. Hence
f is gp continuous.
Remark 5.5. The converse of the above theorem need not be true in general as shown in the following example.
Example 5.5. If a map f : X → Y is gp continuous but not g ∗ sr continuous. Let X = {a, b, c, d}
Y = {a, b, c, d}
τ = {ϕ, {a}, {c}, {d}, {a, d}, {a, c}, {c, d}, {a, c, d}X}
τ c = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ}
σ = {ϕ, {a}, {c}, {d}, {a, d}, {a, c}, {c, d}, {a, c, d}X}
σ c = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ} Let f : (X, τ ) → (Y, σ) be the inverse
mapping. g ∗ sr - closed sets of (X, τ ) are {ϕ, X, {b}, {a, b}, {b, d}, {b, c, d}}. gp closed sets of (X, τ ) are
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D. Sreeja et al. / Generalized star semi...
{ϕ, X, {b, c, d}, {a, b, d}, {b, d}, {b, c}, {a, b}, {b}, ϕ}.
Let
F be
a closed set
in (Y, σ) Let
F = {a, b, c}f −1 (F ) = f −1 (a, b, c) = {a, b, c} is rg closed set in (X, τ ). Therefore f is gp continuous then
f −1 (F ) = f −1 (a, b, c) is not g ∗ sr continuous. Hence here f is gp continuous but not g ∗ sr continuous.
Theorem 5.6. If a map f : X → Y is g ∗ sr continuous then it is gs continuous.
Proof. Let f : X → Y be g ∗ sr continuous. Let F be any closed set in Y . The inverse image of f −1 (F )
is g ∗ sr closed in X. Since every g ∗ sr closed set is gs closed . Hence f −1 (F ) is gs closed in X. Hence
f is gs continuous.
Remark 5.6. The converse of the above theorem need not be true in general as shown in the following example.
Example 5.6. If a map f
X = {a, b, c, d}
:
X
→
Y is gs continuous but not g ∗ sr continuous.
Let
Y = {a, b, c, d}
τ = {ϕ, {b}, {d}, {b, d}, {a, b}, {b, c}, {a, b, c}, {b, c, d}, {a, b, d}, X}
σ = {ϕ, {a}, {c}, {d}, {a, d}, {a, c}, {c, d}, {a, c, d}, X}
σ = {X, {b, c, d}, {a, b, d}, {a, b, c}, {b, d}, {b, c}, {a, b}, {b}, ϕ} Let f : (X, τ ) → (Y, σ) be the inverse
mapping. g ∗ sr - closed sets of (X, τ ) are {ϕ, {d}, {a, c, d}, X}. gs closed sets of (X, τ ) are
{ϕ, {a}, {c}, {d}, {c, d}, {a, c}, {a, d}, {a, c, d}, {a, b, c}, X}.
Let F be a closed set in (Y, σ) Let
F = {a, b, c}f −1 (F ) = f −1 (a, b, c) = {a, b, c} is gs closed set in (X, τ ). Therefore f is gs continuous then
f −1 (F ) = f −1 (a, b, c) is not g ∗ sr continuous. Hence here f is gs continuous but not g ∗ sr continuous.
References
[1] D.Andrijevic, Mat. Vesnik, 38(1986),24-32.
[2] S.P.Arya and T.Nour, Indian J.Pure Appl. Math., 21(1990),717-19.
[3] P.Bhattacharyaa and B.K.Lahiri, Indian J. Math., 29(1987),375-82.
[4] K.Balachandran and Arokia Rani, On generalized preclosed sets, preprint.
[5] W.Dunham, Kyungpook Math J., 17(1977),161-69.
[6] J .Dontchev, Mem. Fac. Sci. Kochi Univ. Ser. A, Math.16(1995),35-48.
[7] D.S. Jankovic, Ann. Soc.Sci. Bruxelles Ser.I., 97(1983), 59-72.
[8] D.S. Jankovic and I.L.Reily, Indian J.Pure Appl. Math., 16(1985), 957-64.
[9] D.S. Jankovic, Acta Math. Hungar., 46(1985), 83-92.
[10] J.E. Joseph and M.K.Kwack, Proc. Am. Math. Soc., 80(1980),341-48.
56
D. Sreeja et al. / Generalized star semi...
[11] N.Levine, Math. Monthly, 70(1963),36-41.
[12] N.Levine, Rend. Circ. Mat. Palermo, 19(2)(1970),89-96.
[13] A.S Mshour, M.E Abd El-Monsef and S.N. El-Deeb, Proc. Math. Phys.Soc. Egypt, 53(1982),47-53.
[14] H.Maki, R.Devi and K.Balachandran, Bull, Fukuoka Univ. Ed. Part III ,42(1993),13-21.
[15] H.Maki, R.Devi and K.Balachandran, Mem. Fac. Sci. Kochi Univ. Ser. A, Math., 15 (1994),51- 63.
[16] O.Nijastad , Pacific J. Math., 15 (1965),961-70.
[17] T.Noiri, J. Austral. Math. Soc,(Series A), 51(1991),300-304.
[18] N.Palaniappan and K.C. Rao, Kyungpook Math, J., 33(1993),211-19.
Received: May 15, 2015; Accepted: June 23, 2015
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