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Ultra Scientist Vol. 28(3)A, 162-171 (2016).
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ISSN 2231-3478 (Print)
2319-8052 (Online)
Investigation on Weak form of Generalized Closed
sets in Ideal Topological Spaces
G.JAYAPARTHASARATHY
Department of Mathematics
St. Jude’s College, Thoothoor Kanya Kumari-62 176,
Tamil Nadu (India)
E-mail : [email protected]
(Acceptance Date 11th July, 2016)
Abstract
This paper proposes a new class of closed sets in ideal
spacescalled
-closed set and shows that this class of closed sets
lies betweenthe class of
-closed sets and the class of
-closed
sets. Furthermore, it shows that the necessary and sufficient conditions
for a set to be a
-closed set and investigatesthe properties of
-
open sets in idealtopological spaces.
Key words:
semi-*-closed set,
,
-closed set,
-closed set,
-closed sets,
-closed sets.
2010 Mathematics subject classification: 54A05, 54D15.
1. Introduction
The concept of ideals in topological
spaces was initiated byKuratowski 7 and
Vaidyanathaswamy14. In 1990, Jankovic and
Hamlett4 also studied the properties of ideal
topological spaces. After that -closedset
was introduced by Noiri et al.6,11. Also Khan
and Noiri5 introducedand studied the properties
of
-closed sets in ideal topological spaces.
Weakform of open sets called semi-open sets
and also the first step of generalizingclosed
sets was done by Levine 9,10 . Recently,
Jayaparthasarathy et al. 8 introduced a new
class of closed sets called
-closed sets and
its propertiesalso discussed. In this paper we
introduce and characterize the properties of
-closed set, and this class of closed sets
G. Jayaparthasarathy
163
lies between the classes of
the class of
-closedsets and
vi)
-closed set13 if
whenever
and is #gs-openin (X, τ).
vii)
-closed set8 if
-closed sets. We also introduce
-open sets andprove the necessary and
sufficient conditions of
-open sets. Further,,
weinvestigate the relationship between
closed sets and other existing sets inideal
topological spaces.
2. Preliminaries :
This section discuss some basic
properties about ideal topological spaces and
weak form of open sets in topological spaces
which are useful in sequel.
Definition 2.1 9 Let (X, τ) be a
topological space. A subset A of X is called
semi-open if
. The complement
of semi-open set iscalled semi-closed.
Definition 2.2 A subset A of a
topological space (X, τ) is called a
i) generalized closed (briefly g-closed) set9
if
wheneverand
and
U is open in (X, τ).
ii) semi-generalized closed (briefly sg-closed)
set3 if
whenever
and
is semi-open in (X, τ).
-closed set (= ω-closed)12 if
whenever
and issemi-open in
(X, τ).
iv)
-closed set13 if
whenever
iii)
and is -open in(X, τ).
v) #g-semi-closed set (briefly #gs-closed)13
if
whenever
and
is
-open in (X, τ).
and
is
whenever
-openin (X, τ).
The complement of the above mentioned
sets are called their respective generalized
open sets.
Definition 2.37,14 An idealIon a topological space (X, τ) is a non-emptycollection
of subsets ofXsatisfying the following two
conditions:
(i) If AI and BA, then BI
(ii) If AI and BI, then ABI.
Let (X, τ) be a topological space
andIan ideal of subsets of X. An idealtopological
space is a topological space (X, τ)with an
idealIonXand is denoted by (X, τ, I).
Definition 2.41,4 For a subset A of X,
(i) A*(I, ) = {xX: UAI for every U
τ(X, x)} is called the local function of A
with respect to ideal I and topology τ,
where τ (X, x) ={Uτ: xU}.
(ii)
(I, τ) ={x  X: U A I for every
USO(X, x)} is called the semi-local
function of A with respect to ideal I and
topology τ, where SO(X,x)={USO(X):
xU}.
In 1,4 very ideal topological space (X,
τ, I), there exists a topology τ* finer than τ,
defined by τ*={UX:Cl*(XU)=XU} which
is generated by the base β(I, τ) ={UJ: Uτ
164
Investigation on Weak form of---Topological Spaces.
andJI} and Cl*(A) =AA* is a Kuratowski
closure operator for the topology τ*. A subset
A of an ideal topological space (X, τ, I) is said
to be *-closed4 if A*A and Int*(A) denote
the interior of the set A in (X, τ*, I). Also we
5
define
a topology
by
(A) A∪
and there exists
(I) finer than τ and τ*, defined by
(I) ={UX:
(XU) =XU} which is
generated by the base s(I, τ) ={UJ: Uτ
and JI}. A subset A of an ideal topological
space (X, τ, I) is said to be semi-*-closed5 if
A and is said to be *-semi dense5 if A
=
=
.
Also note that1 =sCl( =
)A* (A) and
(A) denote the interior of the set A in (X,
Theorem 2.8 5 Let AandBbe two
subsets of an ideal space (X, τ, I). Then
.
3. Weak open sets in ideal topological
spaces :
This section introduces a weak form
of closed sets in ideal spacesand investigates
some properties of it.
Definition 3.1 A subsetAof an ideal
topological space (X, τ, I) is saidto be
closed if
=
  U whenever
and
-open in X. The complement of
set is called
is
-closed
-open set.
,I).
Definition 2.5 A subsetAof an ideal
topological space (X, τ, I) is said to be
(i) Igeneralized closed (briefly Ig-closed)6 if
whenever
and is open
in X.
(ii) semi generalized I (briefly sgI)-closed5 if
=
 U whenever
and
is semi-
open in X.
(iii)
-closed2 if
and
whenever
is semi-open in X.
Theorem 2.65 Let {Aa : aΩ} be a
locally finite family of sets of an ideal space
=(
(X, τ, I). Then
Example 3.2 Let X = {a, b, c, d}, τ
O(X) ={, X,{a},{b, c},{a, b, c}} and I={,
{a}}. Then
-closed sets are , X, {a}, {d},
{a, d}, {b, c},{b, d},{c, d},{a, b, c}, {a, b, d},
{a, c, d}and{b, c, d}.
Remark 3.3 In an ideal topological
space (X, τ, I), then the following are trivially
true.
(i) If =∅ for every A X, then A is
closed.
=is an
(ii)
-closed for every A X.
Theorem 3.4 Let (X, τ, I) be an ideal
topological space. Then the following are true.
.
(i) Every
Theorem 2.75 Let (X, τ, I) be an ideal
space and
. If
, then
and Bis *-semi dense in itself.
-closed set is
-closed, but not
converse.
(ii) Every -closed set is
-closed, but not
converse.
G. Jayaparthasarathy
165
(iii) Every *-closed set is
-closed, but not
converse.
(iv) Every semi-*-closed set is
-closed, but
not converse.
(v) Every
-closed is
, but not converse.
(vi) If
is an element of I, then A is
-closed.
Proof :
(i) Let A be an
(ii)
(iii)
(iv)
(v)
(vi)
=∅⊆A⊆U. Hence A is
closed and
-closed set.
Theorem 3.6 Let (X, τ, I) be an ideal
topological space and AX. Then the following
statements are equivalent:
(i) A is
-closed.
for every
(iii) For all x ∈
-closed.
-closed set, but it is not
-open set U
})∩A≠∅.
(A), sCl({x})
(iv)
(A)A contains no non empty -closed
set.
=
(v)
A contains no non empty
-closed
set.
Proof (i)  (ii): Let A be an
Then
=
U whenever
-closed set.
and
is
-
open in X and implies
whenever
and
is
-open in X.
(ii)  (iii): Let
and suppose
Then
where
,
is semi-open set. Since
every semi-openset is
-open and by
hypothesis,
. This
contradicts the fact that
. Hence,
-
-closed. Again consider X={a,
b, c},τ O(X) ={∅ , X,{a}} and I={∅ ,{a},{b},
{a, b}}. Then {a, c} is
-closed set, but it is
not *-closed. Also consider X = {a, b, c, d},
τ O(X) = {∅ , X,{a},{b, c},{a, b, c}} and
I ={∅ ,{a}}. Then {a, b, d} is
-
containing A.
-closed set and let
Example 3.5 Consider X={a, b, c}, τ
O(X) ={∅ , X,{a},{b},{a, b}}and I={∅ ,{a}}.
Then {b} is
closed set, but it is not
(ii)
where U is open set. Since
and every open set is
-open, then A is
-closed.
Proof is similar to part (i).
Let A be a*-closed set, then A*A. Let
where
is
-open. Hence
, whenever
and is
open.
Proof is trivially by part (i) and part (iii).
Let A be an
-closed and
where
is semi-open. Since every semi-open
set is
-open, then we have,
,
whenever
and is semi-open.
Let
where is
-open. If AI,
then
but it is not semi-*-closed. Also {a, b} is
-closed set,
(iii)(iv): Suppose
F is a
, where
-closed set containinga pointx. Since
and {x}F, we have sCl({x}F
and
and by hypothesis,
is a contradiction.
. Since
This
166
Investigation on Weak form of---Topological Spaces.
every semi-openset is
(iv)(v): Since
-open and by hypothesis,
,this contradictsthe
Therefore
empty
contains no non
F is a
and
Therefore
be a
is a
Therefore
.
and {x}F, we have sCl({x})
F and
. Since
and by hypothesis,
. Therefore,
. Since
, where
-closed set containinga point x. Since
-open.
and
set, so
contained in
and hence
Hence
(iii)(iv): Suppose
-closed set.
(v)(i): Let
fact that
is semi-closed
-closed set
This
is a contradiction.
(iv)(v): Assume that
F is
, where
-closed set and
This gives
This contr adicts the
Theorem 3.7 Let (X, τ, I) be an ideal
topological space and
. Then the following
statements are equivalent:
and U be a
−A. Since
set, so
(iv)
contains no non empty
-
closed set.
is semi-closed
-closed set conta-
−A. Therefore
 U.
-closed
set.
Theorem 3.8 Let (X, τ, I) be an ideal
topological space. Then every
-closed and
Proof (i)(ii): Let A be an
whenever
-closed set.
and
is
-open in X. By Definition 2.4, we have the
desired result.
(ii)(iii): Let
ined in
is a
and hence
contains no non empty
Then
-open.
. Therefore
-open set
(iii) For all
where
(v)(i): Let
Therefore XUXA and
(i) A is
-closed.
(ii)
for every
U containing A.
(v)
hypothesis.
and suppose
. Then
,
is semi-open set. Since
-open set is semi-*-closed.
Proof By hypothesis,
A  A and A is
*-closed.
 A whenever
-open set. Hence A is semi-
Theorem 3.9 Let
locally finite family of
be a
-closed sets of an
G. Jayaparthasarathy
167
ideal topological space (X, τ, I). Then Ui Ai
is
-closedset.
Proof If A is
theorem 3.7,
N=A*s -A
-closed, then by
contains no non-empty
-closed set. If
Proof Let Ui Ai  U where U is
-open. Since each Ai is
for eachi i  Ω,
U and Ui
hence Ui Ai is
U and
-closed set.
Conversely, suppose A=FN where F is semi-
Example 3.10 Intersection of two
-closed sets need not be
-closed.
Consider X={a, b, c, d}, τ O(X) = {∅, X,{a},{b,
c}, {a, b, c}} and I ={∅ ,{a}}. Then {b, c}
-closed sets, but {b, c} 
{c, d} = {c} is not a
-closed set.
Theorem 3.11 LetAbe an
set andB be
-closedset.
Proof Let U be a
-open subset of
X containing AB. Then AU(X-B). Since
A is
-closed, then
and
Now A  F and
so
then
and
By hypothesis, since
is -closed,
and so
Hence A is
-closed.
.
, because B is
-closed.
Proof Since A is
-closed, then by
sCl(
)BsCl(
)B contains no nonempty
set, by theorem 3.7. Hence B is
Theorem 3.12 Let (X, τ, I) be an ideal
-
closed if and only if A=FN where F is semi*-closed and N contains no nonempty
closed set.
closed, then B is
-closed set. Since sCl(
-
-closed.
topological space and AX. Then A is
Theorem 3.13 Let (X, τ, I) be an ideal
topological space and A and B be subsets of
X such that
and A is
-
theorem 3.7, sCl(A*s)A contains no nonempty
. By theorem 2.8,
is
*-closed and Ncontains no nonempty -closed
set. Let U be a
-open set such that AU.
Then FN  U implies
.
-closed
-closed set of an ideal topological
space (X, τ, I). Then AB is
closed. Thus
* -closed such that
-closed set, then
U. By theorem 2.6, (U i
and {c, d}are
then F is semi-
)A,
-closed
-closed.
Theorem 3.14 Let (X, τ, I) be an ideal
topological space. Then every
-closed, but not conversely..
-closed is
Proof Assume A is
-closed and
168
Investigation on Weak form of---Topological Spaces.
since
, then
whenever A  U and U is
open. By theorem 3.7, we get thedesired result.
Example 3.15 Consider X={a, b, c},
τ O(X) = {∅ , X, {a},{b},{a, b}} and I={∅ ,{a}}.
Then {a, b} is
closed.
-closed set, but it is not
Proof SupposeA is
 sCl(
3.13, we get B is
-closed. Since A B 
, then
=
Theorem 3.19 Let (X, τ, I) be an ideal
topological space and A  X. Then A is
-
closed if and only if
-
-closed.
Proof Assume A is
-closed and
be a
Ais *-semi dense initself, A 
A contains no non-empty
and implies
)U whenever A  U and U
-open. Hence Ais
-closed.
Theorem 3.17 Let (X, τ, I) be an ideal
space where I={∅ }. Then A is
-closed if
and only if A is
-closed.
Proof Since I={∅ },
This implies Ais * -semi dense and so it is
-closed. Converse is theorem 3.14.
-closed. Let U
-open set such that
Then
is
is
closed.
*-semi dense set. Let A  U where U is
open, then theorem 3.7 (ii), sCl( )U. Since
sCl(A)sCl(
), by Theorem
. Then Aand B are * -semi
dense and both are
-closed, by theorem
3.16.
-
Theorem 3.16 Let (X, τ, I) be an ideal
topological space and A  X is
-closed and
*-semi dense set in itself, then A is
Since
.
))=
Since A is
-closed, by Theorem 3.7 (v)
A.

-closed set. This
implies
. Thus X is the
only
-open set containing
Thus
Thus
is
-closed. Conversely, assume
is
-closed and let U be a
such that A  U. Let F be any
such that
. Since
we have
and since X F is
-open set
-closed set
-open. Therefore,
Theorem 3.18 Let (X, τ, I) be an ideal
topological space and A and B be subsets of
X such that A B and A is
-closed.
and hence
Then A and B are
empty
-closed set. By theorem 3.7(v), A is
-closed.
-closed.
Proof Let A and B be subsets of X
such that A B A*s and A is
-closed.
implies
But
. Thus
contains no non
Theorem 3.20 Let (X, τ, I) be an ideal
G. Jayaparthasarathy
169
space and A  X. Then
-closed if and only if
is
is
-closed.
Proof Proof is trivially from the fact that
Proof
(i)(ii): If A is semi-*-closed, then
and so
Hence
is a
-closed set.
(ii)(iii): Since
Theorem 3.21 Let (X, τ, I) be an ideal
topological space and if every
-open set in
X is semi-*-closed. Then every subset of X is
-closed.
Proof Assume that every
-open
set is semi-*-closed and let A X and U be
-open set such that A U. Then
Hence Ais
-closed.
Theorem 3.22 Let A be a
-closed
set and B be
-closed set of an ideal
topological space (X, τ, I). Then A  B is
closedset.
so
is
and
-closed set.
(iii)(i): If
is a
-closed set, since
A is
-closed set, then by theorem 3.6,
=∅ and so A is semi-*-closed.
4. Properties of
-open sets :
This section is to investigate the
properties of
-open sets in idealtopological
spaces.
Theorem 4.1 Let (X, , I) be an ideal
topological space and A  B. Then is
-open
if and only if
whenever F is
-closed and F  A.
Proof Let U be a
X containing A  B. Then
Since A is
-closed, then
and
-open subset of
.
. By theorem 2.7,
Proof Suppose A is
-open. If F is
-closed and F  A, then XF  XF and
so
. Then
because B is
-closed. Thus A  B is
-closed.
Theorem 3.23 Let (X, τ, I) be an ideal
topological space and AX be an
-closed
set, then the following are equivalent.
(i) Ais a semi-*-closed set.
(ii)
(iii) A*A is a
is a
-closed set.
-closed set.
and so
. Conversely, let U be a
-open set such that
. Then
and so
. Therefore
and implies
Therefore, XA is
-closed. Hence A is
open.
Theorem 4.2 Let (X, , I) be an ideal
170
Investigation on Weak form of---Topological Spaces.
topological space and
-open
, then B is
and
is
. If A is
-open.
Proof Since A is
-open, then XA
-closed. By theorem 3.6,
contains no nonempty
set. Since
-closed
which implies
that
and XF is
-open. Since
and
so
it follows that
. Since
. Hence A is
-closed.
(iii) (i): Since
and so
Hence B is
-open.
Conclusion
Theorem 4.3 Let (X, τ, I) be an ideal
topological space and A  X, then the followingare equivalent.
(i) A is
-closed.
(ii) A
is
(iii)
is
-closed.
This paper introduces new weak
closed sets in ideal topological spaces and
investigate its basic properties. This set can
be extended in different research fields such as
extended topology, Fuzzy topology, Intuitionistic
topology, digital topology etc.
-open.
References
Proof (i)(ii): Suppose A is
If U is any
-closed.
-open set such that
, then
. Since A is
3.7(v),we get
-closed, by theorem
Therefore
and implies
Hence
is
(ii)(iii): Suppose
closed. If F is any
is
-closed set such that
, then
and
-closed.
and
. Therefore
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