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Journal of Computer and Mathematical Sciences, Vol.7(4), 192-202, April 2016
(An International Research Journal), www.compmath-journal.org
ISSN 0976-5727 (Print)
ISSN 2319-8133 (Online)
On (RW)* Closed Sets in Topological Spaces
R. S. Wali and Bajirao P Kamble
Department of Mathematics,
Bhandari and Rathi College Guledagudd-587203,Karnataka, INDIA.
Department of Mathematics,
Rani Channamma University, Belagavi-591156, Karnataka, INDIA.
(Received on: April 15, 2016)
ABSTRACT
The aim of this paper is to introduce and study the new class of closed sets
called (rw)*-closed sets in topological spaces. A subset A of a topological space (X, 𝜏)
is called (rw)*-closed , if U contains closure of interior of A whenever U contains A
and U is a rw-open in (X, 𝜏). This new class of sets lies between the class of all #rg
closed sets and the class of weakly generalized closed sets and also we study the
fundamental properties of this class of sets.
Mathematics subject classification (2010): 54A05.
Keywords: Topological spaces, generalized closed sets, rw-closed sets, (rw)* - closed
sets.
1. INTRODUCTION
In a topological spaces the concept of closed sets plays on important rule. The
generalization of closed sets has been studied in different ways in previous year by many
topologiests leading to several new ideas. In 1970 Levine15 first introduced the concept of
generalized closed (briefly g-closed) sets were defined and investigated. Regular open sets,
strong regular open sets and rw-open sets have been introduced and investigated by Stone21,
Tong26 and Benchalli and R S Wali4 respectively.
Levin14,15, Biswas7, Cameron8, Sundram and Sheikjohn29, Bhattacharya and Lahiri5,
Nagaveni18, Pushpalatha19, Palaniappan and Rao27 and Maki, Devi and Balachandran28
introduced and investigated semi-open sets, generalized closed sets, regular semi-open sets,
w-closed sets, semi-generalized closed sets, wg-closed sets, strongly generalized closed sets,
regular generalized closed sets and α-generalised closed sets respectively. We introduced a
new class of closed sets called(rw)* -closed sets which is properly placed in between the class
of #rg- closed sets and wg-closed sets.
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2. PRELIMINARIES
Throughout this paper space (X, 𝜏) and (Y, σ) (or simply X and Y) always denote
topological space on which no separation axioms are assumed unless explicitly stated. For a
subset A of a space X , Cl(A), Int(A) and Ac denote the Closure of A, Interior of A and
Compliment of A in X respectively.
Definition 2.1 : A subset A of a topological space (X, 𝜏) is called a
(1) Semi-open set [14] if A cl(int(A)) and semi-closed set if int(cl(A))  A.
(2) Pre-open set [16] if A  int(cl(A)) and pre-closed set if cl(int(A))  A.
(3) α-open set [24] if A int(cl(int(A))) and α -closed set if cl(int(cl(A)))  A.
(4) semi-pre open set [2] (= β-open[1] if Acl(int(cl(A)))) and a semi-pre closed set (=βclosed ) if int(cl(int(A))) A.
(5) Regular open set [21] if A = int(clA)) and a regular closed set if A = cl(int(A)).
(6) δ-closed [22] if A = clδ(A) , where clδ(A) = {x  X : int(cl(U)) ∩A≠ 𝜑,U 𝜖 𝜏 and x 𝜖 U}.
(7) Regular α-open set [25] (briefly, rα-open) if there is a regular open set U s.t UA αcl(U).
(8) Regular semi open set [8] if there is a regular open set U such that U A cl (U).
(9) θ-closed set [22] if A=cl θ(A), where cl θ(A), = {x 𝜖 X : (cl(U))∩A≠ 𝜑,U 𝜖 𝜏 and x 𝜖 U}.
Definition 2.2: Let (X, 𝜏) be a topological space and A  X
(1) The intersection of all semi closed subsets of spaces X containing A is called the Semi
closure of A and denoted by scl(A).
(2) The intersection of all pre closed subsets of spaces X containing A is called the pre closure
of A and denoted by pcl(A) .
(3) The intersection of all α-closed subsets of spaces X containing A is called the α-closure of
A and denoted by cl (A).
(4) The intersection of all semi pre closed subsets of spaces X containing A is called the semi
pre closure of A and denoted by spcl(A).
It is well know that scl(A) =A ∪ int(Cl(A)) , αcl(A) = A ∪ Cl(int(Cl(A)))
pcl(A) =A ∪ Cl(int(A)) , spcl(A)=A ∪ int(Cl(int(A)))
Definition 2.3: [10] Let X be a topological space. The finite union of regular open sets in X is
said to be 𝜋 -open. The complement of a 𝜋 -open set is said to be 𝜋-closed.
Definition 2.4 :A subset A of a topological space (X, 𝜏) is called a
(1) Generalized closed set (briefly g-closed) [15] if cl(A)  U whenever A  U and U is
open in X
(2) Semi-generalized closed set (briefly sg-closed) [5] if scl(A)U whenever AU and U is
semi-open in X.
(3) Regular generalized closed set(briefly rg-closed)[27]if cl(A)U wheneverAUand U is
regular open in X.
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(4) Generalized semi-pre closed set (briefly gsp-closed) [9] if spcl (A)  U whenever A U
and U is open in X.
(5) w-closed set[ 29] if cl(A)  U whenever A U and U is semi-open in X.
(6) Strongly generalized closed set [19] (briefly, g*-closed) if Cl(A)⊆UwheneverA⊆U and U
is g-open in X.
(7) Weakly generalized closed set (briefly, wg-closed)if cl(int(A))⊆UwheneverA⊆U and U
is open in X.
(8) Regular weakly generalized closed set (briefly, rwg-closed)[18] if cl(int(A))⊆ U
whenever A ⊆ U and U is regular open in X.
(9) Regular generalized α-closed set (briefly, rgα-closed)[25] if αcl(A)⊆ U whenever A⊆ U
and U is regular α-open in X.
(10)Regular weakly closed (briefly rw -closed) set [4] if cl(A) ⊆U whenever A⊆ U and U is
regular semi- open in X.
(11)Generalized regular closed (briefly gr–closed) set [6] if rcl(A)⊆ U whenever A⊆ U and U
is open in X.
(12) R*- closed (briefly R*-closed) set [11] if rcl(A) ⊆U whenever A⊆U and U is regular
semi- open in X.
(13) Regular generalized weakly (briefly rgw-closed) set[17] if cl(int(A))⊆ U whenever A⊆
U and U is regular semi-open in X.
(14) Weakly generalized regular α-closed (briefly wgrα-closed) set[12] if cl(int(A))⊆ U
whenever A⊆ U and U is regular α-open in X.
(15) Pre generalized pre regular closed (briefly pgpr-closed) set [3] if pcl(A) ⊆ U whenever
A⊆ U and U is rg- open in X.
(16) Regular pre semi –closed (briefly rps-closed) set [20] if spcl(A)⊆U whenever A⊆ U and
U is rg- open in X.
(17) Generalized pre regular weakly closed (briefly gprw-closed) set [13] if pcl(A)⊆U
whenever A⊆ U and U is regular semi- open in X.
(18) #rg closed [briefly #rg closed] set [23] if cl(A) ⊆U whenever A⊆ U and U is rw- open
in X.
3. (rw)*-CLOSED SETS IN TOPOLOGICAL SPACES
In this paper we introduced (rw)*-closed sets in topological space and studied some
of their properties.
Definition 3.1: A subset A of topological space (X, τ) is called a (rw)*-closed set (briefly
(rω)*-closed set) if Cl(int(A))U whenever AU and U is rw-open in (X, τ).
First we prove that the class of (rw)*closed sets properly lies between the class of #rg
closed sets and class of wg closed sets.
Theorem (3.2): Every #rg closed sets in X is (rw)* closed sets in X, but not conversely
Proof: Suppose A is any arbitrary #rg-closed set in X. To prove A is (rw)* closed set in X, Let
U be any rw-open set in X such that A⊆ U and A is #rg-closed set, then cl(A)⊆ U, since U is
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rw-open in X, but We know that cl(int(A))⊆cl(A)⊆ U i.e. cl(int(A))⊆U and U is rw-open in
X. Hence A is (rw)* closed sets in X.
The converse of the above theorem need not be true, as seen from the following example.
Example 3.3: Let X= {a, b, c, d} and 𝜏={X, ϕ,{a},{b},{a, b},{a, b, c}} . Then the set A={c}
is (rw)*-closed set but not #rg-closed set in X.
Theorem 3.4: Every (rw)* - closed set in X is wg-closed set in X, but not conversely.
Proof: Let A is (rw)* -closed set in X, To prove that A is wg-closed in X, Let U be any open
set in X, such that A⊆ U,as every open set is rw-open set in X. Now A⊆ U and U is rw-open
and A is (rw)*closed in X, i.e cl(int(A))⊆U,U is rw-open. Therefore cl(int(A))⊆ cl(int(A))⊆U
and U is open in X. Thus cl(int(A))⊆U and U is open in X. Hence A is wg-closed set in X.
The converse of the above theoren need not be true as seen from the following example.
Example 3.5: Let X={a, b, c, d} and 𝜏={X, ϕ,{c},{c,d},{b,c,d}} . Then the set A= {a, c, d} is
wg-closed set but not (rw)*-closed set in X.
Remark 3.6: From the theorem (3.4), every (rw)*-closed set in X is wg-closed set in X, but
not conversely. And also from Nagaveni we know that every wg-closed set is rwg-closed set
in X, but not conversely. Hence every (rw)*-closed set is rwg-closed set in X, but not
conversely.
Remark 3.7: From M.Mariasingam we know that every closed set is #rg-closed set in X, but
not conversely and also from theorem (3.2) Every #rg closed sets in X is (rw)*-closed sets in
X, but not conversely. Hence every closed set is (rw)*-closed set in X, but not conversely.
Remark 3.8: From Stone we know that every regular closed set is closed set, but not
conversely and also from remark 3.7 every closed set is (rw)*-closed set in X, but not
conversely. Hence every regular closed set is (rw)*closed set in X, but not conversely.
Remark 3.9: From Velicko we know that every θ-closed set is closed set, but not conversely,
and also from remark (3.7) every closed set is (rw)*-closed set in X, but not conversely. And
hence every θ -closed set is (rw)*-closed set in X, but not conversely.
Remark 3.10: From Velicko we know that every δ -closed set is closed set, but not conversely,
and also from remark (3.7) every closed set is (rw)*-closed set in X. And hence δ -closed set
is (rw)*- closed set in X, but not conversely.
Remark 3.11: from Dontchev and Noiri We know that every π - closed set is closed set but
not conversely. Also from remark (3.7) every closed set is (rw)*- closed set and hence Every
π - closed set is (rw)* - closed set in X, but not conversely.
Theorem 3.12: Every α - closed set is (rw)*- closed set in X, but not conversely
Proof: Let A is α - closed set in X. To prove that A is (rw)*- closed set in X. Let U be any rwopen set in X, Such that AU, Since A is α - closed and cl(int(cl(A))  A and also we know
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that cl(int(A))  cl(int(cl(A))  A  U, and U is rw-open in X. Thus cl(int((A))U and U is
rw-open in X. Hence A is (rw)*- closed set in X.
The converse of the above theorem need not be true as seen from following example.
Example:3.13: Let X= {a, b, c, d} and 𝜏={X, ϕ,{a},{b},{a, b},{a, b, c}} . Then the set A= {a,
b, d} is (rw)*-closed set, but not α - closed set in X .
Theorem: 3.14: Every pre-closed set is (rw)* - closed set in X, but not conversely.
Proof: Let A is pre-closed set in X. To prove that A is (rw)*- closed set in X. Let U be any
rw-open set in X, such that A U, Since A is pre-closed set and cl(int(A))  A and therefore
cl(int(A))  cl(int(A)  A  U and U is rw-open in X. i.e cl(int(A)) U and U is rw-open in
X. Hence A is (rw)*- closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example: 3.15: Let X={a, b, c, d} and 𝜏={X, ϕ,{a},{b},{a, b},{a, b, c}} . Then the set A= {a,
d} is (rw)* closed set, but not pre-closed set in X.
Theorem3.16: Every pgpr-closed set in X is (rw)* - closed set in X, but not conversely.
Proof: Let A is pgpr-closed set in X. To prove that A is (rw)*- closed set in X. Let U be any
rw-open set in X, such that A U, and pcl(A)  U, U is rg-open and A is pgpr-closed set in
X, As every rw-open set is rg-open set in X, We know that cl(int(A)) pcl(A)U,U is rw-open
set in X; i.e cl(int(A)) U, U is rw-open in X. Hence A is (rw)*- closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example: Let X= {a, b, c, d} and 𝜏={X, ϕ,{a},{b},{a, b},{a, b, c}} . Then the set A= {b, c,
d} is (rw)*-closed set in X, but not pgpr-closed set in X.
Remark: The following example shows that (rw)*- closed sets are independent of semi-closed
sets, g-closed sets,g*-closed sets,β -closed sets, gsp-closed sets, gprw-closed sets, rg𝛼-closed
sets, wgrα-closed sets, rw-closed sets,R*-closed sets, rgw-closed sets, sg-closed sets, gr-closed
sets and rps-closed sets.
Example: 3.17 Let X= {a, b, c, d} and 𝜏 ={X, ϕ ,{a},{b},{a, b},{a, b, c}} Then
closed sets are :{X, ϕ, {d}, {c, d}, {a, c ,d}, {b, c, d}}
(rw)*-closed sets are: {X, ϕ, {c}, {d}, {c, d},{a, d},{b, d}, {a, c ,d}, {b, c, d},{a, b, d}}
semi -closed sets are:{ X, ϕ, {a},{b},{c},{d},{a, c} {a, d},{b, d}, {c, d}, {b, c},{a, c ,d},{b,
c, d}}
β-closed sets are:{X, ϕ, {a},{b},{c},{d},{a, c} {a, d},{b, d}, {c, d}, {b, c},{a, c ,d}, {b, c,
d}}
gsp-closed closed sets are :{X, ϕ, {c},{d},{b, c},{c, d}, {a, d},{a, c}, {a, b ,d} ,{a, c ,d}, {b,
c, d}}
gprw-closed sets are:{ X, ϕ, {d}{a, b}, {c, d}, {a, b ,c},{a, b ,d} , {a, c ,d}, {b, c, d}}
R* closed sets are:{X, ϕ, {a, b}, {c d}, {a, b, c},{a, b, d},{a ,c, d}, {b, c, d}}
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Gr-closed sets are :{ X, ϕ, {d},{a, c},{a, d},{b, d}, {c, d}, {a, c ,d}, {b, c, d},{a, b, d}}
Rw-closed sets are:{ X, ϕ, {d}, {a, b},{ c, d}, {a, b, c}, {a, c ,d}, {b, c, d}},{a, b, d}}
Rgw-closed sets are:{X, ϕ, {c},{d},{a, b}, {c, d}, {a, b, c},{a, b, d}{a, c ,d}, {b, c, d}}
Sg-closed ets are:{X,ϕ,{a},{b},{c},{d},{c,d},{b,c},{a,d},{b, d},{a,c},{a,c, d},{b, c, d}}
rgα closed sets are:{ X, ϕ, {d}, {a, b},{c, d},{a, b, c},{a, b, d}, {a, c ,d}, {b, c, d}}
g-closed sets are:{X, ϕ, {d}, {c, d},{a, d},{b, d} {a, c ,d}, {b, c, d},{a, b, d}}
g*-closed csets are :{X, ϕ, {d}, {c, d},{a, d},{b, d}, {a, c ,d}, {b, c, d},{a, b, d}}
wgrα-closed sets are :{X, ϕ, {c},{d}, {c, d},{a, b, c},{a, b, d}, {a, c ,d}, {b, c, d}}
rps-closed sets are: {X, ϕ, {a},{b},{c},{d},{c, d}, {b, d}, {a, c}, {a, c, d}, {b, c, d}}
Example: 3.18 Let X={a, b, c, d} and 𝜏 ={X, ϕ ,{c},{c, d },{b, c, d}}
Closed sets are :{X, ϕ ,{a},{a, b},{a, b, d}}
(rw)*-closed sets are :{X, ϕ, {a},{b}, {d}, {a, b}, {a, d}, {b, d}, {a, b, d}}
g-closed sets are :{X, ϕ, {a}, {a, b}, {a, d}, {a, c}, {a, c ,d}, {a, b, c}, {a, b, d}}
g* closed sets are :{X, ϕ, {a}, {a, b},{a, d},{a, c}, {a, c ,d}, {b, c, d}, {a, b, c}}
Remark 3.18 From the above discussion and know results we have the following implications.
𝜃- Closed
gsp-closed
π-closed
regular-closed
gprw-closed
δ-closed
Closed
rgα-closed
rgw-closed
g*-closed
g-closed
sg-closed
#rg-closed
gr-closed
(rw)* - Closed
α-closed
rps-closed
wgrα-closed
Semi-closed
wg-closed
Pgpr-closed
β-closed
A
A
pre-closed
rω-closed
R*-closed
rwg-closed
B means A & B are independent of each other
B means A implies B but not conversely
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Remark 3.19 The union of two (rw)* closed sets in X is generally not an (rw)* -closed set in
X, as seen from the following example
Example 3.20 Let X={a, b, c, d} and 𝜏 ={X, ϕ, {a}, {c ,d},{a, c, d}} then the set A={c} &
B = {d} are (rw)* closed set in X, but A ∪ B = {c, d} is not (rw)* closed set in X.
Remark 3.21 The intersection of two (rw)* -closed sets in X is generally not an (rw)* -closed
set in X, as seen from the following example
Example 3.22 Let X={a, b, c, d} and 𝜏 ={X, ϕ , {b}, {c},{b, c}} then the set A={a, c} & B
= {c, d} are (rw)* closed set in X, but A∩B= {c} is not (rw)*-closed set in X.
Theorem: 3.23 If A is (rw)*- closed subset of X, such that AB  Cl(int(A)), then B is a also
(rw)*- closed set in X.
Proof: If is given that A is (rw)* closed set in X,then we have to prove that B is also (rw)*
closed set in X. Let U be the rw-open set of X. Such that BU, then AB, since A is (rw)*
closed set, we have Cl(int(A))U and AU.Now BCl(int(A)) = Cl(int(B))
Cl(int(Cl(int(A)))=Cl(int(A)) U.i.e. Cl(int(B)) U , U is rw-open in X and hence B is (rw)*
closed set in X.
Remark 3.24: The converse of the above theorem need not be true in generally, as seen from
the following example
Example 3.25 Let X= {a, b, c, d} and 𝜏 ={X, ϕ ,{a},{b},{a, b},{a, b, c}}, then the set A={c},
B= {a, c, d} such that A & B are (rw)*-closed sets in X, but A B ⊈ Cl(int(A)) Cl(int(A))= ϕ .
Theorem: 3.26 Let (X, 𝜏)be topological space then for each x ϵ X , the set X-{x} ( or {x}c )
is (rw)*-closed or rw-open.
Proof: Let x ϵ X , {x}c is not rw-open in X , then we have to prove that {x}c is (rw)*-closed
in X. And only rw open sets containing {x}c is X (X-{x}X ) and X is rw open set. Therefore
Cl(int(X-{x}) Cl(int(X)) => Cl(int{x}c)  X and hence {x}c is (rw)*- closed in X.
Theorem: 3.27 Let AYX and suppose that A is (rw)*- closed set in X. then A is (rw)*closed relative to Y.
Proof : Given that AYX and A is an (rw)*-closed set in X. To prove that A is (rw)*-closed
relatively to Y. let A Y∩U, where U is rw-open in X. Since A is an (rw)*-closed set in X.
whenever A U and implies Cl(int(A)) U. That is Y∩Cl((int(A))  Y∩U, where
Y∩Cl(int(A)) is closure of interior of A in Y. Thus A is (rw)*- closed set relative to Y.
Theorem: 3.28 If a subset A of topological space X is an (rw)*- closed set in X then
Cl(int(A)) -A does not contain any non empty rw-closed set in X.
Proof: Let A is an (rw)*-closed set in X. We prove result by contradiction. Suppose that F be
an non empty rw-closed subset of Cl(int(A)) – A i.e. F≠ ϕ therefore F Cl(int(A)) – A =>
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F Cl(int(A))∩ (X– A) => F Cl(int(A)) --------(1) & F X– A => A X– F and X –
F is rw-open set and A is an (rw)*- closed set, We have Cl(int(A))  X– F => F  X –
Cl(int(A)) --------------(2) Now from equations (1) and (2) we get FCl(int(A)) ∩ (X –
Cl(int(A)) = ϕ . This shows that F= ϕ. which is contradiction. Hence Cl(int(A)) – A does not
contain any non empty rw-closed set in X.
The converse of the above theorem need not be true as seen from following example.
Example 3.29: Let X= {a, b, c, d, e} and 𝜏 ={X, ϕ ,{a},{d},{e}, {a, d}, {a,e}, {d,e}, {a, d, e}}
then the set A={b,e}, cl(int(A))-A={b, c, e}-{b, e}= {c} does not contain non empty rw-closed
set in X, but A is not (rw)*-closed set in X
Corollary 3.30 :If a subset ‘A’ of topological space (X, τ) is an (rw)* closed set in X, than
Cl(int(A))-A does not contain any non –empty regular open set in X. but converse is not true.
Proof: Proof of this corollary follows theorem (3.28) and the fact that every regular open set
is rw - open set.
Example 3.31:let X= {a, b, c, d} and 𝜏 ={X, ϕ,{a}, {c, d}, {a, c, d}} then the set A= {a},
Cl(int(A))-A={a,b}-{a}={b} does not contain non-empty regular open set in X, but ‘A’ is not
(rw)* closed set in X.
Corollary 3.32 : If a subset A of topological space (X, τ) is an (rw)*closed set in X then
cl(int(A)-A does not contain any non-empty regular closed set in X but converse is not true.
Proof: Proof of this corollary follows from the theorem 3.28 and fact that every reguar open
set is rw-open set.
Example 3.33: let X= {a, b, c, d} and 𝜏 ={X, ϕ,{a}, {c, d}, {a, c, d}} then the set A= {c,d},
Cl(int(A))-A = {b,c,d} – {c,d} = {b} which does not contain non-empty regular closed set in
X, but A is not (rw)* closed set in X.
Theorem 3.34: Let A be (rw)* closed set in X then A is regular closed, iff Cl(int(A))-Ais rw
closed
Proof :-Necessity: suppose that A is regular –closed, than Cl(int(A)) = A and so Cl(int(A))-A
= ϕ,which is rw –closed. Sufficiently:- Suppose A is (rw)*-closed set in X and Cl(int(A))-A is
rw-closed, By the theorem(3.28). Therfore Cl(int(A))-A does not contain any non- empty rw
–closed.That is Cl(int(A))-A= ϕ whıch implies that Cl(int(A))=A,Thus A is regular closed.
Theorem 3.35 :If a subset A of a topological space X is both regular open and (rw)*-closed,
then it is regular closed and hence clopen.
Proof: suppose a subset A of a topological space X is regular open and (rw)*-closed. Then
show that A is regular closed, As Every regular open is rw-open, Now A  A we have
Cl(int(A))  A and also A Cl(int(A)) since A is regular closed and (rw)*-cosed. Therefore
Cl(int(A))=A, Hence A is regular closed and clopen
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Corollary 3.36: If A is both regular open and (rw)*-closed, F is regular-closed in X .Then
A∩F is an (rw)*-closed set in X.
Proof: If A is regular open and (rw)*-closed set in X, By theorem(3.35) A is regular-closed
and F is regular-closed in X. So A∩F is regular -closed and it follows from Remark (3.8),
since every regular closed is (rw)* -closed set in X. Therefore A∩F is an (rw)*-closed set in X.
Theorem 3.37: If A is open and wg-closed set, Then A is (rw)*closed set in X.
Proof: Let A is open and wg-closed set in X. To prove that A is (rw)*-closed set in X, let U
be any rw –open set in X, such that A U,since A is open and wg-closed set,By definition
Cl(int(A))  A and we know that Cl(int(A)) A  U, That is Cl(int(A)) U, U is rw-open in
X and hence A is (rw)*-closed in X.
Theorem 3.38: If A is regular open and rwg-closed then A is (rw)*-closed set in X .
Proof: Let A is regular-open and rwg-closed set in X. To prove that A is (rw)*-closed set in
X, let U be any rw –open set in X, such that AU,since A is regular-open and rwg-closed
set,By definition Cl(int(A))  A and we know that Cl(int(A))A U, That is Cl(int(A)) 
U, U is rw-open in X .Hence A is (rw)*-closed in X.
Theorem 3.39: If A is regular-semiopen and rgw-closed set then it is (rw)*-closed set in X
Proof: Let A is regular semi-open and rgw-closed set in X. To prove thet A is (rw)*-closed
set in X, let U be any rw –open set in X, such that A U,Now A  A and by definition
Cl(int(A))  A.But we know that Cl(int(A))  A  U.That is Cl(int(A))  U, U ıs rw-open
ın X .Hence A is (rw)*-closed in X.
Remark 3.40:If A is both regular-semiopen and (rw)*-closed then A is need not be rgw-closed
set in generally as seen from following example.
Example 3.41: Let X={a,b,c.d} and 𝜏 ={X, ϕ,{a}, {b},{a,b}, {a, b,c}} then A ={a,d} is both
regular-semiopen and (rw)*-closed but not A be rgw-closed set in X.
Theorem 3.42: If subset A of a topological spece X is both regular-semiopen and rw-closed
then it is (rw)*-closed set in X.
Proof: Let A is regular semi-open and rw-closed set in X. To prove thet A is (rw)*-closed set
in X, let U be any rw –open set in X, such that A U,Now A  A and by definition Cl(A) 
A.But we know that Cl(int(A))  Cl(A)  A  U.That is Cl(int(A))  U,U is rw-open in X.
Hence A is (rw)*-closed in X.
Remark 3.43: If A is both regular-semiopen and (rw)*-closed then A is need not be rw-closed
set in generally as seen from following example.
Example 3.44: Let X={a,b,c.d} and 𝜏 ={X, ϕ,{a}, {b},{a,b}, {a, b,c}} then A ={a,d} is both
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regular-semiopen and (rw)*-closed but not A be rw-closed set in X.
Theorem 3.45: If A is both regular α-open and wgrα-closed set in X,then A is (rw)*-closed
set in X.
Proof: Let A is regular α -open and wgrα -closed set in X. To prove thet A is (rw)*-closed set
in X, let U be any rw –open set in X, such that A U,Now A  A and by definition Cl(int(A))
 A and we know that Cl(int(A))  A  U.That is Cl(int(A))  U. Hence A is (rw)*-closed
in X.
Remark 3.46: If A is both regular α-open and (rw)*-closed set in X, But A is need not be
wgrα -closed set in generally as seen from following example.
Example3.47: Let X={a,b,c.d} and 𝜏 ={X, ϕ,{a}, {b},{a,b}, {a, b,c}} then A ={a,d} is both
regular α -open and (rw)*-closed, but not A be wgrα -closed set in X.
Theorem 3.48: If A is both open and g-closed set in X then A is (rw)*-closed set in X
Proof: Let A be an both open and g-closed set in X .Let A  U and U be rw-open in X. Now
A  A by hypothesis cl(A)  A, but we know that cl(int(A))  cl(A)  A  U, that is cl(int(A))
 U. Hence A is (rw)* closed set in X.
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