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International Journal of Mathematical Archive-5(2), 2014, 24-29
Available online through www.ijma.info ISSN 2229 – 5046
A NOTE ON d-LOCALLY CLOSED SETS
H. Jude Immaculate*1 & I. Arockiarani2
1,2Department
of Mathematics, Nirmala College for Women, Coimbatore, (T.N.), India.
(Received on: 23-01-14; Revised & Accepted on: 12-02-14)
ABSTRACT
In this paper we devise a new class of generalized open sets in a topological space called d-open sets. Using this new
class of sets we deduce d-t sets, d-B-sets and locally d-closed sets and some of their properties and related concepts are
investigated.
Keywords: d-open, d-t-set, d-B-set, d-preopen, d-semiopen, locally d-closed.
1. INTRODUCTION
In recent years a number of generalizations of open sets have been developed by many Mathematicians. N. Levine[21]
and M.E Abd El-Monsef [1] introduced semiopen sets and β-sets respectively. In [21]Levine defined a subset A of a
topological space (X, τ) to be semiopen if there exists an open set U in X such that U⊂A⊂𝑈𝑈. β-sets are also called as
semi-preopen sets by Andrijevic [3]. In [15] O. Njastad defined a subset A of a topological space (X, τ) to be a α-set if
A⊂int(cl(int(A)). Andrijevic[6] introduced a class of generalized open sets in a topological space, so called b-open sets
[2]. Tong [16] introduced the concept of t-set and B-set in topological spaces. In this paper, we introduce the notion of
d-open sets, locally d-closed sets, d-t-set and d-B-set. Throughout this paper (X,τ) and (Y,σ) stands for topological
spaces with no separation axioms assumed, unless otherwise stated. The complement of a d-open set is said to be dclosed. The family of all d-open (resp. α-open, semi open, pre open, regular open) subsets of a space X is denoted by
DO(X)(resp. αO(X), SO(X), PO(X), RO(X))respectively.
2. PRELIMINARIES
We present here relevant preliminaries required for the progress of this paper
Definition: 2.1 A subset A of a space X is said to be
1. Preopen [12] if A⊂int(cl(A))
2. Semiopen [11] if A⊂ cl(int(A))
3. α-open[2] if A⊂ int(cl(int(A)))
4. β-open or semi-preopen [3] if A⊂cl(int(cl(A)))
5. Regular open[7] if A= int(cl(A))
Definition: 2.2: A subset A of a space X is called
1. t-set if [16]int(A) = int(cl(A))
2. B-set if [16] A= U∩V, where U∈τ and V is a t-set.
3. Locally closed [7] if A=U∩V where U∈τ and V is a closed set.
Proposition: 2.3 [3] Let A be a subset of a space X. Then
1. scl(A) = A∪int(cl(A)),
sint(A) = A∩cl(int(A)).
2. pcl(A) = A∪cl(int(A)),
pint(A) = A∩int(cl(A)).
3. spcl(A) = A∪int(cl(int(A))), spint(A) = A∩cl(int(cl(A))).
4. 𝑐𝑐𝑐𝑐α (A) = A∪cl(int(cl(A))), 𝑖𝑖𝑖𝑖𝑖𝑖α (A) = A∩int(cl(int(A)))
1,2Department
Corresponding author: H. Jude Immaculate*1
of Mathematics, Nirmala College for Women, Coimbatore, (T.N.), India.
E-mail: [email protected]
International Journal of Mathematical Archive- 5(2), Feb. – 2014
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H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014.
Proposition: 2.4 [2] Let A be a subset of a space X. Then
1. sint(cl(A))=cl(int(cl(A)))
2. scl(int(A))=int(cl(int(A)))
3. 𝑐𝑐𝑐𝑐α (int(A))=cl(int(A))=cl(𝑖𝑖𝑖𝑖𝑖𝑖α (A))= 𝑐𝑐𝑐𝑐α (𝑖𝑖𝑖𝑖𝑖𝑖α (A))
4. int(𝑐𝑐𝑐𝑐α (A))=int (cl(A))= 𝑖𝑖𝑖𝑖𝑖𝑖α(cl(A))
5. spcl(int(A))=int(cl(int(A)))=int(spcl(A))
6. cl(spint(A))=cl(int(cl(A))
3. d-OPEN SETS
Definition: 3.1 A subset A of a space X is called d-open if A⊂scl(int(A)) ∪ sint(cl(A)). The class of all d-open sets in
X is denoted by DO(X). The complement of a d-open set is a d-closed set.
Proposition: 3.2
1. Every open set is d-open set.
2. Every preopen set is d-open set.
3. Every α-open set is d-open set.
4. Every semiopen set is d-open set.
5. Every regular open set is d-open set.
Remark: 3.3 The converse of the above results need not be true may be seen by the following example.
Example: 3.4 Let X= {a, b, c, d}, τ= {X,φ, {a}, {b},{c},{a, b},{a, c},{b, c},{a, b, c}}
A= {a, d} is d-open but not open, preopen ,α-open and regular open.
Example: 3.5 Let X= {a, b, c}, τ= {X,φ, {a}, {b, c}}. A= {b} is d-open but not semiopen.
The following diagram holds for a subset A of a space X.
1.
Open set 2. Regular open set 3. α-open set 4. d-open set 5. Semiopen set 6. Preopen set
Theorem: 3.6 Let X be a topological space and A⊂X. Then A is d-open if and only if A=𝑖𝑖𝑖𝑖𝑖𝑖α A ∪ spintA.
Proof: Let A be d-open. Then A⊂scl(int(A))∪sint(cl(A))
By proposition 2.3
𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA= (A∩int(cl(int(A))))∪(A∩cl(int(cl(A))))
= A ∩(int(cl(int(A)))∪cl(int(cl(A))))
= A ∩(scl(int(A))∪sint(cl(A))) = A
Conversely suppose A= 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA
= (A∩int(cl(int(A))))∪(A∩cl(int(cl(A))))
= (A∩scl(int(A))∪(A∩sint(cl(A))
⊂ scl(int(A))∪sint(cl(A))
Hence A is d-open.
© 2014, IJMA. All Rights Reserved
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H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014.
Theorem: 3.7
1. The union of any family of d-open set is d-open.
2. The intersection of a open set and a d-open set is a d-open set
Remark: 3.8 Every d-open set can be represented as a union of β-open set and a α-open set.
Proposition: 3.9 Let A be a d-open set such that int(A) = φ.Then A is a β-open set.
Definition: 3.10 A subset A of a space X is called d-closed set if X\A is d-open. Thus A is d-closed if and only if
sint(cl(A))∩scl(int(A))⊂A.
Definition: 3.11 If A is a subset of a space X then d-closure of A is denoted by dcl(A) is the smallest d-closed set
containing A. The d-interior of A is denoted by dint (A) is the largest d-open set contained in A.
Theorem: 3.12 The following holds for a subset A of a space X dcl(A)=𝑐𝑐𝑐𝑐α (A)∩spcl(A).
Proof: It is clear that dcl(A)⊂ 𝑐𝑐𝑐𝑐α (A)∩spcl(A).
Conversely, Since dcl(A) is a d-closed set we have
dcl(A) ⊃ sint(cl(dcl(A)))∩scl(int(dcl(A)))
⊃ sint(cl(A))∩scl(int(A))
Hence by proposition 2.3
𝑐𝑐𝑐𝑐α (A)∩spcl(A) = (A∪cl(int(cl(A))))∩(A∪int(cl(int(A))))
= (A∪sint(cl(A)))∩( A∪ (scl(int(A)))
⊂ dcl(A)
Definition: 3.13 Let A be a subset of a space X. The d-interior of A is denoted by dint(A) is defined by the union of all
d-open sets contained in A.
Theorem: 3.14 The following holds for a subset A of a space X dint(A)= 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA.
Proof: It is clear that dint(A) ⊃ 𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA.
Conversely dint(A) ⊂ scl(int(dint(A)))∪sint(cl(dint(A)))
⊂scl(int(A))∪sint(cl(A))
By proposition 2.3
𝑖𝑖𝑖𝑖𝑖𝑖α A∪spintA= A ∩(scl(int(A)))∪(A ∩sint(cl(A)))
⊃ dint(A)
Proposition: 3.15
1. dcl(int(A)) = 𝑐𝑐𝑐𝑐α (int(A))∩spcl(int(A)).
= cl(int(A))∩int(cl(int(A)))
= int(cl(int(A)))
2. int(dcl(A)) = int(𝑐𝑐𝑐𝑐α (A))∩int(spcl(A))
= int(cl(A))∩int(cl(int(A)))
= int(cl(int(A)))
3. 𝑖𝑖𝑖𝑖𝑖𝑖α (dcl(A))= 𝑖𝑖𝑖𝑖𝑖𝑖α (𝑐𝑐𝑐𝑐α (A))∩ 𝑖𝑖𝑖𝑖𝑖𝑖α (spcl(A))
= int(cl(A))∩int(cl(int(A)))
= int(cl(int(A)))
From the above we conclude that dcl(int(A))= int(dcl(A))= 𝑖𝑖𝑖𝑖𝑖𝑖α (dcl(A))= int(cl(int(A)))
Proposition: 3.16
1. cl(dint(A)) = cl 𝑖𝑖𝑖𝑖𝑖𝑖α(A))∪cl(spint(A))
= cl(int(A))∪cl(int(cl(A)))
= cl(int(cl(A)))
2. dint(cl(A)) = 𝑖𝑖𝑖𝑖𝑖𝑖α(cl(A))∪spint(cl(A))
= int(cl(A))∪cl(int(cl(A)))
= cl(int(cl(A)))
© 2014, IJMA. All Rights Reserved
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H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014.
From the above we conclude that cl(dint(A))= dint(cl(A)) = cl(int(cl(A)))
4. d-t-sets
Definition: 4.1 A subset A of a space X is said to be;
1. d-t-set if int(A)=int(dcl(A));
2. d-B-set if A=U∩V, where U∈τ and V is d-t-set;
3. d-semiopen if A⊆cl(dint(A));
4. d-preopen if A⊆int(dcl(A)).
Proposition: 4.2 For subsets A and B of a space (X, τ), the following properties hold;
1. A is a d-t-set if and only if it is d-semi closed.
2. If A is d-closed, then it is a d-t-set.
3. If A and B are d-t-sets, then A∩B is a d-t-set.
Proof:
1. Let A be a d-t-set. Then int(A)=int(dcl(A)). Therefore int(dcl(A)) ⊆int(A) ⊆A and A is d-semi closed.
Conversely if A is d-semi closed, then int(dcl(A) ⊆A thus int(dcl(A)) ⊆int(A). Also A⊆dcl(A) and int(A)⊆
int(dcl(A)). Hence int(A)=int(dcl(A)).
2. Let A be d-closed, then A=dcl(A) and int(A)=int(dcl(A)) therefore A is d-t-set.
3. Let A and B be a d-t-set. Then we have
int(A∩B) ⊆ int(dcl(A∩B))
⊆(int(dcl(A))∩(dcl(B)))
=int(dcl(A)∩Int(dcl(B))
=int(A)∩Int(B)
=int(A∩B)
Then int(A∩B)=int(dcl(A∩B) hence A∩B is a d-t-set.
Proposition: 4.3 For a subset A of a space (X, τ), the following properties hold;
i.
If A is t-set then it is a d-t-set;
ii.
If A is d-t-set then it is a d-B-set;
iii.
If A is B-set then it is a d-B-set.
Theorem: 4.4 For a subset A of a space (X, τ),the following are equivalent;
1. A is open.
2. A is d-preopen and a d-B-set.
Proof:
(1)⇒(2) Let A be open. Then A ⊆dcl(A), A=Int(A) ⊆ Int(dcl(A)) and A is d-preopen. Also A= A∩X hence A is
d-B-set.
(2)⇒(1) Since A is a d-B-set, we have A=U∩V, where U is open set and Int(V)=Int(dcl(V)).By hypothesis, A is
also d-preopen, and we have
A ⊆ int(dcl(A)
= int(dcl(U∩V)
⊆ int(dcl(U)∩dcl(V))
= int(dcl(U)∩int(dcl(V))
= int(dcl(U)∩Int(V)
Hence
A = U∩V= (U∩V)∩U
⊆ (int(dcl(U))∩int(V))∩U
= (int(dcl(U))∩U)∩int(V)
= U∩int(V)
Therefore A= (U∩V) = (U∩int(V)), and A is open.
Remark: 4.5 d-preopen sets and d-B-open sets are independent.
Example: 4.6 Let X= {a, b, c} and τ={X,φ, {c}, {d}, {c, d}, {a, c, d}, {b, c, d}}. It is clear that {b, c} is a d-B-set but it
is not d-preopen, since {b, c} = {b, c, d}∩{b, c}. {b, c} is b-t-set and {b, c}⊄Int(dcl({b, c})))={c}.
© 2014, IJMA. All Rights Reserved
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H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014.
Example: 4.7 Let X= {a, b, c} and τ={X,φ, {b}, {a, b}, {b, c, d}}.It is clear that {b, c} is a d-preopen but it is not a dB-set, since {b, c} ⊆ Int(dcl({b, c}))=X, since {c, b} is not d-t-set and the open set containing {b, c} is X or {b, c, d},
therefore {b, c} is not d-B-set.
Lemma: 4.8 Let A be an open subset of a space X. Then dcl(A)= int(cl(int(A)) and int(dcl(A))=int(cl(int(A))
Proof: Let A be open set, then dcl(A) = 𝑐𝑐𝑐𝑐α (A)∩spcl(A).
= A∪(int(cl(int(A))))∩cl(int(cl(A))))
= A∪( int(cl(int(A))))∩cl(A))
= A∪( int(cl(int(A))))
= Int(cl(int(A)))
5. LOCALLY D-CLOSED SETS
Definition: 5.1 A subset A of a space X is called locally d-closed if A=U∩V where U∈τ and V is a d-closed set.
Theorem: 5.2 Let A be a subset of X, A is locally d-closed if and only if there exists an open set U⊆X such that
A=U∩ dcl(A)
Proof: Since A is locally d-closed, then A=U∩F where U is open and F is d-closed.
So A⊆ U and A ⊆F then A⊆ dcl(A) ⊆dcl(F)= F.
∴A⊆U∩dcl(A)⊆U∩dcl(F)=U∩F=A.
∴ A=U∩ dcl(A)
Conversely, assume that A=U∩ dcl(A).
Since dcl(A) is d-closed A is locally d-closed
Result: 5.3
1. Every closed set is locally d-closed.
2. Every locally closed is locally d-closed.
Example: 5.4 Let X= {a, b, c, d} and τ= {X,φ, {a},{c},{a, c},{b, c},{a, b, c}}.
A= {b} is locally d-closed but not locally closed. And A= {a} is locally d-closed but not closed.
Proposition: 5.5 Let A be a subset of a topological space X if A is locally d-closed then
i.
dcl(A)-A is a d-closed set.
ii.
[A∪(X-dcl(A))] is d-open.
iii.
A ⊆ dint[A∪(X-dcl(A))].
Proof:
i. Let A be a subset of a topological space X and A be locally d-closed then there exists an open set U such that
A=U∩ dcl(A)
dcl(A)-A= dcl(A)-(U∩dcl(A))
= dcl(A)∩(U∩dcl(A)) 𝑐𝑐
= dcl(A)∩[X-(U∩dcl(A))]
= dcl(A)∩[(X-U)∪(X-dcl(A)]
= [dcl(A)∩(X-U)]∪φ
= dcl(A) ∩(X-U)
Therefore dcl(A)-A is d-closed
ii.
Since dcl(A)-A is d-closed then[X-(dcl(A)-A)] is d-open
[X-(dcl(A)-A)] = X-(dcl(A) ∩𝐴𝐴𝑐𝑐 )
= (X-dcl(A))∪(X-𝐴𝐴𝑐𝑐 )
= (X-dcl(A)) ∪A
⇒ [A∪(X-dcl(A))] is d-open.
iii. It is clear that A⊆ [A∪(X-dcl(A))]
= dint[A∪(X-dcl(A))]
Hence A ⊆ dint[A∪(X-dcl(A))]
© 2014, IJMA. All Rights Reserved
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H. Jude Immaculate* & I. Arockiarani / A Note On d-Locally Closed Sets / IJMA- 5(2), Feb.-2014.
Definition: 5.6 A subset A of a topological space is called D(c, d) set if int(A) = dint(A). From the following examples
we can deduce that d-open and D(c, d) sets are independent.
Example: 5.7 Let X= {a, b, c, d}, τ= {X,φ, {a}, {b},{c},{a, b},{a, c},{b, c},{a, b, c}}
DO(X)= {X,φ, {a},{b},{c},{a, b},{a, c}, {a, d},{b, c},{b, d}, {c, d},{a, b, c},{a, b, d},{a, c, d},{b, c, d}}. It is clear
that A= {d} is D(c, d) set but not d-open .Also B= {a, b, d} is d-open but not D(c, d)
Theorem: 5.8 For a subset A of a space X the following are equivalent.
1. A is open
2. A is d-open and D(c, d) set.
Proof: 1⇒ 2 If A is open then A is d-open and A=int(A)=dint(A)
So A is a D(c, d) set.
2⇒1 A is d-open and D(c, d) set. Then A= dint (A) and int(A)=dint(A) and consequently A is open.
Proposition: 5.9 Suppose X is closed under finite union of d-closed sets. Let A and B be locally d-closed. If A and B
are separate then A∪B is locally d-closed.
Proof: Since a and B are locally d-closed A=G ∩ dcl(A) and B=H ∩ dcl(B) where g and H are open in X.
Put U= G ∩(X\cl(B)) and V=H ∩ (X\cl(A)). Then U ∩ dcl(A) = G ∩(X\cl(B)) ∩ dcl(A) = A ∩(X\cl(B)) = A
Since A⊆ X\cl(B), Similarly V∩dcl(B )=B.and U∩dcl(B)⊆ U∩cl(B)=φ and V ∩ dcl(A)=V ∩cl(A) = φ
Since U and V are open
(U ∩ V) ∩ dcl(A∪B) = (U∩V)∩(dcl(A)∪dcl(B))
= (U∩dcl(A))∪(U∩dcl(B))∪(V∩dcl(A))∪(V∩dc l(B))
= A∪ B
Hence A ∪ B is locally d-closed.
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Source of support: Nil, Conflict of interest: None Declared
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