Gen. Math. Notes, Vol. 19, No. 1, November, 2013, pp.16-27
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
On β-Connectedness in Intuitionistic Fuzzy
Topological Spaces
R. Renuka1 and V. Seenivasan2
1,2
Department of Mathematics
University College of Engineering, Panruti
(A Constituent College of Anna University Chennai)
Panruti - 607106, Tamilnadu, India
1
E-mail: renuka.autpc@gmail.com
2
E-mail: seenujsc@yahoo.co.in
(Received: 4-8-13 / Accepted: 17-9-13)
Abstract
In this paper the concept of types of intuitionistic fuzzy β-connected and
intuitionistic fuzzy β-extremally disconnected in intuitionistic fuzzy topological
spaces are introduced and studied. Here we introduce the concepts of intuitionistic fuzzy βC5 -connectedness, intuitionistic fuzzy βCS -connectedness, intuitionistic fuzzy βCM -connectedness, intuitionistic fuzzy β-strongly connectedness, intuitionistic fuzzy β-super connectedness, intuitionistic fuzzy βCi connectedness(i=1,2,3,4), and obtain several properties and some characterizations concerning connectedness in these spaces.
Keywords: Intuitionistic fuzzy connected, intuitionistic fuzzy β-connected,
intuitionistic fuzzy β-stronglyconnected, intuitionistic fuzzy βC5 -connectedness,
intuitionistic fuzzy βCS -connectedness, intuitionistic fuzzy βCM -connectedness,
intuitionistic fuzzy βCi -connectedness(i=1,2,3,4), intuitionistic fuzzy β-super
connectedness.
1
Introduction
Ever since the introduction of fuzzy sets by L.A.Zadeh [11], the fuzzy concept
has invaded almost all branches of mathematics. The concept of fuzzy topological spaces was introduced and developed by C.L.Chang [2]. Atanassov[1]
On β-Connectedness in Intuitionistic Fuzzy...
17
introduced the notion of intuitionistic fuzzy sets, Coker [3] introduced the intuitionistic fuzzy topological spaces. Several types of fuzzy connectedness in
intuitionistic fuzzy topological spaces were defined by Turnali and Coker[10].
In this paper we have introduced some types of intuitionistic fuzzy β-connected
and intuitionistic fuzzy β-extremally disconnected spaces and studied their
properties and characterizations.
2
Preliminaries
Definition 2.1. [1] Let X be a nonempty fixed set and I the closed interval
[0,1]. An intuitionistic fuzzy set (IFS) A is an object of the following form
A = {<x, µA (x), νA (x)>; x∈X}
where the mappings µA (x): X→I and νA (x): X→I denote the degree of membership(namely) µA (x) and the degree of nonmembership(namely) νA (x) for
each element x∈ X to the set A respectively, and 0 ≤ µA (x)+ νA (x) ≤ 1 for
each x∈X.
Definition 2.2. [1] Let A and B are intuitionistic fuzzy sets of the form A =
{<x, µA (x), νA (x)>; x∈X} and B = {<x, µB (x), νB (x)>; x∈X}.Then
(i) A ⊆ B if and only if µA (x) ≤ µB (x) and νA (x) ≥ νB (x);
(ii) Ā(or Ac ) = {<x, νA (x), µA (x)>; x∈X};
(iii) A ∩ B = {<x, µA (x) ∧ µB (x), νA (x) ∨ νB (x)>; x∈X};
(iv) A ∪ B = {<x, µA (x) ∨ µB (x), νA (x)∧ νB (x) >; x∈X};
(v) [ ]A = {<x, µA (x), 1 – µA (x)>; x∈X};
(vi) h iA = {<x, 1 – νA (x), νA (x),>; x∈X};
We will use the notation A= {<x, µA , νA >; x∈X} instead of A= {<x,
µA (x), νA (x)>; x∈X}
Definition 2.3. [3] 0∼ = {<x,0,1>; x∈X} and 1∼ = {<x,1,0>; x∈X}.
Let α, β ∈ [0,1] such that α+β ≤ 1. An intuitionistic fuzzy point (IFP)p( α,β )
(α,β)
if x = p,
is intuitionistic fuzzy set defined by p( α,β ) (x) =
(0, 1)
otherwise
Definition 2.4. [3] An intuitionistic fuzzy topology (IFT) in Coker’s sense on
a nonempty set X is a family τ of intuitionistic fuzzy sets in X satisfying the
following axioms:
18
R. Renuka et al.
(i) 0∼ , 1∼ ∈ τ ;
(ii) G1 ∩ G2 ∈ τ , for any G1 ,G2 ∈ τ ;
(iii) ∪ Gi ∈ τ for any arbitrary family {Gi ; i ∈ J } ⊆ τ .
In this paper by (X,τ ) or simply by X we will denote the intuitionistic fuzzy
topological space (IFTS). Each IFS which belongs to τ is called an intuitionistic
fuzzy open set (IFOS) in X. The complement Ā of an IFOS A in X is called
an intuitionistic fuzzy closed set (IFCS) in X.
Definition 2.5. [6] Let p( α,β ) be an IFP in IFTS X. An IFS A in X is called
an intuitionistic fuzzy neighborhood (IFN) of p( α,β ) if there exists an IFOS B
in X such that p( α,β ) ∈ B ⊆A.
Let X and Y are two non-empty sets and f :(X,τ ) → (Y,σ) be a function.
If B = {<y, µB (y), νB (y)>;y∈Y} is an IFS in Y, then the pre-image of B under
f is denoted and defined by f −1 (B) = {<x, f −1 (µB (x) ), f −1 (νB (x))>; x∈X }
Since µB (x) , νB (x) are fuzzy sets, we explain that f −1 (µB (x)) = µB (x)(f (x)),
f −1 (νB (x)) = νB (x)(f (x)).
Definition 2.6. [3] Let (X,τ ) be an IFTS and A = {<x, µA (x) , νA (x)>; x∈X
} be an IFS in X. Then the intuitionistic fuzzy closure and intuitionistic fuzzy
interior of A are defined by
T
(i) cl(A) = {C:C is an IFCS in X and C ⊇ A};
S
(ii) int(A) = {D:D is an IFOS in X and D ⊆ A};
It can be also shown that cl(A) is an IFCS, int(A) is an IFOS in X and A is
an IFCS in X if and only if cl(A) = A ; A is an IFOS in X if and only int(A)
= A.
Proposition 2.1. [3] Let (X,τ ) be an IFTS and A,B be intuitionistic fuzzy
sets in X. Then the following properties hold:
(i) clA = (int(A)), int(A) = (cl(A));
(ii) int(A) ⊆A ⊆cl(A).
Definition 2.7. [5] An IFS A in an IFTS X is called an intuitionistic fuzzy
β-open set (IFβOS) if and only if A ⊆ cl(int(clA)). The complement of an
IFβOS A in X is called intuitionistic fuzzy β-closed(IFβCS) in X.
Definition 2.8. [5] Let f be a mapping from an IFTS X into an IFTS Y. The
mapping f is called:
On β-Connectedness in Intuitionistic Fuzzy...
19
(i) intuitionistic fuzzy continuous if and only if f −1 (B) is an IFOS in X, for
each IFOS B in Y;
(ii) intuitionistic fuzzy β-contiunous if and only if f −1 (B) is an IFβOS in X,
for each IFOS B in Y.
Definition 2.9. [9] Let (X,τ ) be an IFTS and A = {<x, µA (x), νA (x)>; x∈X}
be an IFS in X.Then the intuitionistic fuzzy β-closure and intuitionistic fuzzy
β-interior of A are defined by
T
(i) βcl(A) = {C:C is an IFβCS in X and C ⊇ A};
S
(ii) βint(A) = {D:D is an IFβOS in X and D ⊆ A}.
Definition 2.10. [7] A function f :(X,τ )→(Y,σ) from an intuitionistic fuzzy
topological space (X,τ ) to another intuitionistic fuzzy topological space (Y,σ)
is said to be intuitionistic fuzzy β-irresolute if f −1 (B) is an IFβOS in (X,τ )
for each IFβOS B in (Y,σ).
3
Types of Intuitionistic Fuzzy β-Connectedness
in Intuitionistic Fuzzy Topological Spaces
Definition 3.1. An IFTS (X, τ ) is IFβ-disconnected if there exists intuitionistic fuzzy β-open sets A , B in X, A 6= 0∼ , B 6= 0∼ such that A∪B = 1∼ and
A∩B = 0∼ . If X is not IFβ-disconnected then it is said to be IFβ-connected.
Example 3.1. LetX = {a, b}, τ = {0∼ , 1∼ , A} where
b
a
b
a
b
a
b
a
, 0.2
),( 0.3
, 0.5
)>; x∈X}, B = {<x,( 0.4
, 0.4
),( 0.2
, 0.5
)>; x∈X},
A = {<x,( 0.4
A and B are intuitionistic fuzzy β-open sets in X , A 6= 0∼ , B 6= 0∼ and A∪B
= B 6= 1∼ , A∩B = A 6= 0∼ . Hence X is IFβ-connected.
a
b
a
b
Example 3.2. LetX = {a, b}, τ = {0∼ , 1∼ , A} where A = {<x,( 0.4
, 0.2
),( 0.3
, 0.5
)>;
a b
a b
a b
a b
x∈X}, B = {<x,( 1 , 0 ),( 0 , 1 )>; x∈X}, C = {<x,( 0 , 1 ),( 1 , 0 )>; x∈X}, B and C
are intuitionistic fuzzy β-open sets in X , B 6= 0∼ ,C 6= 0∼ and B∪C = 1∼ ,
B∩C = 0∼ . Hence X is IFβ-disconnected.
Definition 3.2. An IFTS (X, τ ) is IFβC 5 -disconnected if there exists IFS A
in X, which is both IFβOS and IFβCS such that A 6= 0∼ , and A 6= 1∼ . If X is
not IFβC5 - disconnected then it is said to be IFβC5 -connected.
a
b
a
b
Example 3.3. LetX = {a, b}, τ = {0∼ , 1∼ , A} where A = {<x,( 0.4
, 0.2
),( 0.3
, 0.5
)>;
x∈X} A is an IFβOS in X, But A is not IFβCS since int(cl(intA)) * A, and
1∼ 6= A 6= 0∼ . Thus X is IFβC5 -connected.
20
R. Renuka et al.
Example 3.4. LetX = {a, b}, τ = {0∼ , 1∼ , A} where
a
b
a
b
A = {<x,( 0.4
, 0.2
),( 0.3
, 0.5
)>; x∈X}, B = {<x,( a1 , 0b ),( a0 , 1b )>; x∈X},
B is an intuitionistic fuzzy β-open sets in X . Also B is IFβCS since int(cl(intB))
= 0∼ ⊆ B. Hence there exists an IFS B in X such that 1∼ 6= B 6= 0∼ which is
both IFβOS and IFβCS in X. Thus X is IFβC5 -disconnected.
Proposition 3.1. IFβC5 - connectedness implies IFβ-connectedness.
Proof. Suppose that there exists nonempty intuitionistic fuzzy β-open sets A
and B such that A∪B = 1∼ and A∩B = 0∼ (IFβ-disconnected) then µA ∨ µB
= 1, νA ∧ νB = 0 and µA ∨ µB = 0, νA ∧ νB = 1. In other words B̄ = A. Hence
A is IFβ-clopen which implies X is IFβC5 -disconnected.
But the converse need not be true as shown by the following example.
Example 3.5. LetX = {a, b}, τ = {0∼ , 1∼ , A} where
a
b
a
b
a
b
a
b
A = {<x,( 0.5
, 0.5
),( 0.5
, 0.5
)>; x∈X}, B = {<x,( 0.4
, 0.2
),( 0.3
, 0.5
)>; x∈X},
A is an IFβOS in X. And B is an IFβOS inX since B ⊆ cl(int(clB)). Also 1∼
a
b
a
b
a
b
a
b
6= A∪B = {<x,( 0.5
, 0.5
),( 0.3
, 0.5
)>; x∈X}, 0∼ 6= A∩B = {<x,( 0.4
, 0.2
),( 0.5
, 0.5
)>;
x∈X}. Hence X is IFβ-connected. Since IFS A is both IFβOS and IFβCS in
X, X is IFβC5 -disconnected.
Proposition 3.2. Let f :(X,τ )→(Y,σ) be a IFβ-irresolute surjection, (X, τ )
is an IFβ-connected, then (Y,σ) is IFβ-connected.
Proof. Assume that (Y,σ) is not IFβ-connected then there exists nonempty
intuitionistic fuzzy β-open sets A and B in (Y,σ) such that A∪B = 1∼ and
A∩B = 0∼ . Since f is IFβ-irresolute mapping, C = f −1 (A) 6= 0∼ , D = f −1 (B)
6= 0∼ which are intuitionistic fuzzy β-open sets in X. And f −1 (A)∪f −1 (B) =
f −1 (1∼ ) = 1∼ which implies C∪D = 1∼ .
f −1 (A)∩f −1 (B) = f −1 (0∼ ) = 0∼ which implies C∩D = 0∼ .Thus X is IFβdisconnected , which is a contradiction to our hypothesis. Hence Y is IFβconnected.
Proposition 3.3. (X,τ ) is IFβC5 -connected iff there exists no nonempty intuitionistic fuzzy β-open sets A and B in X such that A = B̄.
Proof. Suppose that A and B are intuitionistic fuzzy β-open sets in X such
that A 6= 0∼ 6= B and A = B̄. Since A = B̄, B̄ is an IFβOS and B is an IFβCS.
And A 6= 0∼ implies B 6= 1∼ . But this is a contradiction to the fact that X is
IFβC5 -connected.
Conversely, let A be both IFβOS and IFβCS in X such that 0∼ 6= A 6= 1∼ .
Now take B = Ā. B is an IFβOS and A 6= 1∼ which implies B = Ā 6= 0∼ which
is a contradiction.
On β-Connectedness in Intuitionistic Fuzzy...
21
Definition 3.3. An IFTS (X, τ ) is IFβ-strongly connected if there exists no
nonempty IFβCS A and B in X such that µA + µB ⊆ 1, νA + νB ⊇ 1.
In otherwords, an IFTS (X, τ ) is IFβ-strongly connected if there exists no
nonempty IFβCS A and B in X such that A ∩ B = 0∼ .
Proposition 3.4. An IFTS (X, τ ) is IFβ-strongly connected if there exists no
IFβOS A and B in X, A 6= 1∼ 6= B such that µA + µB ⊇ 1, νA + νB ⊆ 1.
Example 3.6. LetX = {a, b}, τ = {0∼ , 1∼ , A} where
a
b
a
b
a
b
a
b
A = {<x,( 0.5
, 0.5
),( 0.5
, 0.5
)>; x∈X}, B = {<x,( 0.4
, 0.2
),( 0.6
, 0.8
)>; x∈X},
A is an IFβOS in X. And B is an IFβOS inX since B ⊆ cl(int(clB)). Also µA
+ µB ⊆ 1, νA + νB ⊇ 1. Hence X is IFβ-strongly connected.
Proposition 3.5. Let f :(X,τ )→(Y,σ) be a IFβ-irresolute surjection. If X is
an IFβ-strongly connected, then so is Y.
Proof. Suppose that Y is not IFβ-strongly connected then there exists IFβCS
C and D in Y such that C 6= 0∼ , D 6= 0∼ , C∩D = 0∼ . Since f is IFβirresolute, f −1 (C), f −1 (D) are IFβCSs in X and f −1 (C)∩f −1 (D) = 0∼ , f −1 (C)
6= 0∼ , f −1 (D) 6= 0∼ . (If f −1 (C) = 0∼ then f (f −1 (C)) = C which implies
f (0∼ ) = C. So C = 0∼ a contradiction) Hence X is IFβ-stronglydisconnected,
a contradiction. Thus (Y,σ) is IFβ-strongly connected.
IFβ-strongly connected does not imply IFβC5 -connected, and IFβC5 -connected
does not imply IFβ-strongly connected. For this purpose we see the following
examples:
Example 3.7. LetX = {a, b}, τ = {0∼ , 1∼ , A} where
a
b
a
b
a
b
a
b
A = {<x,( 0.5
, 0.5
),( 0.5
, 0.5
)>; x∈X}, B = {<x,( 0.4
, 0.2
),( 0.6
, 0.8
)>; x∈X},
A is an IFβOS in X. And B is an IFβOS inX since B ⊆ cl(int(clB)). Also µA
+ µB ⊆ 1, νA + νB ⊇ 1. Hence X is IFβ-strongly connected. But X is not
IFβC5 -connected, since A is both IFβOS and IFβCS in X.
Example 3.8. LetX= {a,b}, τ = { 0∼ , 1∼ , A, B, A∪B, A∩B } where
b
a
b
a
b
a
b
a
A = {<x,( 0.5
, 0.6
),( 0.4
, 0.3
)>; x∈X}, B = {<x,( 0.5
, 0.4
),( 0.2
, 0.4
)>; x∈X},
X is IFβC5 -connected. But X is not IFβ-strongly connected since A and B are
intuitionistic fuzzy β-open sets in X such that µA + µB ⊇ 1, νA + νB ⊆ 1.
Lemma 3.1. [10](i) A ∩ B = 0∼ ⇒ A⊆B̄. (ii) A * B̄ ⇒ A ∩ B 6= 0∼
22
R. Renuka et al.
Definition 3.4. A and B are non-zero intuitionistic fuzzy sets in (X,τ ). Then
A and B are said to be
(i) IFβ-weakly separated if βclA ⊆ B̄ and βclB ⊆ Ā
(ii) IFβ-q-separated if (βclA)∩B = 0∼ = A∩(βclB).
Definition 3.5. An IFTS (X, τ ) is said to be IFβCS -disconnected if there
exists IFβ-weakly separated non-zero intuitionistic fuzzy sets A and B in (X,
τ ) such that A∪B = 1∼ .
a
b
a
b
Example 3.9. LetX = {a, b}, τ = {0∼ , 1∼ , A} where A = {<x,( 0.4
, 0.2
),( 0.3
, 0.5
)>;
a b
a b
a b
a b
x∈X}, B = {<x,( 1 , 0 ),( 0 , 1 )>; x∈X}, C = {<x,( 0 , 1 ),( 1 , 0 )>; x∈X}, B and C
are intuitionistic fuzzy β-open sets in X, βclB ⊆ C̄ and βclC ⊆ B̄ Hence B and
C are IFβ-weakly separated and B ∪ C = 1∼ . So X is IFβCS -disconnected.
Definition 3.6. An IFTS (X, τ ) is said to be IFβCM -disconnected if there
exists IFβ-q-separated non-zero IFS’s A and B in (X, τ ) such that A∪B = 1∼ .
a
b
a
b
Example 3.10. LetX = {a, b}, τ = {0∼ , 1∼ , A} where A = {<x,( 0.4
, 0.2
),( 0.3
, 0.5
)>;
x∈X}, B = {<x,( a1 , 0b ),( a0 , 1b )>; x∈X}, C = {<x,( a0 , 1b ),( a1 , 0b )>; x∈X}, B and C
are intuitionistic fuzzy β-open sets in X, (βclB)∩C = 0∼ and B ∩ (βclC) =
0∼ which implies B and C are IFβ-q-separated and B ∪ C = 1∼ . Hence X is
IFβCM -disconnected.
Remark 3.1. An IFTS (X, τ ) is be IFβCS -connected if and only if (X, τ ) is
IFβCM -connected
Definition 3.7. An IFS A in (X.τ ) is said to be IFβ-regular open set if
βint(βcl(A)) = A and IFβ-regular closed set if βcl(βint(A)) = A.
Definition 3.8. An IFTS (X, τ ) is said to be IFβ-super disconnected if there
exists an IFβ-regular open set A in X such that 0∼ 6= A 6= 1∼ . X is called
IFβ-super connected if X is not IFβ-super disconnected.
a
b
a
b
Example 3.11. LetX = {a, b}, τ = {0∼ , 1∼ , A} where A = {<x,( 0.4
, 0.2
),( 0.3
, 0.5
)>;
x∈X}, B = {<x,( a1 , 0b ),( a0 , 1b )>; x∈X}, C = {<x,( a0 , 1b ),( a1 , 0b )>; x∈X}, B and C
are intuitionistic fuzzy β-open sets in X,and βint(βclB) = B. This implies B
is an IFβ-regular open set in X. Hence X is an IFβ-super disconnected.
Proposition 3.6. Let (X, τ ) be an IFTS. Then the following are equivalent:
(a) X is IFβ-super connected.
(b) For each IFβOS A 6= 0∼ in X, we have βclA = 1∼
(c) For each IFβCS A 6= 1∼ in X, we have βintA = 0∼
(d) There exists no IFβOS s A and B in X such that A 6= 0∼ 6= B and A ⊆ B̄.
(e) There exists no IFβOS s A and B in X such that A 6= 0∼ 6= B, B = βclA
and A = βclB
(f ) There exists no IFβCS s A and B in X such that A 6= 1∼ 6= B, B = βintA
and A = βintB.
On β-Connectedness in Intuitionistic Fuzzy...
23
Proof. Proof: (a) ⇒ (b) Assume that there exists an A 6= 0∼ such that βclA
6= 1∼ . Take A = βint(βclA). Then A is proper β-regular open set in X which
contradicts that X is IFβ-super connectedness.
(b) ⇒ (c) Let A 6= 1∼ . be an IFβCS in X. If we take B = Ā then B is an
IFβOS in X and B 6= 0∼ . Hence by (b) βclB = 1∼ ⇒ βclB = 0∼ ⇒ βint(B̄)
= 0∼ ⇒ βintA = 0∼ .
(c) ⇒ (d) Let A and B are IFβOS in x such that A 6= 0∼ 6= B and A ⊆ B̄.
Since B̄ is an IFβCS in X, B̄ 6= 1∼ by (c) βintB̄ = 0∼ . But A ⊆ B̄ implies 0∼
6= A = βint(A) ⊆ βint(B̄) = 0∼ which is a contradiction.
(d) ⇒ (a) Let 0∼ 6= A 6= 1∼ be an IFβ-regular open set in X. If we take B =
βclA, we get B 6= 0∼ . (If not B = 0∼ implies βclA = 0∼ ⇒ βclA = 1∼ ⇒ A =
βint(βclA) = βint(1∼ ) = 1∼ ⇒ A = 1∼ a contradiction to A 6= 1∼ .) We also
have A ⊆ B̄ which is also a contradiction. Therefore X is IFβ-super connected.
(a) ⇒ (e) Let A and B be two IFβOS in (X, τ ) such that A 6= 0∼ 6= B, B =
βclA and A = βclB. Now we have βint(βclA) = βint(B̄) = βclB = A, A 6=
0∼ and A 6= 1∼ , since if A = 1∼ then 1∼ = βclB ⇒ βclB = 0∼ ⇒ B = 0∼ .
But B 6= 0∼ Therefore A 6= 1∼ ⇒ A is proper IFβ-regular open set in (X, τ )
which is contradiction to (a). Hence (e) is true.
(e) ⇒ (a) Let A be IFβOS in X such that A = βint(βclA), 0∼ 6= A 6= 1∼ . Now
take B = βclA . In this case, we get B 6= 0∼ and B is an IFβOS in X and B
= βclA and βclB = βcl(βclA) = (βint(βclA)) = βint(βclA) = A. But this is
a contradiction to (e). Therefore (X, τ ) is IFβ-super connected space.
(e) ⇒ (f) Let A and B be IFβ-closed sets in (X, τ ) such that A 6= 1∼ 6= B, B =
βintA and A = βintB. Taking C = Ā and D = B̄, C and D become IFβ-open
sets in (X, τ ) and C 6= 0∼ 6= D, βclC = βcl(A) = (βintA) = βintA = B̄ = D
and similarly βclD = C. Bu this is a contradiction to (e). Hence (f) is true.
(f) ⇒ (e) We can prove this by the similar way as in (e) ⇒(f).
Proposition 3.7. Let f :(X,τ )→(Y,σ) be a IFβ-irresolute surjection. If X is
an IFβ-super connected, then so is Y.
Proof. Suppose that Y is IFβ-super disconnected. Then there exists IFβOS’s
C and D in Y such that C 6= 0∼ 6= D, C ⊆ D̄. Since f is IFβ-irresolute, f −1 (C)
and f −1 (D) are IFβOSs in X and C ⊆ D̄ implies f −1 (C) ⊆ f −1 (D̄) = f −1 (D).
Hence f −1 (C) 6= 0∼ 6= f −1 (D̄) which means that X is IFβ-super disconnected
which is a contradiction.
Definition 3.9. [4] Two intuitionistic fuzzy sets A and B are said to be qcoincident (AqB) if and only if there exists an element x ∈ X such that µA (x)
> νB (x) or νA (x) < µB (x).
Definition 3.10. [4] Two intuitionistic fuzzy sets A and B in X are said to
be not q-coincident (Aq̄B) if and only if A ⊆ B̄
24
R. Renuka et al.
Definition 3.11. [8] An IFTS (X, τ ) is called intuitionistic fuzzy C5 -connected
between two intuitionistic fuzzy sets A and B if there is no IFOS E in (X, τ )
such that A ⊆ E and Eq̄B.
Definition 3.12. An IFTS (X, τ ) is called intuitionistic fuzzy β-connected
between two intuitionistic fuzzy sets A and B if there is no IFβOS E in (X, τ )
such that A ⊆ E and Eq̄B.
a
b
Example 3.12. LetX = {a, b}, τ = {0∼ , 1∼ , M } where M = {<x,( 0.5
, 0.5
),
b
a
( 0.4 , 0.4 )>; x∈X}, (X,τ ) be IFTS. Consider the intuitionistic fuzzy sets A =
a
b
a
b
a
b
a
b
{<x,( 0.2
, 0.4
), ( 0.7
, 0.6
)>; x∈X}, B = {<x,( 0.5
, 0.5
), ( 0.5
, 0.4
)>; x∈X} M is IFβOS
in (X, τ ). Then (X, τ ) is intuitionistic fuzzy β-connected between A an B.
Theorem 3.1. If an IFTS (X, τ ) is an intuitionistic fuzzy β-connected between
two intuitionistic fuzzy sets A and B, then it is intuitionistic fuzzy C5 -connected
between two intuitionistic fuzzy sets A and B.
Proof. Suppose (X, τ ) is not intuitionistic fuzzy C5 -connected between two
intuitionistic fuzzy sets A and B then there exists an IFOS E in (X, τ ) such
that A ⊆ E and Eq̄B. Since every IFOS is IFβOS, there exists an IFβOS E in
(X, τ ) such that A ⊆ E and Eq̄B which implies (X, τ ) is not intuitionistic fuzzy
β-connected between A and B, a contradiction to our hypothesis. Therefore,
(X, τ ) is intuitionistic fuzzy C5 -connected between A and B.
However, the converse of the above Theorem is need not be true, as shown
by the following example.
b
a
, 0.5
),
Example 3.13. LetX = {a, b}, τ = {0∼ , 1∼ , M } where M = {<x,( 0.5
b
a
( 0.4 , 0.4 )>; x∈X}, (X,τ ) be IFTS. Consider the intuitionistic fuzzy sets A =
a
b
a
b
a
b
a
b
{<x,( 0.2
, 0.4
), ( 0.7
, 0.6
)>; x∈X}, B = {<x,( 0.5
, 0.5
), ( 0.5
, 0.4
)>; x∈X} M is IFOS
in (X, τ ). Then (X, τ ) is intuitionistic fuzzy C5 -connected between A an B.
b
a
b
a
Consider IFS C = {<x,( 0.5
, 0.4
),( 0.5
, 0.6
)>; x∈X} C is an IFβOS such that A
⊆ C and C ⊆ B̄ which implies (X, τ ) is intuitionistic fuzzy β-disconnected
between A an B.
Theorem 3.2. Let (X, τ ) be an IFTS and A and B be intuitionistic fuzzy sets
in (X, τ ). If AqB then (X, τ ) is intuitionistic fuzzy β-connected between A
and B.
Proof. Suppose (X, τ ) is not intuitionistic fuzzy β-connected between A and
B. Then there exists an IFβOS E in (X, τ ) such that A ⊆ E and E ⊆ B̄. This
implies that A ⊆ B̄. That is Aq̄B which is a contadiction to our hypothesis.
Therefore (X, τ ) is intuitionistic fuzzy β-connected between A and B.
However, the converse of the above Theorem is need not be true, as shown
by the following example.
On β-Connectedness in Intuitionistic Fuzzy...
25
a
b
Example 3.14. LetX = {a, b}, τ = {0∼ , 1∼ , M } where M = {<x,( 0.5
, 0.5
),
b
a
( 0.4 , 0.4 )>; x∈X}, (X,τ ) be IFTS. Consider the intuitionistic fuzzy sets A =
a
b
a
b
a
b
a
b
{<x,( 0.2
, 0.4
), ( 0.7
, 0.6
)>; x∈X}, B = {<x,( 0.5
, 0.5
), ( 0.5
, 0.4
)>; x∈X}. M is
IFβOS in (X, τ ). Then (X, τ ) is intuitionistic fuzzy β-connected between
A an B. But A is not q-coincident with B, since µA (x) < νB (x)
Definition 3.13. Let N be an IFS in IFTS (X, τ )
(a) If there exists intuitionistic fuzzy β-open sets M and W in X satisfying the
following properties, then N is called IFβCi -disconnected(i = 1,2,3,4):
C1 : N ⊆ M ∪ W, M ∩ W ⊆ N̄, N ∩ M 6= 0∼ , N∩ W 6= 0∼ ,
C2 : N ⊆ M ∪ W, N ∩ M ∩ W = 0∼ , N ∩ M 6= 0∼ , N ∩ W 6= 0∼ ,
C3 : N ⊆ M ∪ W, M ∩ W ⊆ N̄, M * N̄, W * N̄,
C4 : N ⊆ M ∪ W, N ∩ M ∩ W = 0∼ , M * N̄, W * N̄,
(b) N is said to be IFβCi -connected(i = 1,2,3,4) if N is not IFβCi -disconnected(i
= 1,2,3,4).
Obviously, we can obtain the following implications between several types
of IFβCi -connected(i = 1,2,3,4):
IFβC1 -connectedness −→ IFβC2 -connectedness
↓
↓
IFβC3 -connectedness −→ IFβC4 -connectedness
Example 3.15. Let X = {a, b, c}, τ = {0∼ , 1∼ , M, W }
b
c
a
b
b
a
b
c
a
b
c
a
, 0.1
, 0.3
),( 0.6
, 0.9
, 0.7
)>; x∈X}, W = {<x,( 0.5
, 0.3
, 0.4
),( 0.5
, 0.7
, 0.6
)>;
where M = {<x,( 0.4
b
c
a
b
c
a
x∈X}. (X,τ ) be IFTS. Consider the IFS N = {<x,( 0.3 , 0.1 , 0.2 ), ( 0.7 , 0.9 , 0.8 )>;
x∈X}, N is IFβC2 -connected IFβC3 -connected, IFβC4 -connected but IFβC1 disconnected.
Example 3.16. Let X = {a, b}, τ = {0∼ , 1∼ , M, W, M ∪ W, M ∩ W }
a
b
a
b
a
b
a
b
where M = {<x,( 0.3
, 0.9
),( 0.7
, 0.1
)>; x∈X}, W = {<x,( 0.9
, 0.7
),( 0.1
, 0.3
)>; x∈X},
a
b
a
b
(X,τ ) be IFTS. Consider the IFS N = {<x,( 0.2
, 0.2
), ( 0.8
, 0.8
)>; x∈X}, N is
IFβC4 -connected but IFβC3 -disconnected.
Example 3.17. Let X = {a, b}, τ = {0∼ , 1∼ , M, W, M ∪ W }
b
b
a b
a b
where M = {<x,( a0 , 0.2
),( a1 , 0.8
)>; x∈X}, W = {<x,( 0.2
, 0 ),( 0.8
, 1 )>; x∈X}, (X,τ )
a
b
a
b
be IFTS. Consider the IFS N = {<x,( 0.1 , 0.1 ), ( 0.9 , 0.9 )>; x∈X}, M and W are
intuitionistic fuzzy β-open sets in X. Then N is IFβC4 -connected but IFβC2 disconnected.
26
4
R. Renuka et al.
Intuitionistic Fuzzy β-Extremally Disconnectedness in Intuitionistic Fuzzy Topological Spaces
Definition 4.1. Definition. Let (X,τ ) be any IFTS. X is called IFβ- extremally
disconnected if the β-closure of every IFβOS in X is IFβOS.
Theorem 4.1. For an IFTS (X, τ ) the following are equivalent:
(i) (X, τ ) is an IFβ-extremally disconnected space.
(ii) For each IFβCS A, βint(A) is an IFβCS.
(iii) For each IFβOS A, βcl(A) = βcl(βcl(A))
(iv) For each intuitionistic fuzzy β-open sets A and B with βcl(A) = B̄, βcl(A)
= βclB
Proof. (i)⇒(ii) Let A be any IFβCS. Then Ā is an IFβOS. So βcl(Ā) = βintA
is an IFβOS. Thus βint(A) is an IFβCS in (X, τ ).
(ii)⇒(iii) Let A be an IFβOS. Then βcl(βcl(A)) = βcl(βint(A)). βcl(βcl(A))
= βcl(βint(A)). Since A is an IFβOS, Ā is an IFβCS. So by (ii) βint(A) is
an IFβCS.. That is βcl(βint(A)) = βint(A). Hence βcl(βint(A)) = βint(A)
= βcl(A).
(iii)⇒(iv) Let A and B be any two intuitionistic fuzzy β-open sets in (X, τ )
such that βcl(A) = B̄. (iii) implies βcl(A) = βcl(βcl(A)) = βcl(B) = βclB.
(iv)⇒(i) Let A be any IFβOS in (X, τ ). Put B = βclA. Then βcl(A) = B̄.
Hence by (iv) βcl(A) = βclB. Therefore βcl(A) is IFβOS in (X, τ ). That is
(X, τ ) is an IFβ-extremally disconnected space.
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