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Gen. Math. Notes, Vol. 19, No. 2, December, 2013, pp. 59-70
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On Intuitionistic Fuzzy Soft Topology
İsmail Osmanoğlu1 and Deniz Tokat2
1,2
Department of Mathematics, Faculty of Arts and Sciences
Nevşehir Hacı Bektaş Veli University, Nevşehir, Turkey
1
E-mail: [email protected]
2
E-mail: [email protected]
(Received: 16-9-13 / Accepted: 22-10-13)
Abstract
In this paper, we define subspace, separation axioms, compactness and connectedness on intuitionistic fuzzy (IF, in short) soft topological spaces. We
also give base theorems of these concepts.
Keywords: IF soft set, IF soft topology, IF soft subspace, IF soft separation axioms, IF soft compactness and IF soft connectedness.
1
Introduction
Classical mathematical methods are not enough to solve the problems of daily
life and also are not enough to meet the new requirements. Therefore, some
theories such as Fuzzy set theory [14], Rough set theory [9] and Soft set theory
[8] have been developed to solve these problems.
Applications of these theories appear in topology and many areas of mathematics. Topological structure on concepts of fuzzy, soft, fuzzy soft, intuitionistic fuzzy and intuitionistic fuzzy soft was defined by Chang [4], Shabir and
Naz [10], Tanay et al. [11], D.Coker [5] and Z. Li et al [6], respectively. Z. Li
et al [6] studied the concepts of intuitionistic fuzzy (IF, in short) soft open set,
IF soft interior point, IF soft closure point and IF soft base.
We will examine the concepts of subspace of a IF soft topological space,
IF soft separation axioms, IF soft compactness and IF soft connectedness with
some base theorems.
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İsmail Osmanoğlu et al.
Preliminaries
In this section, we present the basic definitions of soft set, IF set and IF soft
set.
Definition 2.1 (Soft set). [8] A pair (F, A) is called a soft set (over U )
where F is a mapping F : A → P (U ). In other words, the soft set is a
parameterized family of subsets of the set U . Every set F (e), e ∈ E , from this
family may be considered as the set of e-elements of the soft set (F, A), or as
the set of e-approximate elements of the soft set.
Definition 2.2 (IF set). [1] An intuitionistic fuzzy (IF, in short) set A over
the universe U can be defined as follows
A = {(x, µA (x), νA (x)) : x ∈ U }
where µA : U → [0, 1], νA : U → [0, 1] with the property 0 ≤ µA (x)+νA (x) ≤
1, ∀x ∈ U . The values µA (x) and νA (x) represent the degree of membership
and nonmembership of x to A respectively.
Definition 2.3. [1] Let A = {(x, µA (x), νA (x)) : x ∈ U } and B = {(x, µB (x),
νB (x)) : x ∈ U } be intuitionistic fuzzy sets of U .
(1) A ⊆ B if and only if µA (x) ≤ µB (x) and νA (x) ≥ νB (x) for all x ∈ U,
(2) A ∩ B = {(x, min{µA (x), µB (x)}, max{νA (x), νB (x)}) : x ∈ U } ,
(3) A ∪ B = {(x, max{µA (x), µB (x)}, min{νA (x), νB (x)}) : x ∈ U } .
Definition 2.4. [1] An intuitionistic fuzzy set A over the universe U defined
as A = (x, 0, 1) : x ∈ U ) is said to be intuitionistic fuzzy null set and is denoted
by 0̃.
Definition 2.5. [1] An intuitionistic fuzzy set A over the universe U defined
as A = (x, 1, 0) : x ∈ U ) is said to be intuitionistic fuzzy absolute set and is
denoted by 1̃.
Definition 2.6 (IF soft set). [7] Let U be an initial universe set and E be
the set of parameters. Let IF U denote the collection of all IF subsets of U .
Let A⊆E. A pair (F, A) is called an IF soft set over U where F is a mapping
given by F : A → IF U .
In general, for every e ∈ A, F (e) is an IF set of U and it is called IF value
set of parameter e. Clearly, F (e) can be written as an IF set such that
F (e) = {(x, µA (x), νA (x)) : x ∈ U }.
The set of all IF soft sets over U with parameters from E is called an IF
soft class and it is denoted by IF S(UE ).
On Intuitionistic Fuzzy Soft Topology
61
Example 2.7. [2] Let (F, A) describe the character of the students with respect to the given parameters, for finding the best student of an academic year.
Let the set of students under consideration be U = {s1 , s2 , s3 , s4 }. Let E =
{good result (r), conduct (c), games and sports perf ormances (g), sincerity
(s), pleasing personality (p)} be the set of parameters framed to choose the
best student. Suppose Mr. X has the parameter set A = {r, c, p} ⊆ E choose
the best student. Then
(F, A) = {F (r) = {(s1 , 0.8, 0.1), (s2 , 0.7, 0.05), (s3 , 0.9, 0.1), (s4 , 0.7, 0.2)},
F (c) = {(s1 , 0.6, 0.2), (s2 , 0.7, 0.1), (s3 , 0.5, 0.3), (s4 , 0.3, 0.6)}, F (p) = {(s1 , 0.6,
0.2), (s2 , 0.7, 0.1), (s3 , 0.5, 0.3), (s4 , 0.3, 0.6)}}
Definition 2.8. [7] Let (F, A) and (G, B) be two IF soft sets over U .
˜ (G, B) = (H, C) where C = A ∪ B and ∀e ∈ C,
Union (F, A)∪

if e ∈ A − B
 F (e)
H(e)
if e ∈ B − A
H(e) =

F (e) ∪ G(e) if e ∈ A ∩ B
˜ (G, B) = (H, C) where C = A ∩ B and ∀e ∈ C, H(E) =
Intersection (F, A)∩
F (e) ∩ G(e).
˜
Subset (F, A)⊆(G,
B) where i) A ⊆ B,
ii) for all e ∈ A, F (e) ⊆ G(e)
Complement The complement of an intuitionistic fuzzy soft set (F, A) is
denoted by (F, A)c and is defined by (F, A)c = (F c , A), where F c : A →
IF U is a mapping given by F c (e) = [F (e)]c for all e ∈ A. Thus if
F (e) = (x, µF (e) (x), νF (e) (x)) : x∈ U ,
then ∀e ∈ A, F c (e) = (F (e))c = (x, νF (e) (x), µF (e) (x)) : x ∈ U .
Absolute IF soft set A soft set (F, A) over U is said to be null intuitionistic
fuzzy soft set denoted by à if ∀e ∈ A, F (e) is the absolute intuitionistic
fuzzy set 1̃ of U where 1̃(x) = 1, ∀x ∈ U .
We would use the notation ŨA to represent the absolute intuitionistic
fuzzy soft set with respect to the set of parameters A.
Null IF soft set A soft set (F, A) over U is said to be null intuitionistic fuzzy
soft set denoted by Φ if ∀e ∈ A, F (e) is the null intuitionistic fuzzy set
0̃ of U where 0̃(x) = 0, ∀x ∈ U .
We would use the notation ΦA to represent the null intuitionistic fuzzy
soft set with respect to the set of parameters A.
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İsmail Osmanoğlu et al.
Definition 2.9. [3] Let (F, A) and (G, B) be two IF soft sets over U . We
define the difference of (F, A) and (G, B) as the IF soft (H, C) written as
(F, A) − (G, B) = (H, C) where C = A ∩ B and ∀e ∈ C, x ∈ U,
µH(e) (x) = min(µF (e) (x), νG(e) (x)),
νH(e) (x) = max(νF (e) (x), µG(e) (x)).
Definition 2.10. [13] Let IF S(XE ) and IF S(YK ) be two intuitionistic
fuzzy soft classes, and let ϕ : X → Y and φ : E → K be mappings. Then a
mapping (ϕ, φ) : IF S(XE ) → IF S(YK ) is defined as: for (F, A) ∈ IF S(XE ),
the image of (F, A) under (ϕ, φ), denoted by (ϕ, φ)(F, A) = (ϕ(F ), φ(A)), is
an intuitionistic fuzzy soft set in IF S(YK ) given by
(
sup
µF (e) (x) if ϕ−1 (y) 6= ∅
−1
−1
µϕ(F )(k) (y) = e∈φ (k)∩A,x∈ϕ (y)
0
otherwise,
and
(
inf
νF (e) (x) if ϕ−1 (y) 6= ∅
νϕ(F )(k) (y) = e∈φ−1 (k)∩A,x∈ϕ−1 (y)
1
otherwise,
for all k ∈ φ(A) and y ∈ Y . For (G, B) ∈ IF S(YK ), the inverse image
of (G, B) under (ϕ, φ), denoted by (ϕ, φ)−1 (G, B) = (ϕ−1 (G), φ−1 (B)) is an
intuitionistic fuzzy soft set in IF S(XE ) given by
µϕ−1 (G)(e) (x) = µG(φ(e)) (ϕ(x)) and νϕ−1 (G)(e) (x) = νG(φ(e)) (ϕ(x))
for all e ∈ φ−1 (B) and x ∈ X.
3
Some Topological Structures on IF Soft Topology
Definition 3.1. [6] Let τ ⊆ IF S(XE ), then τ is said to be a IF soft topology
on X if the following conditions hold.
i. ΦE , X̃E belong to τ .
ii. The union of any number of IF soft sets in τ belongs to τ .
iii. The intersection of any two IF soft sets in τ belongs to τ .
τ is called a IF soft topology over X and the binary (XE , τ ) is called a IF
soft topological space over X.
The members of τ are said to be IF soft open sets in X.
A IF soft set (F, E) over X is said to be a IF soft closed set in X, if its
complement (F, E)c belongs to τ .
On Intuitionistic Fuzzy Soft Topology
63
Example 3.2. Let X = {x1 , x2 }, E = {e1 , e2 } and τ = {ΦE , X̃E , (F1 , E),
(F2 , E), (F3 , E)} where (F1 , E), (F2 , E), (F3 , E) are IF soft sets over X, defined
as follows
F1 (e1 ) = {(x1 , 0.3, 0.4), (x2 , 1, 0.3)}, F1 (e2 ) = {(x1 , 0.3, 0.2), (x2 , 1, 0.1)},
F2 (e1 ) = {(x1 , 0.9, 0.2), (x2 , 1, 0.1)}, F2 (e2 ) = {(x1 , 0.9, 0.1), (x2 , 1, 0.1)},
F3 (e1 ) = {(x1 , 0, 0.4), (x2 , 0.6, 0.3)}, F3 (e2 ) = {(x1 , 0, 0.4), (x2 , 0.6, 0.2)}.
Then τ defines a IF soft topology on X and hence (XE , τ ) is a IF soft
topological space over X.
Definition 3.3. Let (XE , τ1 ) and (XE , τ2 ) be IF soft topological spaces.
Then, the following hold.
If τ2 ⊇ τ1 , then τ2 is IF soft finer than τ1 .
If τ2 ⊃ τ1 , then τ2 is IF soft strictly finer than τ1 .
If either τ2 ⊇ τ1 or τ2 ⊆ τ1 , then τ1 is comparable with τ2 .
Definition 3.4. Let X be the universal set, E be the set of parameters.
• Let τ be the collection of all IF soft sets which can be defined over X.
Then τ is called the IF soft discrete topology on X and (XE , τ ) is said to
be a IF soft discrete space over X.
• τ = {Φ, X̃} is called the IF soft indiscrete topology on X and (XE , τ ) is
said to be a IF soft indiscrete space over X.
Definition 3.5. Let (XE , τ ) be a IF soft topological space over X and Y be
a non-empty subset of X. Then
τY = {(Y F, E) : (F, E) ∈ T }
is said to be the IF soft topology on Y and (YE , τY ) is called a IF soft subspace
˜ (F, E)
of (XE , τ ). Where (Y F, E) = ỸE ∩
We can easily verify that τY is, in fact, a IF soft topology on Y .
Example 3.6. Any IF soft subspace of a IF soft discrete topological space
is a IF soft discrete topological space.
Also, any IF soft subspace of a IF soft indiscrete topological space is a IF
soft indiscrete topological space
3.1
Separation Axioms
Definition 3.7. A IF soft set (F, A) is said to be a IF soft point, denoted
by eF , if for the element e ∈ A, F (e) 6= 0̃ and F (é) = 0̃, ∀é ∈ A − {e}.
Definition 3.8. The complement of a IF soft point eF is a IF soft point ecF
such that F c (e) = (F (e))c .
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Example 3.9. Let X = {x1 , x2 , x3 } and E = {e1 , e2 }. Then e2F = {(x1 ,
0.8, 0.1), (x2 , 0.7, 0.05), (x3 , 0.9, 0.1)} is a IF soft point whose complement is
ec2F = {(x1 , 0.1, 0.8), (x2 , 0.05, 0.7), (x3 , 0.1, 0.9)}.
Definition 3.10. A IF soft point eF is said to be in a IF soft set (G, A),
denoted by eF ∈ (G, A) if for the element e ∈ A, F (e) ≤ G(e).
Definition 3.11. We define the separation axioms as follows.
T0 Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈ X̃E such that
eK 6= eS . If there exist IF soft open sets (F, E) and (G, E) such that
eK ∈ (F, E) and eS ∈
/ (F, E) or eS ∈ (G, E) and eK ∈
/ (G, E),
then (XE , τ ) is called a IF soft T0 -space.
T1 Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈ X̃E such that
eK 6= eS . If there exist IF soft open sets (F, E) and (G, E) such that
eK ∈ (F, E), eS ∈
/ (F, E) and eS ∈ (G, E), eK ∈
/ (G, E),
then (XE , τ ) is called a IF soft T1 -space.
T2 (Hausdorf f ) Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈
X̃E such that eK 6= eS . If there exist IF soft open sets (F, E) and
˜ (G, E) = ΦE ,
(G, E) such that eK ∈ (F, E), eS ∈ (G, E) and (F, E)∩
then (XE , τ ) is called a IF soft T2 -space.
Regular Let (XE , τ ) be a IF soft topological space over X, (G, E) be a IF soft
closed set in X and eK ∈ X̃E such that eK ∈
/ (G, E). If there exist IF
soft open sets (F1 , E) and (F2 , E) such that
˜ (F2 , E) = ΦE ,
˜ 2 , E) and (F1 , E)∩
eK ∈ (F1 , E), (G, E)⊂(F
then (XE , τ ) is called a IF soft regular space.
T3 Let (XE , τ ) be a IF soft topological space over X. Then (XE , τ ) is said to
be a IF soft T3 -space if it is IF soft regular and IF soft T1 -space.
N ormal Let (XE , τ ) be a IF soft topological space over X, (F, E) and (G, E)
˜ (G, E) = ΦE . If there exist
IF soft closed sets over X such that (F, E)∩
IF soft open sets (F1 , E) and (F2 , E) such that
˜ (F2 , E) = ΦE ,
˜ 1 , E), (G, E)⊂(F
˜ 2 , E) and (F1 , E)∩
(F, E)⊂(F
then (XE , τ ) is called a IF soft normal space.
T4 Let (XE , τ ) be a IF soft topological space over X. Then (XE , τ ) is said to
be a IF soft T4 -space if it is IF soft normal and IF soft T1 -space.
On Intuitionistic Fuzzy Soft Topology
65
Proposition 3.12. Let (XE , τ ) be a IF soft topological space over X and
Y be a non-empty subset of X.
i. If (XE , τ ) is a IF soft T0 -space then (YE , τY ) is a IF soft T0 -space.
ii. If (XE , τ ) is a IF soft T1 -space then (YE , τY ) is a IF soft T1 -space.
iii. If (XE , τ ) is a IF soft T2 -space then (YE , τY ) is a IF soft T2 -space.
iv. If (XE , τ ) is a IF soft regular space then (YE , τY ) is a IF soft regular
space.
Proof. We give here the proof of i. The others can be proved in a similar way.
i. Let eK , eS ∈ ỸE such that eK 6= eS . If (XE , τ ) is a IF soft T0 -space then
there exist IF soft open sets (F, E) and (G, E) such that eK ∈ (F, E),
˜ (F, E) =
eS ∈
/ (F, E) or y ∈ (G, E), eK ∈
/ (G, E). Hence eK ∈ ỸE ∩
Y
Y
Y
Y
( F, E), eS ∈
/ ( F, E) or eS ∈ ( G, E) and eK ∈
/ ( G, E). Thus (YE , τY )
is a IF soft T0 -space.
Proposition 3.13. i. Every IF soft T1 -space is a IF soft T0 -space.
ii. Every IF soft T2 -space is a IF soft T1 -space.
iii. Every IF soft T3 -space is a IF soft T2 -space.
iv. Every IF soft T4 -space is a IF soft T3 -space.
Proof. We give here the proof of i. The others can be proved in a similar way.
Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈ X̃E such
that eK 6= eS .
i. If (XE , τ ) is a IF soft T1 -space then there exist IF soft open sets (F, E) and
(G, E) such that eK ∈ (F, E) and eS ∈
/ (F, E), and eS ∈ (G, E) and
eK ∈
/ (G, E). Obviously then we have eK ∈ (F, E) and eS ∈
/ (F, E), or
eS ∈ (G, E) and eK ∈
/ (G, E). Thus (XE , T ) is a IF soft T0 -space.
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İsmail Osmanoğlu et al.
3.2
Compactness
IF soft compactness was defined by Turanlı et al in [12]. However, we will give
a different definition for IF soft compactness.
Definition 3.14. [12] Let (XE , τ ) and (YK , σ) be two IF soft topological
spaces.
(1) A IF soft function f : (XE , τ ) → (YK , σ) is called IF soft continuous if
for all (G, B) ∈ σ, f −1 (G, B) ∈ τ.
(2) A IF soft function f : (XE , τ ) → (YK , σ) is called IF soft open if for all
(F, A) ∈ τ, f (F, A) ∈ σ.
Definition 3.15. A family Ψ of IF soft sets is a cover of a IF softset (F, A)
if
˜∪
˜ {(Fi , A) : (Fi , A) ∈ Ψ, i ∈ I} .
(F, A) ⊆
It is a IF soft open cover if each member of Ψ is a IF soft open set. A subcover
of Ψ is a subfamily of Ψ which is also a cover.
Definition 3.16. Let (XE , τ ) be IF soft topological space and (F, A) ∈ IF S(
XE ). IF soft (F, A) is called compact if each IF soft open cover of (F, A) has a
finite subcover. Also IF soft topological space (XE , τ ) is called compact if each
IF soft open cover of X̃E has a finite subcover.
Example 3.17. A IF soft topological space (XE , τ ) is compact if X is finite.
Example 3.18. Let (XE , τ ) and (YK , σ) be two IF soft topological spaces
and τ ⊂ σ. Then, IF soft topological space (XE , τ ) is compact if (YK , σ) is
compact.
Proposition 3.19. Let (G, B) be a IF soft closed set in IF soft compact
space (XE , τ ). Then (G, B) is also compact.
Proof. Let (Fi , A) be any open covering of (G, B). Then
˜ (∪
˜ i∈I (Fi , A)) ∪ (G, B)c ; that is, (Fi , A) together with IF soft open set
X̃E ⊆
(G, B)c is a open covering of X̃E . Therefore there exists a finite subcovering
(F1 , A) , (F2 , A) , ..., (Fn , A) , (G, B)c .
So
˜ (F1 , A) ∪
˜ (F2 , A) ∪
˜ ...∪
˜ (Fn , A) ∪
˜ (G, B)c .
X̃E ⊆
Therefore
˜ (F1 , A) ∪
˜ (F2 , A) ∪
˜ ...∪
˜ (Fn , A) ∪
˜ (G, B)c
(G, B) ⊆
which clearly implies
˜ (F1 , A) ∪
˜ (F2 , A) ∪
˜ ...∪
˜ (Fn , A)
(G, B) ⊆
c
˜ (G, B) = ΦE . Hence (G, B) has a finite subcovering and so
since (G, B) ∩
is compact.
On Intuitionistic Fuzzy Soft Topology
67
Proposition 3.20. Let (G, B) be a IF soft compact set in IF soft Hausdorff
space (XE , τ ). Then (G, B) is closed.eK , eS ∈ X̃E
Proof. Let eK ∈ (G, B)c . For each eS ∈ (G, B), we have eK 6= eS , so there
are disjoint IF soft open sets (FeS , A) and (Fy , A) so that eK ∈ (FeS , A) and
eS ∈ (HeS , A). Then {(HeS , A) : eS ∈ (G, B)} is an IF soft open cover of
(G, B) Let {(HeS 1 , A) , (HeS 2 , A) , ..., (HeS n , A)} be a finite subcover. Then
˜ ni=1 (FeS i , A) is an open set containing eK and contained in (G, B)c . Thus
∩
(G, B)c is IF soft open and (G, B) is closed.
Theorem 3.21. Let (XE , τ ) and (YK , σ) be IF soft topological spaces and
f : (XE , τ ) → (YK , σ) continuous and onto IF soft function. If (XE , τ ) is IF
soft compact, then (YK , σ) is IF soft compact,
Proof. We will use Theorem 5.2. and Theorem 5.4. of
[13] . Let (Fi , A) be any
˜∪
˜ −1 (∪
˜ i∈I (Fi , A). Then f −1 ỸK ⊆f
˜ i∈I (Fi , A));
open covering of ỸK ; i.e., YK ⊆
˜∪
˜ i∈I f −1 ((Fi , A)). So f −1 ((Fi , A)) is an open covering of X̃E . As
and X̃E ⊆
(X, τ ) is compact, there are 1, 2, ..., n in I such that
˜ −1 ((F1 , A)) ∪
˜ f −1 ((F2 , A)) ∪
˜ ...∪
˜ f −1 ((Fn , A)).
X̃E ⊆f
Since (ϕ, ψ)
we have
is surjective,
ỸK = f X̃E
˜ (f −1 ((F1 , A)) ∪
˜ ...∪
˜ f −1 ((Fn , A)))
⊆f
˜ ...∪
˜ f (f −1 ((Fn , A)))
= f (f −1 ((F1 , A))) ∪
˜ (F2 , A) ∪
˜ ...∪
˜ (Fn , A).
= (F1 , A) ∪
˜ (F1 , A) ∪
˜ (F2 , A) ∪
˜ ...∪
˜ (Fn , A); i.e., ỸK is covered by a finite
So we have ỸK ⊆
number of (Fi , A).
Hence (Y, σ) is compact.
Definition 3.22. Let (XE , τ ) and (YK , σ) be two IF soft topological spaces.
A IF soft function f : (XE , τ ) → (YK , σ) is called IF soft closed if f ((F, A))
is IF soft closed set in (YK , σ), for all IF soft closed set (F, A) in (XE , τ ).
Theorem 3.23. Let (XE , τ ) be a IF soft topological space and (YK , σ) be
a IF soft Hausdorff space. IF soft function f is closed if IF soft function
f : (XE , τ ) → (YK , σ) is continuous.
Proof. Let (G, B) be any IF soft closed set in (XE , τ ). By proposition 3.19
we have (G, B) is compact. Since IF soft function f is continuous, IF softset
f ((G, B)) is compact in (YK , σ). As (YK , σ) is IF soft Hausdorff space, IF soft
set f ((G, B)) is closed. Then IF soft function f is closed.
Definition 3.24. A family Ψ of IF soft sets has the finite intersection
property if the intersection of the members of each finite subfamily of Ψ is not
the null IF soft set.
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İsmail Osmanoğlu et al.
Theorem 3.25. A IF soft topological space is compact if and only if each
family of IF soft closed sets with the finite intersection property has a nonnull
intersection.
˜ {(Fi , A) : (Fi , A) ∈
Proof. ⇒: Let Ψ be any family of IF soft closed such that ∩
c
Ψ, i ∈ I} = ΦE . Consider Ω = {(Fi , A) : (Fi , A) ∈ Ψ, i ∈ I}. So Ω is
a IF soft open cover of X̃E . As IF soft topological space is compact, there
˜ ni=1 (Fi , A) =
exists a finite subcovering (F1 , A)c , (F2 , A)c , ..., (F3 , A)c .Then ∩
˜ ni=1 (Fi , A)c = X̃E − X̃E = ΦE . Thence Ψ cannot have finite intersection
X̃E − ∪
property.
⇐: Assume that a IF soft topological space is not compact. Then any IF
soft open cover of X̃E has not a finite subcover. Let {(Fi , A) : i ∈ I} be IF soft
˜ ni=1 (Fi , A) 6= X̃E . Therefore ∩
˜ ni=1 (Fi , A)c 6= ΦE . Thus,
open cover of X̃E . So ∪
{(Fi , A)c : i = 1, ..., n} have finite intersection property. By using hypothesis,
˜ (Fi , A)c 6= ΦE and so ∪
˜ (Fi , A) 6= X̃E . This is a contradiction. Thus IF soft
∩
topological space is compact.
3.3
Connectedness
Definition 3.26. Let (XE , τ ) be a IF soft topological space over X. A IF
soft separation of X̃E is a pair (F, E), (G, E) of no-null IF soft open sets such
that
˜ (G, E), (F, E)∩
˜ (G, E) = Φ.
X̃E = (F, E)∪
Definition 3.27. A IF soft topological space (XE , τ ) is said to be IF soft
connected if there does not exist a IF soft separation of X̃E .Otherwise, (XE , τ )
is said to be IF soft disconnected.
Example 3.28. Let us consider the IF soft topological space (XE , τ ) that
is given in Example 3.2.
˜ (F2 , E) 6= ΦE , (F1 , E)∩
˜ (F3 , E) 6= ΦE and (F2 , E)∩
˜ (F3 , E) 6=
Since (F1 , E)∩
˜
˜
˜
ΦE also (F1 , E)∪(F2 , E) 6= X̃E , (F1 , E)∪(F3 , E) 6= X̃E and (F2 , E)∪(F3 , E) 6=
X̃E , IF soft topological space (XE , τ ) is IF soft connected.
Theorem 3.29. IF soft topological space (XE , τ ) is IF soft connected if and
only if the only IF soft sets in IF S(XE ) that are both IF soft open and IF soft
closed are Φ and X̃
Proof. Let (XE , τ ) be IF soft connected. Suppose to the contrary that (F, E) is
both IF soft open and IF soft closed different from ΦE and X̃E . Clearly, (F, E)c
˜ (F, E)c = X̃E and
is a IF soft open set different from ΦE and X̃E . Also (F, E)∪
˜ (F, E)c = ΦE . Therefore we have (F, E), (F, E)c is a IF soft separation
(F, E)∩
of X̃E . This is a contradiction. Thus the only IF soft closed and open sets in
XE are ΦE and X̃E .
On Intuitionistic Fuzzy Soft Topology
69
Conversely, let (F, E), (G, E) be a IF soft separation of X̃E . Let (F, E) =
X̃E . Then (G, E) = ΦE . This is a contradiction. Hence, (F, E) 6= X̃E .
Therefore (F, E) = (G, E)c . This shows that (F, E) is both IF soft open and
IF soft closed different from ΦE and X̃E . This is a contradiction. Therefore,
(XE , τ ) is soft connected.
Example 3.30. Since the only IF soft sets in IF S(XE ) that are both IF
soft open and IF soft closed are X̃E and ΦE , IF soft indiscrete topological space
(XE , τ ) is IF soft connected.
Example 3.31. IF Soft discrete topological space (XE , τ ) is IF soft disconnected. Because for at least one IF soft set (F, E) in IF S(XE ), IF soft set
(F, E) is both IF soft open set and IF soft closed.
Corollary 3.32. Let (XE , τ ) be a IF soft topological space over X. Then
following statements are equivalent.
1. (XE , τ ) is IF soft connected.
˜ (G, E) but
2. No-null IF soft open sets (F, E), (G, E) and X̃E = (F, E)∪
˜
(F, E)∩(G, E) 6= ΦE .
3. The only IF soft sets in IF S(XE ) that are both IF soft open and IF soft
closed in XE are Φ and X̃
˜ (G, E) and (F, E)∩
˜ (G, E) = ΦE then (F, E) = ΦE or
4. If X̃E = (F, E)∪
(G, E) = ΦE .
˜ (G, E) and (F, E)∩
˜ (G, E) = ΦE then (F, E) = X̃E or
5. If X̃E = (F, E)∪
(G, E) = X̃E .
4
Conclusion
In this work, we introduced IF soft subspace, IF soft separation axioms, IF soft
compactness and IF soft connectedness. Also we gave some basic properties
of these concepts.
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