In this paper, we introduce certain types of continuous functions and intuitionistic fuzzy
θ
-compactness in intuitionistic fuzzy topological spaces. We show that intuitionistic fuzzy
θ
-compactness is strictly weaker than intuitionistic fuzzy compactness. Furthermore, we show that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to intuitionistic fuzzy
θ
-compactness in the original intuitionistic fuzzy topology. This characterization shows that intuitionistic fuzzy
θ
-compactness can be related to an appropriated notion of intuitionistic fuzzy convergence.
1. Introduction
The concept of an intuitionistic fuzzy set as a generalization of fuzzy sets was introduced by Atanassov
[1]
. Coker and his colleagues
[2
?
4]
introduced an intuitionistic fuzzy topology using intuitionistic fuzzy sets.
Many researchers studied continuity and compactness in fuzzy topological spaces and intuitionistic fuzzy topological spaces
[5
?
8]
. Recently, Hanafy et al.
[9]
introduced an intuitionistic fuzzy
θ
-closure operator and intuitionistic fuzzy
θ
-continuity.
In this paper, we introduce certain types of continuous functions and intuitionistic fuzzy
θ
-compactness in intuitionistic fuzzy topological spaces. We show that intuitionistic fuzzy
θ
-compactness is strictly weaker than intuitionistic fuzzy compactness. Moreover, we show that the sufficient condition in Theorem 4.5 holds for intuitionistic fuzzy
θ
-compact spaces; however, in general, it fails for intuitionistic fuzzy compact spaces. Furthermore, we show that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to the intuitionistic fuzzy
θ
-compactness in the original intuitionistic fuzzy topology described in Theorem 4.6. This characterization shows that the intuitionistic fuzzy
θ
-compactness can be related to an appropriated notion of intuitionistic fuzzy convergence.
2. Preliminaries
Let
X
and
I
denote a nonempty set and unit interval [0, 1], respectively. An
intuitionistic fuzzy set A
in
X
is an object of the form
where the functions
μA
:
X
→
I
and
γA
:
X
→
I
denote the degree of membership and the degree of non-membership, respectively, and
μA
+
γA
≤ 1. Obviously, every fuzzy set
μA
in
X
is an intuitionistic fuzzy set of the form (
μA
, 1 –
μA
).
Throughout this paper,
I
(
X
) denotes the family of all intuitionistic fuzzy sets in
X
and intuitionistic fuzzy is abbreviated as IF.
Definition 2.1.
[1]
Let
X
denote a nonempty set and let intuitionistic fuzzy sets
A
and
B
be of the form
A
= (
μA
,
γA
),
B
= (
μB
,
γB
). Then,
-
(1)A≤BiffμA(x) ≤μB(x) andγA(x) ≥γB(x) for allx∈X,
-
(2)A=BiffA≤BandB≤A,
-
(3)Ac= (γA,μA),
-
(4)A∩B= (μA∧μB,γA∨γB),
-
(5)A∪B= (μA∨μB,γA∧γB),
Definition 2.2.
[2]
An
intuitionistic fuzzy topology
on
X
is a family
Τ
of intuitionistic fuzzy sets in
X
that satisfy the following axioms.
-
(2)G1∩G2∈Τfor anyG1,G2∈Τ,
-
(3) ⋃Gi∈Τfor any {Gi:i∈J} ⊆Τ.
In this case, the pair (
X, Τ
) is called an
intuitionistic fuzzy topological space
and any intuitionistic fuzzy set in
Τ
is known as an
intuitionistic fuzzy open set
in
X
.
Definition 2.3.
[2]
Let (
X, Τ
) and
A
denote an intuitionistic fuzzy topological space and intuitionistic fuzzy set in
X
, respectively. Then, the
intuitionistic fuzzy interior
of
A
and the
intuitionistic fuzzy closure
of
A
are defined by
and
Theorem 2.4.
[2]
For any IF set
A
in an IF topological space (
X, Τ
), we have
-
cl(Ac) = (int(A))cand int(Ac) = (cl(A))c.
Definition 2.5.
[3
,
4]
Let
α,β
∈ [0, 1] and
α
+
β
≤ 1. An
intuitionistic fuzzy point x
(α,β)
of
X
is an intuitionistic fuzzy set in
X
defined by
In this case,
x,α,
and
β
are called the
support, value, and nonvalue
of
x
(α,β)
, respectively. An intuitionistic fuzzy point
x
(α,β)
is said to
belong
to an intuitionistic fuzzy set
A
= (
μA
,
γA
) in
X
, denoted by
x
(α,β)
∈
A
, if
α
≤
μA
(
x
) and
β
≥
γA
(
x
).
Remark 2.6.
If we consider an IF point
x
(α,β)
as an IF set, then we have the relation
x
(α,β)
∈
A
if and only if
x
(α,β)
≤
A
.
Definition 2.7.
[4
,
10]
Let (
X,Τ
) denote an intuitionistic fuzzy topological space.
-
(1) An intuitionistic fuzzy pointx(α,β)is said to bequasicoincidentwith the intuitionistic fuzzy setU= (μU,γU), denoted byx(α,β)qU, ifα>γU(x) or β <μU(x).
-
(2) LetU= (μU,γU) andV= (μV,γV) denote two intuitionistic fuzzy sets inX. Then,UandVare said to bequasi-coincident, denoted byUqV, if there exists an elementx∈Xsuch thatμU(x) >γV(x) orγU(x) <μV(x).
The word ‘not quasi-coincident’ will be abbreviated as
herein.
Proposition 2.8.
[4]
Let
U, V
, and
x
(α,β)
denote IF sets and an IF point in
X
, respectively. Then,
Definition 2.9.
[4]
Let (
X,Τ
) denote an intuitionistic fuzzy topological space and let
x
(α,β)
denote an intuitionistic fuzzy point in
X
. An intuitionistic fuzzy set
A
is said to be an
intuitionistic fuzzy ϵ-neighborhood
(
q-neighborhood
) of
x
(α,β)
if there exists an intuitionistic fuzzy open set
U
in
X
such that
x
(α,β)
∈
U
≤
A
(
x
(α,β)
qU
≤
A
, respectively).
Theorem 2.10.
[10]
Let
x
(α,β)
and
U
= (
μU
,
γU
) denote an IF point in
X
and an IF set in
X
, respectively. Then,
x
(α,β)
∈ cl(
U
) if and only if
UqN
, for any IF
q
-neighborhood
N
of
x
(α,β)
.
Definition 2.11.
[9]
An intuitionistic fuzzy point
x
(α,β)
is said to be an
intuitionistic fuzzy θ-cluster point
of an intuitionistic fuzzy set
A
if for each intuitionistic fuzzy
q
-neighborhood
U
of
x
(α,β)
,
Aq
cl(
U
). The set of all intuitionistic fuzzy
θ
-cluster points of
A
is called
intuitionistic fuzzy θ-closure
of
A
and is denoted by cl
θ
(
A
). An intuitionistic fuzzy set
A
is called an
intuitionistic fuzzy θ-closed set
if
A
= cl
θ
(
A
). The complement of an intuitionistic fuzzy
θ
-closed set is said to be an
intuitionistic fuzzy θ-open set
.
Definition 2.12.
[11]
Let (
X,Τ
) and
U
denote an intuitionistic fuzzy topological space and an intuitionistic fuzzy set in
X
, respectively. The
intuitionistic fuzzy θ-interior
of
U
is denoted and defined by
Definition 2.13.
[2]
Let (
X,Τ
) and (
Y,U
) denote two intuitionistic fuzzy topological spaces and let
f
:
X
→
Y
denote a function. Then,
f
is said to be
intuitionistic fuzzy continuous
if the inverse image of an intuitionistic fuzzy open set in
Y
is an intuitionistic fuzzy open set in
X
.
Definition 2.14.
[2]
An intuitionistic fuzzy topological space (
X,Τ
) is said to be
intuitionistic fuzzy compact
if every open cover of
X
has a finite subcover.
Definition 2.15.
[9]
A function
f
:
X
→
Y
is said to be
intuitionistic fuzzy θ-continuous
if for each intuitionistic fuzzy point
x
(a,b)
in
X
and each intuitionistic fuzzy open
q
-neighborhood
V
of
f
(
x
(a,b)
), there exists an intuitionistic fuzzy open
q
-neighborhood
U
of
x
(a,b)
such that
f
(cl(
U
)) ≤ cl(
V
).
Proposition 2.16.
[12]
Let
f
: (
X,Τ
) → (
Y,T'
) and
x
(α,β)
denote a function and an IF point in
X
, respectively.
-
(1) Iff(x(α,β))qV, thenx(α,β)qf–1(V) for any IF setVinY.
-
(2) Ifx(α,β)qU, thenf(x(α,β))qf(U) for any IF setUinX.
Remark 2.17.
Intuitionistic fuzzy sets have some different properties compared to fuzzy sets, and these properties are shown in the subsequent examples.
-
1.x(α,β)∈A∪B⇏x(α,β)∈Aorx(α,β)∈B.
-
2.x(α,β)qAandx(α,β)qB⇏x(α,β)q(A∩B).
Thus, we have considerably different results in generalizing concepts of fuzzy topological spaces to the intuitionistic fuzzy topological space.
Example 2.18.
Let
A, B
denote IF sets on the unit interval [0, 1] defined by
In addition, let
x
= ¼, (
α,β
) = (¼,½). Then,
x
(α,β)
∈
A
∪
B
. However,
x
(α,β)
∉
A
and
x
(α,β)
∉
B
.
Example 2.19.
Let
A, B
denote IF sets on the unit interval [0, 1] defined by
In addition, let
x
= ¼, (
α,β
) = (½,¼). Then,
x
(α,β)
qA
and
x
(α,β)
qB
; however,
For the notions that are not mentioned in this section, refer to
[11]
.
3. Intuitionistic Fuzzy θ-Irresolute and Weakly θ-Continuity
Definition 3.1.
Let (
X,Τ
) and (
Y,U
) be IF topological spaces. A mapping
f
: (
X,Τ
) → (
Y,U
) is said to be
intuitionistic fuzzy θ-irresolute
if the inverse image of each IF
θ
-open set in
Y
is IF
θ
-open in
X
.
Theorem 3.2.
Let (
X,Τ
) and (
Y,U
) be IF topological spaces. Let
Τθ
be an IF topology on
X
generated using the subbase of all the IF
θ
-open sets in
X
, and let
Uθ
be an IF topology on
Y
generated using the subbase of all the IF
θ
-open sets in
Y
. Then a function
f
: (
X,Τ
) → (
Y,U
) is IF
θ
-irresolute if and only if
f
: (
X,Τθ
) → (
Y,Uθ
) is IF continuous.
Proof.
Trivial.
Recall that a fuzzy set
A
is said to be a
fuzzy θ-neighborhood
of a fuzzy point
xα
if there exists a fuzzy closed
q
-neighborhood
U
of
xα
, such that
[13]
.
Definition 3.3.
An intuitionistic fuzzy set
A
is said to be an
intuitionistic fuzzy θ-neighborhood
of intuitionistic fuzzy point
x
(α,β)
if there exists an intuitionistic fuzzy open
q
-neighborhood
U
of
x
(α,β)
such that cl(
U
) ≤
A
.
Recall that a function
f
: (
X,Τ
) → (
Y,T'
) is said to be a
fuzzy weakly θ-continuous
function if for each fuzzy point
xα
in
X
and each fuzzy open
q
-neighborhood
V
of
f
(
xα
), there exists a fuzzy open
q
-neighborhood
U
of
xα
such that
f
(
U
) ≤ cl(
V
)
[13]
.
Definition 3.4.
A function
f
: (
X,Τ
) → (
Y,T'
) is said to be
intuitionistic fuzzy weakly θ-continuous
if for each intuitionistic fuzzy point
x(α,β)
in
X
and each intuitionistic fuzzy open
q
-neighborhood
V
of
f
(
x(α,β)
), there exists an intuitionistic fuzzy open
q
-neighborhood
U
of
x(α,β)
such that
f
(
U
) ≤ cl(
V
).
Theorem 3.5.
A function
f
: (
X,Τ
) → (
Y,T'
) is IF weakly
θ
-continuous if and only if for each IF point
x(α,β)
in
X
and each IF open
θ
-neighborhood
N
of
f
(
x(α,β)
) in
Y
,
f
–1
(
N
) is an IF
q
-neighborhood of
x(α,β)
.
Proof.
Let
f
be an IF weakly
θ
-continuous function, and let
x(α,β)
be an IF point in
X
. Let
N
be an IF
θ
-neighborhood of
f
(
x(α,β)
) in
Y
. Then there exists an IF open
q
-neighborhood
V
of
f
(
x(α,β)
) such that cl(
V
) ≤
N
. Since
f
is IF weakly
θ
-continuous, there exists an IF
q
-neighborhood
U
of
x(α,β)
such that
f
(
U
) ≤ cl(
V
) ≤
N
. Thus
U
≤
f
–1
(
N
). Therefore, there exists an IF
q
-neighborhood
U
of
x(α,β)
such that
U
≤
f
–1
(
N
). Hence
f
–1
(
N
) is an IF
q
-neighborhood of
x(α,β)
.
Conversely, let
x(α,β)
be an IF point in
X
, and let
V
be an IF open
q
-neighborhood of
f
(
x(α,β)
). Then cl(
V
) is an IF
θ
-neighborhood of
f
(
x(α,β)
). By the hypothesis,
f
–1
(cl(
V
)) is an an IF
q
-neighborhood of
x(α,β)
. Then there exists an IF open set
U
such that
x(α,β)
qU
≤
f
–1
(cl(
V
)). Thus
f
(
U
) ≤ cl(
V
). Therefore there exists an IF open
q
-neighborhood
U
of
x(α,β)
such that
f
(
U
) ≤ cl(
V
). Hence
f
is an IF weakly
θ
-continuous function.
Theorem 3.6.
If a function
f
: (
X,Τ
) → (
Y,T'
) is IF weakly
θ
-continuous, then
-
(1)f(cl(A)) ≤ clθ(f(A)) for each IF setAinX,
-
(2)f(cl(int(cl(f–1(B))))) ≤ clθ(B) for each IF setBinY.
Proof.
(1) Let
x(α,β)
∈ cl(
A
), and let
V
be an IF open
q
-neighborhood of
f
(
x(α,β)
). Since
f
is IF weakly
θ
-continuous, there exists an IF open
q
-neighborhood
U
of
x(α,β)
such that
f
(
U
) ≤ cl(
V
). Since
x(α,β)
∈ cl(
A
),
UqA
. Thus
f
(
U
)
qf
(
A
). Since
f
(
U
) ≤ cl(
V
), we have cl(
V
)
qf
(
A
). Thus for each IF open
q
-neighborhood
V
of
f
(
x(α,β)
), cl(
V
)
qf
(
A
). Hence
f
(
x(α,β)
) ∈ cl
θ
(
f
(
A
)).
(2) Let
B
be an IF set in
Y
and
x(α,β)
∈ cl(int(cl(
f
–1
(
B
)))). Let
V
be an IF open
q
-neighborhood of
f
(
x(α,β)
). Since
f
is IF weakly
θ
-continuous, there exists an IF open
q
-neighborhood
U
of
x(α,β)
such that
f
(
U
) ≤ cl(
V
). Since int(cl(
f
–1
(
B
))) ≤ cl(
f
–1
(
B
)),
-
cl(int(cl(f–1(B)))) ≤ cl(cl(f–1(B))) = cl(f–1(B)).
Since
x(α,β)
∈ cl(int(cl(
f
–1
(
B
)))),
x(α,β)
∈ cl(
f
–1
(
B
)). Thus
f
–1
(
B
)
qU
, or
Bqf
(
U
). Since
f
(
U
) ≤ cl(
V
), we have cl(
V
)
qB
. Therefore
f
(
x(α,β)
) ∈ cl
θ
(
B
). Hence we obtain
f
(cl(int(cl(
f
–1
(
B
))))) ≤ cl
θ
(
B
), for each IF set
B
in
Y
.
Theorem 3.7.
Let
f
: (
X,Τ
) → (
Y,T'
) be a function. Then the following statements are equivalent:
-
(1)fis an IF weaklyθ-continuous function.
-
(2) For each IF open setUwithx(α,β)qf–1(U),x(α,β)qint(f–1(cl(U))).
Proof. (1) ⇒ (2). Let
f
be an IF weakly
θ
-continuous function, and let
U
be an IF open set with
x(α,β)
qf
–1
(
U
). Then
f
(
x(α,β)
)
qU
. By the definition of IF weakly
θ
-continuous, there exists an IF open
q
-neighborhood
V
of
x(α,β)
such that
f
(
V
) ≤ cl(
U
). Thus
V
≤
f
–1
(cl(
U
)), i.e.
Therefore,
x(α,β)
∉ cl((
f
–1
(cl(
U
)))
c
) = (int(
f
–1
(cl(
U
))))
c
. Hence we have
x(α,β)
q
(int(
f
–1
(cl(
U
)))).
(2) ⇒ (1). Let the condition hold, and let
x(α,β)
be any IF point in
X
and
V
an IF open
q
-neighborhood of
f
(
x(α,β)
). Then
x(α,β)
qf
–1
(
V
). By the hypothesis,
Put
U
= int(
f
–1
(cl(
V
))). Then
U
is an IF open
q
-neighborhood of
x(α,β)
. Since int(
f
–1
(cl(
V
))) ≤
f
–1
(cl(
V
)),
-
f(int(f–1(cl(V)))) ≤f(f–1(cl(V))) ≤ cl(V).
Thus
f
(
U
) ≤ cl(
V
). Therefore there exists an IF open
q
-neighborhood
U
of
x(α,β)
such that
f
(
U
) ≤ cl(
V
). Hence
f
is an IF weakly
θ
-continuous function.
4. Intuitionistic Fuzzy θ-Compactness
Definition 4.1.
A collection {
Gi
|
i
∈
I
} of intuitionistic fuzzy
θ
-open sets in an intuitionistic fuzzy topological space (
X,Τ
) is said to be an
intuitionistic fuzzy θ-open cover
of a set
A
if
A
≤
˅
{
Gi
|
i
∈
I
}.
Definition 4.2.
An intuitionistic fuzzy topological space (
X,Τ
) is said to be
intuitionistic fuzzy θ-compact
if every intuitionistic fuzzy
θ
-open cover of
X
has a finite subcover.
Definition 4.3.
A subset
A
of an intuitionistic fuzzy topological space (
X,Τ
) is said to be
intuitionistic fuzzy θ-compact
if for every collection {
Gi
|
i
∈
I
} of intuitionistic fuzzy
θ
-open sets of
X
such that
A
≤
˅
{
Gi
|
i
∈
I
}, there is a finite subset
I
0
of
I
such that
A
≤
˅
{
Gi
|
i
∈
I
0
}.
Remark 4.4.
Since every IF
θ
-open set is IF open, it follows that every IF compact space is IF
θ
-compact.
Theorem 4.5.
An IF topological space (
X,Τ
) is IF
θ
-compact if and only if every family of IF
θ
-closed subsets of
X
with the finite intersection property has a nonempty intersection.
Proof.
Let
X
be IF
θ
-compact and let
F
= {
Fi
|
i
∈
I
} denote any family of IF
θ
-closed subsets of
X
with the finite intersection property. Suppose that
Then,
is an IF
θ
-open cover of
X
. Since
X
is IF
θ
-compact, there is a finite subset
I
0
of
I
such that
This implies that
which contradicts the assumption that
F
has a finite intersection property. Hence,
Let
g
= {
Gi
|
i
∈
I
} denote an IF
θ
-open cover of
X
and consider the family
of an IF
θ
-closed set. Since
g
is a cover of
X
,
Hence,
g'
does not have the finite intersection property, i.e., there are finite numbers of IF
θ
-open sets {
G
1
,
G
2
, … ,
Gn
} in
g
such that
This implies that {
G
1
,
G
2
, … ,
Gn
} is a finite subcover of
X
in
g
. Hence,
X
is IF
θ
-compact.
Theorem 4.6.
Let (
X,Τ
) denote an IF topological space and
Τθ
denote the IF topology on
X
generated using the subbase of all IF
θ
-open sets in
X
. Then, (
X,Τ
) is IF
θ
-compact if and only if (
X,Τθ
) is IF compact.
Proof.
Let (
X,Τθ
) be IF compact and let
g
= {
Gi
|
i
∈
I
} denote an IF
θ
-open cover of
X
in
T
. Since for each
i
∈
I, Gi
∈
Τθ, g
is an IF open cover of
X
in
Τθ
. Since (
X,Τθ
) is IF compact,
g
has a finite subcover of
X
. Hence, (
X,Τ
) is IF
θ
-compact.
Let (
X,Τ
) be IF
θ
-compact and let
g
= {
Gi
|
Gi
∈
Τθ, i
∈
I
} denote an IF open cover of
X
in
Τθ
. Since for each
i
∈
I, Gi
∈
Τθ
,
Gi
is an IF
θ
-open set in (
X,Τ
). Therefore,
g
is an IF
θ
-open cover of
X
in
T
. Since (
X,Τ
) is IF
θ
-compact,
g
has a finite subcover of
X
. Hence, (
X,Τθ
) is IF compact.
Theorem 4.7.
Let
A
be an IF
θ
-closed subset of an IF
θ
-compact space
X
. Then,
A
is also IF
θ
-compact.
Proof.
Let
A
denote an IF
θ
-closed subset of
X
and let
g
= {
Gi
|
i
∈
I
} denote an IF
θ
-open cover of
A
. Since
Ac
is an IF
θ
-open subset of
X, g
= {
Gi
|
i
∈
I
} ∪
Ac
is an IF
θ
-open cover of
X
. Since
X
is IF
θ
-compact, there is a finite subset
I
0
of
I
such that
Hence,
A
is IF
θ
-compact relative to
X
.
Theorem 4.8.
An IF topological space (
X,Τ
) is IF
θ
-compact if and only if every family of IF closed subsets of
X
in
Τθ
with the finite intersection property has a nonempty intersection.
Proof.
Trivial by Theorem 4.5.
Theorem 4.9.
Let (
X,Τ
) and (
Y,U
) denote IF topological spaces. Let
Τθ
denote an IF topology on
X
generated by the subbase of all IF
θ
-open sets in
X
and let
Uθ
denote an IF topology on
Y
generated by the subbase of all IF
θ
-open sets in
Y
. Then, a function
f
: (
X,Τ
) → (
Y,U
) is IF
θ
-irresolute if and only if
f
: (
X,Τθ
) → (
Y,Uθ
) is IF continuous.
Proof.
Trivial.
Recall that a function
f
: (
X,Τ
) → (
Y,T'
) is said to be
intuitionistic fuzzy strongly θ-continuous
if for each IF point
x(α,β)
in
X
and for each IF open
q
-neighborhood
V
of
f
(
x(α,β)
), there exists an IF open
q
-neighborhood
U
of
x(α,β)
such that
f
(cl(
U
)) ≤
V
(
[9]
).
-
Theorem 4.10.(1) An IF stronglyθ-continuous image of an IFθ-compact set is IF compact.
-
(2) Let (X,Τ) and (Y,U) denote IF topological spaces and letf: (X,Τ) → (Y,U) be IFθ-irresolute. If a subsetAofXis IFθ-compact, then imagef(A) is IFθ-compact.
Proof.
(1) Let
f
: (
X,Τ
) → (
Y,U
) denote an IF strongly
θ
-continuous mapping from an IF
θ
-compact space
X
onto an IF topological space
Y
. Let
g
= {
Gi
|
i
∈
I
} be an IF open cover of
Y
. Since
f
is an IF strongly
θ
-continuous function,
f
: (
X,Τθ
) → (
Y,U
) is an IF continuous function (Theorem 4.2 of
[11]
). Therefore, {
f
–1
(
Gi
) |
i
∈
I
} is an IF
θ
-open cover of
X
. Since
X
is IF
θ
-compact, there is a finite subset
I
0
of
I
such that
Since
f
is onto, {
Gi
|
i
∈
I
0
} is a finite subcover of
Y
. Hence,
Y
is IF compact.
(2) Let
g
= {
Gi
|
i
∈
I
} be an IF
θ
-open cover of
f
(
A
) in
Y
. Since
f
is an IF
θ
-irresolute, for each
Gi
,
f
–1
(
Gi
) is an IF
θ
-open set. Moreover, {
f
–1
(
Gi
) |
i
∈
I
} is an IF
θ
-open cover of
A
. Since
A
is IF
θ
-compact relative to
X
, there exists a finite subset
I
0
of
I
such that
A
≤
˅
{
f
–1
(
Gi
) |
i
∈
I
0
}. Therefore,
f
(
A
) ≤
˅
{
Gi
|
i
∈
I
0
}. Hence,
f
(
A
) is IF
θ
-compact relative to
Y
.
Theorem 4.11.
Let
A
and
B
be subsets of an IF topological space (
X,Τ
). If
A
is IF
θ
-compact and
B
is IF
θ
-closed in
X
, then
A
∧
B
is IF
θ
-compact.
Proof.
Let
g
= {
Gi
|
i
∈
I
} be an IF
θ
-open cover of
A
˄
B
in
X
. Since
Bc
is IF
θ
-open in
X
, (
˅
{
Gi
|
i
∈
I
}) ∨
Bc
is an IF
θ
-open cover of
A
. Since
A
is IF
θ
-compact, there is a finite subset
I
0
of
I
such that
A
≤ (
˅
{
Gi
|
i
∈
I
0
}) ∨
Bc
. Therefore,
A
∧
B
≤ (
˅
{
Gi
|
i
∈
I
0
}). Hence,
A
∧
B
is IF
θ
-compact.
5. Conclusion
We introduced IF
θ
-irresolute and weakly
θ
-continuous functions, and intuitionistic fuzzy
θ
-compactness in intuitionistic fuzzy topological spaces. We showed that intuitionistic fuzzy
θ
-compactness is strictly weaker than intuitionistic fuzzy compactness. Moreover, we showed that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to intuitionistic fuzzy
θ
-compactness in the original intuitionistic fuzzy topology.
- Conflict of Interest
No potential conflict of interest relevant to this article was reported.
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