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
μ_{A}
:
X
→
I
and
γ_{A}
:
X
→
I
denote the degree of membership and the degree of nonmembership, respectively, and
μ_{A}
+
γ_{A}
≤ 1. Obviously, every fuzzy set
μ_{A}
in
X
is an intuitionistic fuzzy set of the form (
μ_{A}
, 1 –
μ_{A}
).
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
= (
μ_{A}
,
γ_{A}
),
B
= (
μ_{B}
,
γ_{B}
). 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
= (
μ_{A}
,
γ_{A}
) in
X
, denoted by
x
_{(α,β)}
∈
A
, if
α
≤
μ_{A}
(
x
) and
β
≥
γ_{A}
(
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 bequasicoincident, denoted byUqV, if there exists an elementx∈Xsuch thatμU(x) >γV(x) orγU(x) <μV(x).
The word ‘not quasicoincident’ 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 ϵneighborhood
(
qneighborhood
) 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
= (
μ_{U}
,
γ_{U}
) 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 θcluster 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 θclosure
of
A
and is denoted by cl
_{θ}
(
A
). An intuitionistic fuzzy set
A
is called an
intuitionistic fuzzy θclosed set
if
A
= cl
_{θ}
(
A
). The complement of an intuitionistic fuzzy
θ
closed set is said to be an
intuitionistic fuzzy θopen 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 θinterior
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 θcontinuous
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 θIrresolute and Weakly θContinuity
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 θirresolute
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
Τ_{θ}
be an IF topology on
X
generated using the subbase of all the IF
θ
open sets in
X
, and let
U_{θ}
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,Τ_{θ}
) → (
Y,U_{θ}
) is IF continuous.
Proof.
Trivial.
Recall that a fuzzy set
A
is said to be a
fuzzy θneighborhood
of a fuzzy point
x_{α}
if there exists a fuzzy closed
q
neighborhood
U
of
x_{α}
, such that
[13]
.
Definition 3.3.
An intuitionistic fuzzy set
A
is said to be an
intuitionistic fuzzy θneighborhood
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 θcontinuous
function if for each fuzzy point
x_{α}
in
X
and each fuzzy open
q
neighborhood
V
of
f
(
x_{α}
), there exists a fuzzy open
q
neighborhood
U
of
x_{α}
such that
f
(
U
) ≤ cl(
V
)
[13]
.
Definition 3.4.
A function
f
: (
X,Τ
) → (
Y,T'
) is said to be
intuitionistic fuzzy weakly θcontinuous
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 weaklyθcontinuous 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 θCompactness
Definition 4.1.
A collection {
G_{i}

i
∈
I
} of intuitionistic fuzzy
θ
open sets in an intuitionistic fuzzy topological space (
X,Τ
) is said to be an
intuitionistic fuzzy θopen cover
of a set
A
if
A
≤
˅
{
G_{i}

i
∈
I
}.
Definition 4.2.
An intuitionistic fuzzy topological space (
X,Τ
) is said to be
intuitionistic fuzzy θcompact
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 θcompact
if for every collection {
G_{i}

i
∈
I
} of intuitionistic fuzzy
θ
open sets of
X
such that
A
≤
˅
{
G_{i}

i
∈
I
}, there is a finite subset
I
_{0}
of
I
such that
A
≤
˅
{
G_{i}

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
= {
F_{i}

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
= {
G_{i}

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}
, … ,
G_{n}
} in
g
such that
This implies that {
G
_{1}
,
G
_{2}
, … ,
G_{n}
} 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
Τ_{θ}
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,Τ_{θ}
) is IF compact.
Proof.
Let (
X,Τ_{θ}
) be IF compact and let
g
= {
G_{i}

i
∈
I
} denote an IF
θ
open cover of
X
in
T
. Since for each
i
∈
I, G_{i}
∈
Τ_{θ}, g
is an IF open cover of
X
in
Τ_{θ}
. Since (
X,Τ_{θ}
) 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
= {
G_{i}

G_{i}
∈
Τ_{θ}, i
∈
I
} denote an IF open cover of
X
in
Τ_{θ}
. Since for each
i
∈
I, G_{i}
∈
Τ_{θ}
,
G_{i}
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,Τ_{θ}
) 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
= {
G_{i}

i
∈
I
} denote an IF
θ
open cover of
A
. Since
A^{c}
is an IF
θ
open subset of
X, g
= {
G_{i}

i
∈
I
} ∪
A^{c}
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
Τ_{θ}
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
Τ_{θ}
denote an IF topology on
X
generated by the subbase of all IF
θ
open sets in
X
and let
U_{θ}
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,Τ_{θ}
) → (
Y,U_{θ}
) is IF continuous.
Proof.
Trivial.
Recall that a function
f
: (
X,Τ
) → (
Y,T'
) is said to be
intuitionistic fuzzy strongly θcontinuous
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 stronglyθcontinuous image of an IFθcompact set is IF compact.

(2) Let (X,Τ) and (Y,U) denote IF topological spaces and letf: (X,Τ) → (Y,U) be IFθirresolute. If a subsetAofXis IFθcompact, then imagef(A) is IFθcompact.
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
= {
G_{i}

i
∈
I
} be an IF open cover of
Y
. Since
f
is an IF strongly
θ
continuous function,
f
: (
X,Τ_{θ}
) → (
Y,U
) is an IF continuous function (Theorem 4.2 of
[11]
). Therefore, {
f
^{–1}
(
G_{i}
) 
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, {
G_{i}

i
∈
I
_{0}
} is a finite subcover of
Y
. Hence,
Y
is IF compact.
(2) Let
g
= {
G_{i}

i
∈
I
} be an IF
θ
open cover of
f
(
A
) in
Y
. Since
f
is an IF
θ
irresolute, for each
G_{i}
,
f
^{–1}
(
G_{i}
) is an IF
θ
open set. Moreover, {
f
^{–1}
(
G_{i}
) 
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}
(
G_{i}
) 
i
∈
I
_{0}
}. Therefore,
f
(
A
) ≤
˅
{
G_{i}

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
= {
G_{i}

i
∈
I
} be an IF
θ
open cover of
A
˄
B
in
X
. Since
B^{c}
is IF
θ
open in
X
, (
˅
{
G_{i}

i
∈
I
}) ∨
B^{c}
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
≤ (
˅
{
G_{i}

i
∈
I
_{0}
}) ∨
B^{c}
. Therefore,
A
∧
B
≤ (
˅
{
G_{i}

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.
Çoker D.
1996
“An introduction to fuzzy subspaces in intuitionisticfuzzy topological spaces”
The Journal of Fuzzy Mathematics
4
(4)
749 
764
Çoker D.
,
Demirci M.
1995
“On intuitionistic fuzzy points”
Notes on Intuitionistic Fuzzy Sets
1
(2)
79 
84
Min W. K.
,
Kim Y. K.
2011
“A note on fuzzy rm precontinuityand fuzzy rminimal compactness on fuzzyrminimal spaces”
Journal of The Korean Institute of Intelligent Systems
21
(1)
128 
131
DOI : 10.5391/JKIIS.2011.21.1.128
Lim P. K.
,
Kim S. R.
,
Hur K.
2010
“Intuitionistic smoothtopological spaces”
Journal of The Korean Institute of Intelligent Systems
20
(6)
875 
883
DOI : 10.5391/JKIIS.2011.20.6.875
Kim J. I.
,
Min W. K.
,
Yoo Y. H.
2009
“Fuzzy γcompactness on fuzzy γminimal spaces”
International Journal of Fuzzy Logic and Intelligent Systems
9
(4)
281 
284
DOI : 10.5391/IJFIS.2009.9.4.281
Kandil O.
,
Tantawy M.Yakout
,
Saleh S.
2009
“C*compactness in lfuzzy topological spaces”
International Journal of Fuzzy Logic and Intelligent Systems
9
(4)
261 
268
DOI : 10.5391/IJFIS.2009.9.4.261
Hanafy I. M.
,
Abd ElAziz A. M.
,
Salman T. M.
2006
“Intuitionistic fuzzy θclosure operator”
Boletn de la Asociacion Matematica Venezolana
13
(1)
27 
39
Hanafy I. M.
2003
“Intuitionistic fuzzy functions”
International Journal of Fuzzy Logic and Intelligent Systems
3
(2)
200 
205
DOI : 10.5391/IJFIS.2003.3.2.200
Lee S. J.
,
Eoum Y. S.
2010
“Intuitionistic fuzzy θclosureand θinterior”
Communications of the Korean Mathematical Society
25
(2)
273 
282
DOI : 10.4134/CKMS.2010.25.2.273
Hanafy I. M.
2002
“On fuzzy γopen sets and fuzzy γcontinuity in intuitionistic fuzzy topological spaces”
The Journal of Fuzzy Mathematics
10
(1)
9 
19
Mukherjee M. N.
,
Sinha S. P.
1990
“On some nearfuzzycontinuous functions between fuzzy topological spaces”
Fuzzy Sets and Systems
34
(2)
245 
254
DOI : 10.1016/01650114(90)90163Z