In this paper, we characterize the intuitionistic fuzzy
δ
continuous, intuitionistic fuzzy weakly
δ
continuous, intuitionistic fuzzy almost continuous, and intuitionistic fuzzy almost strongly
θ
continuous functions in terms of intuitionistic fuzzy
δ
closure and interior or
θ
closure and interior.
1. Introduction and Preliminaries
By using the intuitionistic fuzzy sets introduced by Atanassov
[1]
, Çoker and his colleagues
[2

4]
introduced the intuitionistic fuzzy topological space, which is a generalization of the fuzzy topological space. Moreover, many researchers have studied about this space
[5

12]
.
In the intuitionistic fuzzy topological spaces, Hanafy et al.
[13]
introduced the concept of intuitionistic fuzzy
θ
closure as a generalization of the concept of fuzzy
θ
closure by Mukherjee and Sinha
[14
,
15]
, and characterized some types of functions. In the previous papers
[16
,
17]
, we also introduced and investigated some properties of the concept of intuitionistic fuzzy
θ
interior and
δ
closure in intuitionistic fuzzy topological spaces.
In this paper, we characterize the intuitionistic fuzzy
δ
continuous, intuitionistic fuzzy weakly
δ
continuous, intuitionistic fuzzy almost continuous, and intuitionistic fuzzy almost strongly
θ
continuous functions in terms of intuitionistic fuzzy
δ
closure and interior, or
θ
closure and interior.
Let
X
be a nonempty set and
I
the unit interval [0, 1]. An
intuitionistic fuzzy set A
in
X
is an object of the form
A
= (
μ_{A}, ϓ_{A}
), where the functions
μ_{A}
:
X
→
I
and
ϓ_{A}
:
X
→
I
denote the degree of membership and the degree of nonmembership, respectively, and
μ_{A}
+
ϓ_{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 “IF” stands for “intuitionistic fuzzy.” For the notions which are not mentioned in this paper, refer to
[17]
.
Theorem 1.1
(
[7]
). The following are equivalent:

(1) An IF setAis IF semiopen inX.

(2)A≤ cl(int(A)).
Corollary 1.2
(
[17]
). If
U
is an IF regular open set, then
U
is an IF
δ
open set.
Theorem 1.3
(
[17]
). For any IF semiopen set
A
, we have cl(
A
) = cl
_{δ}
(
A
).
Lemma 1.4
(
[17]
). (1) For any IF set
U
in an IF topological space (
X, T
), int(cl)undefined(
U
)) is an IF regular open set.

(2) For any IF open setUin an IF topological space (X, T) such thatx(αβ)qU, int(cl(U)) is an IF regular openqneighborhood ofx(α,β).
Theorem 1.5
(
[12]
). Let
x(_{αβ})
be an IF point in
X
, and
U
= (μU, ϓU) an IF set in
X
. Then
x
_{(α,β)}
∈ cl(
U
) if and only if
UqN
, for any IF
q
neighborhood
N
of
x
_{(α,β)}
.
2. Intuitionistic Fuzzy δcontinuous andWeakly δcontinuous Functions
Recall that a fuzzy set
N
in (
X, T
) is said to be a
fuzzy δneighborhood
of a fuzzy point
x_{α}
if there exists a fuzzy regular open
q
neighborhood
V
of
x_{α}
such that
or equivalently
V
≤
N
(See
[14]
). Now, we define a similar definition in the intuitionistic fuzzy topological spaces.
Definition 2.1.
An intuitionistic fuzzy set
N
in (
X, T
) is said to be an
intuitionistic fuzzy δneighborhood
of an intuitionistic fuzzy point
x
_{(α,β)}
if there exists an intuitionistic fuzzy regular open
q
neighborhood
V
of
x
_{(α,β)}
such that
Lemma 2.2.
An IF set
A
is an IF
δ
open set in (
X, T
) if and only if for any IF point
x
_{(α,β)}
with
x_{(α,β)}qA, A
is an IF
δ
neighborhood of
x
_{(α,β)}
.
Proof
. Let
A
be an IF
δ
open set in (
X, T
) such that
x_{(α,β)}qA
. Then
Since
A^{c}
is an IF
δ
closed set, we have
x
_{(α,β)}
∉
A^{c}
= cl
_{δ}
(
A^{c}
). Then there exists an IF regular open
q
neighborhood
U
of
x
_{(α,β)}
such that
Thus
U
≤
A
. Hence
A
is an IF
δ
neighborhood of
x
_{(α,β)}
.
Conversely, to show that
A^{c}
is an IF
δ
closed set, take any
x
_{(α,β)}
∉
A^{c}
. Then we have
x_{(α,β)}qA
. Thus
A
is an IF
δ
neighborhood of
x
_{(α,β)}
. Therefore there exists an IF regular open
q
neighborhood
V
of
x
_{(α,β)}
such that
V
≤
A^{c}
, i.e.
x
_{(α,β)}
∉ cl
_{δ}
(
A^{c}
). Since cl
_{δ}
(
A^{c}
) ≤
A^{c}
, we have
A^{c}
is an IF
δ
closed set. Hence
A
is an IF
δ
open set.
Recall that a function
f
: (
X, T
) → (
Y, T’
) is said to be a
fuzzy δcontinuous
function if for each fuzzy point
x_{α}
in
X
and for any fuzzy regular open
q
neighborhood
V
of
f(x_{(α)})
, there exists an fuzzy regular open
q
neighborhood
U
of
x_{(α)}
such that
f(U) ≤ V
(See
[18]
). We define a similar definition in the intuitionistic fuzzy topological spaces as follows.
Definition 2.3.
A function
f : (X, T) → (Y, T’)
is said to be
intuitionistic fuzzy δcontinuous
if for each intuitionistic fuzzy point
x
_{(α,β)}
in
X
and for any intuitionistic fuzzy regular open
q
neighborhood
V
of
f
(
x
_{(α,β)}
), there exists an intuitionistic fuzzy regular open
q
neighborhood
U
of
x
_{(α,β)}
such that
Now, we characterize the intuitionistic fuzzy
δ
continuous function in terms of IF
δ
closure and IF
δ
interior.
Theorem 2.4.
Let
f : (X, T) → (Y, T’)
be a function. Then the following statements are equivalent:

(1)fis an IFδcontinuous function.

(2)f(clδ(U)) ≤ clδ(f(U)) for each IF setUinX.

(3) clδ(f1(V)) ≤f1(clδ(V)) for each IF setVinY.

(4)f1(intδ(V)) ≤ intδ(f1(V)) for each IF setVinY.
Proof
. (1) ⇒ (2). Let
x
_{(α,β)}
∈ cl
_{δ}
(U), and let B be an IF regular open
q
neighborhood of
f
(
x
_{(α,β)}
) in
Y
. By (1), there exists an IF regular open
q
neighborhood
A
of
x
_{(α,β)}
such that
f
(
A
) ≤
B
. Since
x
_{(α,β)}
∈ cl
_{δ}
(
U
) and
A
is an IF regular open
q
neighborhood of
x
_{(α,β)}
,
AqU
. So
f(A)qf(U)
. Since
f(A)
≤
B, Bqf(U)
. Then
f
(
x
_{(α,β)}
) ∈ cl
_{δ}
(
f(U)
). Hence
f(clδ(U)))
≤ cl
_{δ}
(
f
(
U
)).
(2) ⇒ (3). Let
V
be an IF set in
Y
. Then
f^{1}
(
V
) is an IF setin
X
. By (2),
f
(cl
_{δ}
(
f^{1}
(
V
))) ≤ cl
_{δ}
(
f
(
f^{1}
(
V
))) ≤ cl
_{δ}
(
V
).Thus cl
_{δ}
(
f^{1}
(
V
)) ≤
f^{1}
(cl
_{δ}
(
V
)).
(3) ⇒ (1). Let
x
_{(α,β)}
be an IF point in
X
, and let
V
be an IFregular open
q
neighborhood of
f
(
x
_{(α,β)}
in
Y
. Since
V^{c}
is anIF regular closed set,
V^{c}
is an IF semiopen set. By Theorem1.3, cl(
V^{c}
) = cl
_{δ}
(
V^{c}
). Since
f
(
x_{(α,β)})qV
,
f
(
x
_{(α,β)}
) ∉
V^{c}
= cl(
V^{c}
) = cl
_{δ}
(
V^{c}
). Therefore
x
_{(α,β)}
∉
f^{1}
(cl
_{δ}
(
V^{c}
)). By (3),
x
_{(α,β)}
∉ cl
_{δ}
(
f^{1}
(
V^{c}
)). Then there exists an IF regular open
q
neighborhood
U
of
x
_{(α,β)}
such that
So
U
≤
f^{1}
(
V
), i.e.
f
(
U
) ≤
V
. Hence
f
is an IF
δ
continuous function.
(3) ⇒ (4). Let
V
be an IF set in
Y
. By (3), cl
_{δ}
(
f^{1}
(
V^{c}
)) ≤
f^{1}
(cl
_{δ}
(
V^{c}
)). Thus
(4) ⇒ (3). Let
V
be an IF set in
Y
. Then
V^{c}
is an IF set in
Y
. By the hypothesis,
f^{1}
(int
_{δ}
(
V^{c}
)) ≤ int
_{δ}
(
f^{1}
(
V^{c}
)). Thus
Hence cl
_{δ}
(
f^{1}
(
V
)) ̤
f^{1}
(cl
_{δ}
(
V
)).
The intuitionistic fuzzy
δ
continuous function is also characterized in terms of IF
δ
open and IF
δ
closed sets.
Theorem 2.5.
Let
f : (X, T) → (Y, T’)
be a function. Then the following statements are equivalent:

(1)fis an IFδcontinuous function.

(2)f1(A) is an IFδclosed set for each IFδclosed setAinX.

(3)f1(A) is an IFδopen set for each IFδopen setAinX.
Proof
. (1) ⇒ (2). Let
A
be an IF
δ
closed set in
X
. Then
A
= cl
_{δ}
(
A
). By Theorem 2.4, cl
_{δ}
(
f^{1}
(
A
)) ≤
f^{1}
(cl
_{δ}
(
A
)) =
f^{1}
(
A
). Hence
f^{1}
(
A
) = cl
_{δ}
(
f^{1}
(
A
)). Therefore,
f^{1}
(
A
) is an IF
δ
closed set.
(2) ⇒ (3). Trivial.
(3) ⇒ (1). Let
x
_{(α,β)}
be an IF point in
X
, and let
V
be an IF regular open
q
neighborhood of
f
(
x
_{(α,β)}
). By Corollary 1.2,
V
is an IF
δ
open set. By the hypothesis,
f^{1}
(
V
) is an IF
δ
open set. Since
x_{(α,β)}qf^{1}
(
V
), by Lemma 2.2, we have that
f^{1}
(
V
) is an IF
δ
neighborhood of
x
_{(α,β)}
. Therefore, there exists an IF regular open
q
neighborhood
U
of
x
_{(α,β)}
such that
U
≤
f^{1}
(
V
). Hence
f
(
U
) ≤
V
.
The intuitionistic fuzzy
δ
continuous function is also characterized in terms of IF
δ
neighborhoods.
Theorem 2.6.
A function
f : (X, T) → (Y, T’)
is IF
δ
continuous if and only if for each IF point
x
_{(α,β)}
of
X
and each IF
δ
neighborhood
N
of
f
(
x
_{(α,β)}
), the IF set
f^{1}
(
N
) is an IF
δ
neighborhood of
x
_{(α,β)}
.
Proof
. Let
x
_{(α,β)}
be an IF point in
X
, and let
N
be an IF
δ
neighborhood of
f
(
x
_{(α,β)}
). Then there exists an IF regular open
q
neighborhood
V
of
f
(
x
_{(α,β)}
) such that
V
≤
N
. Since
f
is an an IF
δ
continuous function, there exists an IF regular open qneighborhood
U
of
x
_{(α,β)}
such that
f
U
) ≤
V
. Thus,
U
≤
f^{1}
(
V
) ≤
N
. Hence
f^{1}
(
N
) is an IF
δ
neighborhood of
x
_{(α,β)}
.
Conversely, let
x
_{(α,β)}
be an IF point in
X
, and
V
an IF regular open
q
neighborhood of
f
(
x
_{(α,β)}
). Then
V
is an IF
δ
neighborhood of
f
(
x
_{(α,β)}
). By the hypothesis,
f^{1}
(
V
) is an IF
δ
neighborhood of
x
_{(α,β)}
. By the definition of IF
δ
neighborhood, there exists an IF regular open
q
neighborhood
U
of
x
_{(α,β)}
such that
U
≤
f^{1}
(
V
). Thus
f
(
U
) ≤
V
. Hence
f
is an IF
δ
continuous function.
Theorem 2.7.
Let
f : (X, T) → (Y, T’)
be a bijection. Then the following statements are equivalent:

(1) f is an IFδcontinuous function.

(2) intδ(f(U)) ≤f(intδ(U)) for each IF setUinX.
Proof
. (1) ⇒ (2). Let
U
be an IF set in
X
. Then
f
(
U
) is an IF set in
Y
. By Theorem 2.4,
f^{1}
int
_{δ}
(
f
(
U
))) ≤ int
_{δ}
(
f^{1}
(
f
(
U
))). Since
f
is onetoone,
Since
f
is onto,
(2) ⇒ (1). Let
V
be an IF set in
Y
. Then
f^{1}
(
V
) is an IF set in
X
. By the hypothesis, int
_{δ}
(
f
(
f^{1}
(
V
))) ≤
f
(int
_{δ}
(
f^{1}
(
V
))). Since
f
is onto,
Since
f
is onetoone,
Hence by Theorem 2.4,
f
is an IF
δ
continuous function.
Recall that a function
f : (X, T) → (Y, T’)
is said to be
fuzzy weakly δcontinuous
if for each fuzzy point
x_{α}
, in
X
and each fuzzy open
q
neighborhood
V
of
f
(
x_{α}
), there exists an fuzzy open
q
neighborhood
U
of
x_{α}
, such that
f
(int(cl(
U
))) ≤ cl(
V
) (See
[14]
). We define a similar definition in the intuitionistic fuzzy topological spaces as follows.
Definition 2.8.
A function
f : (X, T) → (Y, T’)
is said to be
intuitionistic fuzzy weakly δcontinuous
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
Theorem 2.9.
Let
f : (X, T) → (Y, T’)
be a function. Then the following statements are equivalent:

(1)fis an IF weaklyδcontinuous function.

(2)f(clδ(A)) ≤ clθ(f(A)) for each IF setAinX.

(3) clδ(f1(B)) ≤f1(clθ(B)) for each IF setBinY.

(4)f1(intθ(B)) ≤ intδ(f1(B)) for each IF setBinY.
Proof
. (1) ⇒ (2): Let
x
_{(α,β)}
∈ cl
_{δ}
(
A
), and let
V
be an IF open
q
neighborhood of
f
(
x
_{(α,β)}
) in
Y
. Since
f
is an IF weakly
δ
continuous function, there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
f
(int(cl(
U
))) ≤ cl(
V
). Since int(cl(
V
)) is an IF regular open
q
neighborhood of
x
_{(α,β)}
and
x
_{(α,β)}
∈ cl
_{δ}
(
A
), we have
Aq
int(cl(
V
)). Thus
f
(
A
)
qf
(int(cl(
V
))). Since
f
(int(cl(
V
))) ≤ cl(
V
), we have
f
(
A
)
q
cl(
V
). Thus
f
(
x
_{(α,β)}
) ∈ cl
_{θ}
(
f
(
A
)). Hence
f
(cl
_{δ}
(A)) ≤ cl
_{θ}
(
f
(
A
)).
(2) ⇒ (3): Let
B
be an IF set in
Y
. Then
f^{1}
(
B
) is an IF set in
X
. By (2),
f
cl
_{δ}
(
f^{1}
(
B
))) ≤ cl
_{θ}
(
f
(
f^{1}
(
B
))) ≤ cl
_{θ}
(
B
). Hence cl
_{δ}
(
f^{1}
(
B
)) ≤
f^{1}
(cl
_{θ}
(
B
)).
(3) ⇒ (1): Let
x
_{(α,β)}
be an IF point in
X
, and let
V
be an IF open
q
neighborhood of
f
(
x
_{(α,β)}
) in
Y
. Since cl(
V
) ≤ cl(
V
),
Thus
f
(
x
_{(α,β)}
) ∉ cl
_{θ}
((cl(
V
))
^{c}
). By (3),
f
(
x
_{(α,β)}
) ∉ cl
_{δ}
(
f^{1}
((cl(
V
))
^{c}
)). Then there exists an intuitionistic fuzzy regular open
q
neighborhood
U
of
x
_{(α,β)}
such that
Thus int(cl(
U
)) ≤
f^{1}
(cl(
V
)). Therefore, there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
f
(int(cl(
U
))) ≤ cl(
V
). Hence
f
is an IF weakly
δ
continuous function.
(3) ⇒ (4): Let
B
be an IF set in
Y
. Then
B^{c}
is an IF set in
Y
. By (3), cl
_{δ}
(
f^{1}
(
B^{c}
)) ≤
f^{1}
(cl
_{θ}
(
B^{c}
)). Hence we have int
_{δ}
(
f^{1}
(
B
)) = (cl
_{θ}
(
f^{1}
(
B^{c}
))) ≥ (
f^{1}
(cl
_{θ}
(
B^{c}
)))
^{c}
= int
_{θ}
(
f^{1}
(
B
)).
(4) ⇒ (3): Similarly.
Theorem 2.10.
A function
f : (X, T) → (Y, T’)
is IF weakly
δ
continuous if and only if for each IF point
x
_{(α,β)}
in
X
and each IF
θ
neighborhood
N
of
f
(
x
_{(α,β)}
), the IF set
f^{1}
N
) is an IF
δ
neighborhood of
x
_{(α,β)}
.
Proof
. Let
x
_{(α,β)}
be an IF point in
X
, and 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 an IF weakly
δ
continuous function, there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
f
(int(cl(
U
))
x
_{(α,β)}
cl(
V
). Since cl(
V
)
x
_{(α,β)}
N
, int(cl(
U
))
x
_{(α,β)}
f^{1}
(
N
). Hence
f^{1}
(
N
) is an IF
δ
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)}
). Since cl(
V
) ≤ cl(
V
), cl(
V
) is an IF
θ
neighborhood of
f
(
x
_{(α,β)}
). By the hypothesis,
f^{1}
(cl(
V
)) is an IF
θ
neighborhood of
x
_{(α,β)}
. Then there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that int(cl(
V
)) ≤
f^{1}
(cl(
V
)). Thus int(cl(
V
)) ≤
f^{1}
(cl(
V
)). Hence
f
is IF almost strongly
δ
continuous.
Theorem 2.11.
Let
f : (X, T) → (Y, T’)
be an IF weakly
δ
continuous function. Then the following statements are true:

(1)f1(V) is an IFθclosed set inXfor each IFδclosed setVinY.

(2)f1(V) is an IFθopen set inXfor each IFδopen setVinY.
Proof
. (1) Let
B
be an IF
θ
closed set in
Y
. Then cl
_{θ}
(
B
) =
B
. Since
f
is an IF weakly
δ
continuous function, by Theorem 2.9, cl
_{δ}
(
f^{1}
(
B
)) ≤
f^{1}
(cl
_{θ}
(
B
)) =
f^{1}
(
B
). Hence
f^{1}
(
B
) is an IF
δ
closed set in
X
.
(2) Trivial.
Theorem 2.12.
Let
f : (X, T) → (Y, T’)
be a bijection. Then the following statements are equivalent:

(1)fis an IF weaklyδcontinuous function.

(2) intθ(f(A)) ≤fintδ(A)) for each IF setAinX.
Proof
. (1) ⇒ (2): Let
A
be an IF set in
X
. Then
f
(
A
)is an IF set in
Y
. By Theorem 2.9(4),
f^{1}
(int
_{θ}
(
f
(
A
))) ≤ int
_{δ}
(
f^{1}
(
f
(
A
))). Since
f
is onetoone,
Since
f
is onto,
Hence int
_{θ}
(
f
(
A
)) ≤ f(int
_{δ}
(
A
)).
(2) ⇒ (1): Let
B
be an IF set in
Y
. Then
f
^{1}
(
B
) is an IF set in
X
. By (2) int
_{θ}
(
f
(
f
^{1}
(
B
))) ≤
f
(int
_{δ}
(
f
^{1}
(
B
))). Since
f
is onto,

intθ(B) = intθ(f(f1(B)) ≤f(intδ(f1(B)):
f
is onetoone,

f1(intθ(B) ≤f1(f(intδ(f1(B))) = intδ(<(f1(B)).
By Theorem 2.9,
f
is an IF weakly δcontinuous function.
3. IF Almost Continuous and Almost Strongly θcontinuous Functions
Definition 3.1
(
[7]
). A function
f
: (
X, T
) → (
Y, T’
) is said to be
intuitionistic fuzzy almost continuous
if for any intuitionistic fuzzy regular open set
V
in
Y , f^{1}
(
V
) is an intuitionistic fuzzy open set in
X
.
Theorem 3.2
(
[12]
). A function
f
: (
X, T
) → (
Y, T’
) is IF almost continuous if and only if for each IF point
x
_{(α,β)}
in
X
and for any IF open
q
neighborhood
V
of (
x
_{(α,β)}
), there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
Theorem 3.3.
Let
f
: (
X, T
) → (
Y, T’
) be a function. Then the following statements are equivalent:
(1)
f
is an IF almost continuous function.
(2)
f
(cl(
U
)) ≤ cl
_{δ}
(
f(U)
) for each IF set
U
in
X
.
(3) f
^{1}
(
V
) is an IF closed set in
X
for each IF δclosed set
V
in
Y
.
(4) f
^{1}
(
V
) is an IF open set in
X
for each IF δopen set
V
in
Y
.
Proof.
(1) ⇒ (2). Let
x
_{(α,β)}
∈ cl(
U
). Suppose that
f
(
x
_{(α,β)}
) ∉ cl
_{δ}
(
f(U)
). Then there exists an IF open
q
neighborhood
V
of
f
(
x_{(α,β)})
such that
Since
f
is an IF almost continuous function,
f
^{1}
(
V
) is an IF open set in
X
. Since
V_{q}f
(
x
_{(α,β))}
, we have
f
^{1}
(
V
)
qx
_{(α,β)}
. Thus
f
^{1}
(
V
) is an IF open
q
neighborhood of
x
_{(α,β)}
. Since
x
_{(α,β)}
. ∈ cl(
U
), by Theorem 1.5, we have
f
^{1}
(
V
)
qU
. Thus
f(f1(V))qf(U)
. Since
f
(
f
^{1}
(
V
)) ≤ V, we have
Vqf(U)
. This is a contradiction. Hence
f(cl(U))
≤ cl
_{δ}
(
f(U
)).
(2) ⇒ (3). Let
V
be an IF δclosed set in
Y
. Then
f
^{1}
(
V
) is an IF set in
X
. By the hypothesis,

f(cl(f1(V)))) ≤ clδ(f(f1(V))) ≤ clδ(V) =V.
Thus cl(
f
^{1}
(
V
)) ≤
f
^{1}
(
V
). Hence
f
^{1}
(
V
) is an IF closed set in
X
.
(3) ⇒(4). Let
V
be an IF δopen set in
Y
. Then
V^{c}
is an IF δclosed set in
Y
. By the hypothesis,
f
^{1}
(
V^{c}
) = (
f
^{1}
(
V
))
^{c}
is an IF closed set in
X
. Hence
f
^{1}
(
V
) is an IF open set in
X
.
(4) ⇒ (1). Let
x
_{(α,β)}
be an IF point in
X
, and let
V
be an IF open
q
neighborhood of
f(x_{(α,β)})
in
Y
. Then int(cl(
V
)) is an IF regular open
q
neighborhood
f(x_{(α,β)})
. By Theorem 1.2, int(cl(
V
)) is an IF δopen set in
Y
. By the hypothesis,
f
^{1}
(int(cl(
V
))) is IF open in
X
. Since int(cl(
V
))
qf
(
x
_{(α,β)}
), we have
x_{(α,β)}q
)
f
^{1}
(int(cl(
V
))). Thus
x
_{(α,β)}
does not belong to the set (
f
^{1}
(int(cl(
V
))))
^{c}
. Put
B
= (
f
^{1}
(int(cl(
V
))))
^{c}
. Since
B
is an IF closed set and
x
_{(α,β)}
∉
B
= cl(
B
), there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
Then
x
_{(α,β)}
q
^{U}
≤
B^{c}
=
f
^{1}
(int(cl(
V
))). Thus
f(U)
≤ int(
cl(V
)). Hence,
f
is an IF almost continuous function.
Theorem 3.4.
Let
f
: (
X, T
) → (
Y, T’
) be a function. Then the following statements are equivalent:
(1)
f
is an IF almost continuous function.
(2) cl(
f
^{1}
(
V
)) ≤
f
^{1}
(cl
_{δ}
(
V
)) for each IF set
V
in
Y
.
(3) int
_{δ}
(
f
^{1}
(
V
)) ≤
f
^{1}
(int(
V
)) for each IF set
V
in
Y
.
Proof.
(1) ⇒ (2). Let
V
be an IF set in
Y
. Then
f
^{1}
(
V
) is an IF set in
X
. By Theorem 3.3,

f(cl(f1(V))) ≤ clδ(f(f1(V))) ≤ clδ(V).
Thus cl(
f
^{1}
(
V
)) ≤
f
^{1}
(cl
_{δ}
(
V
)).
(2) ⇒ (1). Let
U
be an IF set in
X
. Then
f(U)
is an IF set in
Y
. By the hypothesis, cl(
f
^{1}
(
f(U
))) ≤
f
^{1}
(cl
_{δ}
(
f(U
))). Then

cl(U) ≤ cl(f1(f(U))) ≤f1(clδ(f(U))).
Thus
f(cl(U
)) ≤ cl
_{δ}
(
f(U
)). By Theorem 3.3,
f
is an IF almost continuous function.
(2) ⇒ (3). Let
V
be an IF set in
Y
. Then
V^{c}
is an IF set in
Y
. By the hypothesis, cl(
f
^{1}
(
V^{c}
)) ≤
f
^{1}
(cl
_{δ}
(
V^{c}
)). Thus
(3) ⇒ (2). Let
V
be an IF set in
Y
. Then
V^{c}
is an IF set in
Y
. By the hypothesis,
f
^{1}
(int
_{δ}
(
V^{c}
)) ≤ int(
f
^{1}
>(
V^{c}
)). Thus
Hence cl(
f
^{1}
(
V
)) ≤
f
^{1}
(cl
_{δ}
(
V
)).
Corollary 3.5.
A function
f
: (
X, T
) → (
Y, T’
) is IF almost continuous if and only if for each IF point
x
_{(α,β)}
in
X
and each IF δneighborhood N of
f(x_{(α,β)})
, the IF set
f
^{1}
(
N
) is an IF
q
neighborhood of
x
_{(α,β)}
.
Proof
. Let
x
_{(α,β)}
be an IF point in
X
, and let
N
be an IF δneighborhood of
f(x_{(α,β)})
. Then there exists an IF regular open
q
neighborhood
V
of
f(x_{(α,β)})
such that
V
≤
N
. Since
f
is an IF almost continuous function, there exists an IF open
q
neighborhood
U
of
f(x_{(α,β)})
such that
f(U)
≤ int(cl(
V
)) =
V
≤
N
. Thus there exists an IF open set
U
such that
x
_{(α,β)}
q
^{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
q
neighborhood of f(x(;)). Then int(cl(V )) is an IF regular open qneighborhood of
f(x_{(α,β)})
. Also, int(cl(
V
)) is an IF δneighborhood of
f(x_{(α,β)})
. By the hypothesis,
f
^{1}
(int(cl(
V
)))is an IF
q
neighborhood of
x
_{(α,β)}
. Since
f
^{1}
(int(cl(
V
))) is an IF
q
neighborhood of
x
_{(α,β)}
, there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
U
≤
f
^{1}
(int(cl(
V
))). Thus there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
f(U)
≤ int(cl(
V
)). Hence
f
is an IF almost continuous function.
Theorem 3.6.
Let
f
: (
X, T
) → (
Y, T’
) be a bijection. Then the following statements are equivalent:
(1)
f
is an IF almost continuous function.
(2)
f
(int
_{δ}
(
U
)) ≤ int(f(U)) for each IF set
U
in
X
.
Proof
. Trivial by Theorem 3.4.
Recall that a function
f
: (
X, T
) → (
Y, T’
) is said to be a
fuzzy almost strongly θcontinuous
function if for each fuzzy point
x_{α}
in
X
and each fuzzy open
q
neighborhood
V
of
f(x_{α})
, there exists an fuzzy open
q
neighborhood
U
of
x_{α}
such that
f
(cl(
U
)) ≤ int(cl(
V
)) (See [14]).
Definition 3.7
. A function
f
: (
X, T
) → (
Y, T’
) is said to be
intuitionistic fuzzy almost strongly 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
Theorem 3.8. Let
f
: (
X, T
) → (
Y, T’
) be a function. Then the following statements are equivalent:
(1)
f
is an IF almost strongly
θ
continuous function.
(2)
f
(cl
_{θ}
(
A
)) ≤ cl
_{δ}
(
f(A
)) for each IF set
A
in
X
.
(3)cl
_{θ}
(
f
^{1}
(
B
)) ≤
f
^{1}
(cl
_{δ}
;(
B
)) for each IF set
B
in
Y
.
(4)
f
^{1}
(int
_{δ}
(
B
)) ≤ int
θ
(
f
^{1}
(
B
)) for each IF set
B
in
Y
.
Proof
. (1) ⇒ (2): Let
x
_{(α,β)}
∈ cl
_{θ}
(
A
). Suppose
f(x_{(α,β)})
∉ cl
_{δ}
(
f(A
)). Then there exists an IF open
q
neighborhood
V
of
f(x_{(α,β)})
Since
f
is an IF almost strongly
θ
continuous function, there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
f
(cl(
U
)) ≤ int(cl(
V
)) =
V
. Since
f(A)
≤
V^{c}
≤ (
f
(cl(
U
)))
^{c}
, we have
A
≤ (
f
^{1}
(
f
(cl(
U
))))
^{c}
. Thus
Since
x
_{(α,β)}
∈ cl
_{θ}
(
A
), we have
Aq
cl(
U
). This is a contradiction.
(2) ⇒ (3): Let
B
be an IF set in
Y
. Then
f
^{1}
(
B
) is an IF set in X. By (2), f(cl
_{θ}
(
f
^{1}
(
B
))) ≤ cl
_{θ}
(
f
(
f
^{1}
(
B
))) ≤ cl
_{θ}
(
B
). Thus we have
f
(cl
_{θ}
(
f
^{1}
(
B
))) ≤ cl
_{θ}
(
f
(
f
^{1}
(
B
))) ≤ cl
_{θ}
(
B
). Hence cl
_{θ}
(
f
^{1}
(
B
)) ≤
f
^{1}
(cl
_{δ}
(
B
)).
(3) ⇒ (4): Let
B
be an IF set in
Y
. Then
B^{c}
is an IF set in
Y
. By (3), cl
_{θ}
(
f
^{1}
(
B^{c}
)) ≤
f
^{1}
(cl
_{δ}
(
B^{c}
)) for each IF set
B
in
Y
. Therefore
f
^{1}
(int(B)) = (cl
_{δ}
(
f
^{1}
(
B^{c}
)))
^{c}
≥ (
f
^{1}
(cl
_{δ}
(
B^{c}
)))
^{c}
= int
_{θ}
(
f
^{1}
(
B
)).
(4) ⇒ (1): Let
B
be an IF set in
Y
. Then
B^{c}
is an IF set in
Y
. By (4),
f
^{1}
(int
_{δ}
(
B^{c}
)) ≤ int
_{θ}
(
f
^{1}
(
B^{c}
)). Thus cl
_{θ}
(
f
^{1}
(
B^{c}
)) ≤
f
^{1}
(cl
_{δ}
(
B^{c}
)). Hence
f
is an IF almost strongly
θ
continuous function.
Theorem 3.9.
Let
f
: (
X
,
T
) → (
Y
,
T’
) be a function. Then the following statements are equivalent:
(1)
f
is an IF almost strongly
θ
continuous function.
(2) The inverse image of every IF δclosed set in
Y
is an IF
θ
closed set in
X
.
(3) The inverse image of every IF δopen set in
Y
is an IF
θ
open set in
X
.
(4) The inverse image of every IF regular open set in
Y
is an IF
θ
open set in
X
.
Proof
. (1) ⇒ (2): Let
B
be an IF δclosed set in
Y
. Then cl
_{δ}
(
B
) =
B
. Since
f
is an IF almost strongly
θ
continuous function, by Theorem 3.8, cl
_{θ}
(
f
^{1}
(
B
)) ≤
f
^{1}
cl
_{δ}
(
B
)) =
f
^{1}
(
B
). Thus cl
_{θ}
(
f
^{1}
(
B
)) =
f
^{1}
(
B
). Hence
f
^{1}
(
B
) is an IF δclosed set in
X
.
(2) ⇒ (3): Let
B
be an IF δopen set in
Y
. Then
B^{c}
is anIF δclosed set in
Y
. By (4),
f
^{1}
(
B^{c}
) = (
f
^{1}
(
B
))
^{c}
is an IF
θ
closed set in
X
. Hence
f
^{1}
(
B
) is an IF
θ
open set in
X
.
(3) ⇒ (4): Immediate since IF regular open sets are IF
θ
open sets.
(4) ⇒ (1): 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 int(cl(
V
)) is an IF regular open
q
neighborhood of
f
(x(α,β)). By (4),
f
^{1}
(int(cl(
V
))) is an IF
θ
open set in
X
. Then
Put int(cl(
V
)) =
D
. Suppose
x
_{(α,β)}
∈ (
f
^{1}
(int(cl(
V
))))
^{c}
=
f
^{1}
(
D^{c}
). Then
Let
f
(
x
_{(α,β)}
) =
y
_{(α0,β0)}
. Then α
_{0}
≤ ϓ
D
(
y
) and
β
_{0}
≥ μ
D
(
y
). Since
V
is an IF open set,
V
≤ int(cl(
V
)) =
D
. Thus μ
_{V}
≤ μ
_{D}
and
ϓ
_{u}
≥
ϓ
_{D}
. Thus α
_{0}
≤
ϓ
_{V}
(
y
) and α
_{0}
≥ μ
_{V}
(
y
). Since
V
is an IF open
q
neighborhood of
f
(x
_{(α,β)}
), we have
f
(x
_{(α,β)}
)
_{q}
V
. Thus
y
_{(α0,β0)}
≰
V
^{c}
= (
ϓ
_{V}
, μ
_{V}
). Hence α
_{0}
> ϓ
_{V}
(
y
) and β
_{0}
< μ
_{V}
(
y
). This is a contradiction. Therefore there exists an IF open qneighborhood
U
of x
_{(α, β)}
such that
i.e. cl(
U
) ≤
f
^{1}
(int(cl(
V
))). Then
f
(cl(
U
)) ≤ int(cl(
V
)). Hence
f
is an IF almost strongly
θ
continuous function.
Theorem 3.10.
A function
f
: (
X
,
T
) → (
Y
,
T
’) is IF almost strongly
θ
continuous if and only if for each IF point
x
_{(α,β)}
in
X
and each IF
δ
neighborhood
N
of
f
(
x
_{(α,β)}
), the IF set
f
^{－1}
(
N
) is an IF
θ
neighborhood of
x
_{(α,β)}
.
Proof
. Let
x
_{(α,β)}
be an IF point in
X
, and let
N
be an IF
δ
neighborhood of
f
(
x
_{α,β)}
). Then there exists an an IF regular open
q
neighborhood
V
of
f
(
x
_{(α,β)}
) such that
V
≤
N
. Thus int(cl(
V
)) ≤
N
. Since
f
is an IF almost strongly
θ
continuous function, there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that
f
(cl(
U
)) ≤ int(cl(
V
)). Thus
f
(cl(
U
)) ≤
N
. Therefore, there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that cl(
U
) ≤
f
^{－1}
(
N
). Hence
f
^{－1}
(
N
) is an IF
θ
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)}
). Since int(cl(
V
)) is an IF regular open
q
neighborhood of
f
(
x
_{(α,β)}
) and int(cl(
V
)) ≤ int(cl(
V
)), int(cl(
V
)) is an IF
δ
neighborhood of
f
(
x
_{(α,β)}
). By the hypothesis,
f
^{－1}
(int(cl(
V
))) is an IF
θ
neighborhood of
x
_{(α,β)}
. Then there exists an IF open
q
neighborhood
U
of
x
_{(α,β)}
such that cl(
U
) ≤
f
^{－1}
(int(cl(V))). Therefore f(cl(U)) ≤ int(cl(
V
)). Hence
f
is IF almost strongly
θ
continuous.
Theorem 3.11.
Let
f
: (
X, T
) → (
Y, T’
) be a bijection. Then the following statements are equivalent:
(1)
f
is an IF almost strongly
θ
continuous function.
(2) int
_{δ}
(
f(A
)) ≤ f(int
_{θ}
(
A
)) for each IF set
A
in
X
.
Proof.
(1) ⇒ (2): Let
A
be an IF set in
X
. Then
f(A)
is an IF set in
Y
. By Theorem 3.9,
f
^{1}
(int
_{θ}
(
f(A
))) ≤ int
_{θ}
(
f
^{1}
(
f(A
))). Since
f
is onetoone,

f1(intδ(f(A))) ≤ intθ(f1(f(A))) = intθ(A):
Since
f
is onto,

intδ(f(A)) =f(f1(intδ(f(A)))) ≤f(intθ(A)):
(2) ⇒ (1): Let
B
be an IF set in
Y
. Then
f
^{1}
(
B
) is an IF set in
X
. By (2), int
_{δ}
(
f
^{1}
(
B
))) ≤
f
(int
_{θ}
(
f
^{1}
(
B
))). Since
f
is onto,

intδ(B) = intδ(f(f1(B))) ≤f(intθ(f1(B))):
Since
f
is onetoone,

f1(intδ(B)) ≤f1(f(intθ(f1(B)))) = intθ(f1(B)):
By Theorem 3.9,
f
is an IF almost strongly
θ
continuous function.
4. Conclusion
We characterized the intuitionistic fuzzy δcontinuous functions in terms of IF δclosure and IF δinterior, or IF δopen and IF δclosed sets, or IF δneighborhoods.
Moreover, we characterized the IF weakly δcontinuous, IF almost continuous, and IF almost strongly δcontinuous functions in terms of closure and interior.
 Conflict of Interest
No potential conflict of interest relevant to this article was reported.
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