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
= (
μA, ϓA
), where the functions
μA
:
X
→
I
and
ϓA
:
X
→
I
denote the degree of membership and the degree of nonmembership, respectively, and
μA
+
ϓA
≤ 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 “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 semi-open 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 semi-open 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 openq-neighborhood 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 δ-continuous andWeakly δ-continuous Functions
Recall that a fuzzy set
N
in (
X, T
) is said to be a
fuzzy δ-neighborhood
of a fuzzy point
xα
if there exists a fuzzy regular open
q
-neighborhood
V
of
xα
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 δ-neighborhood
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
Ac
is an IF
δ
-closed set, we have
x
(α,β)
∉
Ac
= cl
δ
(
Ac
). 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
Ac
is an IF
δ
-closed set, take any
x
(α,β)
∉
Ac
. 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
≤
Ac
, i.e.
x
(α,β)
∉ cl
δ
(
Ac
). Since cl
δ
(
Ac
) ≤
Ac
, we have
Ac
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 δ-continuous
function if for each fuzzy point
xα
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 δ-continuous
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 IFδ-continuous function.
-
(2)f(clδ(U)) ≤ clδ(f(U)) for each IF setUinX.
-
(3) clδ(f-1(V)) ≤f-1(clδ(V)) for each IF setVinY.
-
(4)f-1(intδ(V)) ≤ intδ(f-1(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
Vc
is anIF regular closed set,
Vc
is an IF semi-open set. By Theorem1.3, cl(
Vc
) = cl
δ
(
Vc
). Since
f
(
x(α,β))qV
,
f
(
x
(α,β)
) ∉
Vc
= cl(
Vc
) = cl
δ
(
Vc
). Therefore
x
(α,β)
∉
f-1
(cl
δ
(
Vc
)). By (3),
x
(α,β)
∉ cl
δ
(
f-1
(
Vc
)). 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
(
Vc
)) ≤
f-1
(cl
δ
(
Vc
)). Thus
(4) ⇒ (3). Let
V
be an IF set in
Y
. Then
Vc
is an IF set in
Y
. By the hypothesis,
f-1
(int
δ
(
Vc
)) ≤ int
δ
(
f-1
(
Vc
)). 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 IFδ-continuous function.
-
(2)f-1(A) is an IFδ-closed set for each IFδ-closed setAinX.
-
(3)f-1(A) is an IFδ-open set for each IFδ-open 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 q-neighborhood
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 IFδ-continuous 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 one-to-one,
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 one-to-one,
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 δ-continuous
if for each fuzzy point
xα
, in
X
and each fuzzy open
q
-neighborhood
V
of
f
(
xα
), there exists an fuzzy open
q
-neighborhood
U
of
xα
, 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 δ-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
Theorem 2.9.
Let
f : (X, T) → (Y, T’)
be a function. Then the following statements are equivalent:
-
(1)fis an IF weaklyδ-continuous function.
-
(2)f(clδ(A)) ≤ clθ(f(A)) for each IF setAinX.
-
(3) clδ(f-1(B)) ≤f-1(clθ(B)) for each IF setBinY.
-
(4)f-1(intθ(B)) ≤ intδ(f-1(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
Bc
is an IF set in
Y
. By (3), cl
δ
(
f-1
(
Bc
)) ≤
f-1
(cl
θ
(
Bc
)). Hence we have int
δ
(
f-1
(
B
)) = (cl
θ
(
f-1
(
Bc
))) ≥ (
f-1
(cl
θ
(
Bc
)))
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)f-1(V) is an IFθ-closed set inXfor each IFδ-closed setVinY.
-
(2)f-1(V) is an IFθ-open set inXfor each IFδ-open 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 weaklyδ-continuous 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 one-to-one,
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(f-1(B)) ≤f(intδ(f-1(B)):
f
is one-to-one,
-
f-1(intθ(B) ≤f-1(f(intδ(f-1(B))) = intδ(<(f-1(B)).
By Theorem 2.9,
f
is an IF weakly δ-continuous 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 δ-closed set
V
in
Y
.
(4) f
-1
(
V
) is an IF open set in
X
for each IF δ-open 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
Vqf
(
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(f-1(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 δ-closed set in
Y
. Then
f
-1
(
V
) is an IF set in
X
. By the hypothesis,
-
f(cl(f-1(V)))) ≤ clδ(f(f-1(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 δ-open set in
Y
. Then
Vc
is an IF δ-closed set in
Y
. By the hypothesis,
f
-1
(
Vc
) = (
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 δ-open 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
≤
Bc
=
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(f-1(V))) ≤ clδ(f(f-1(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(f-1(f(U))) ≤f-1(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
Vc
is an IF set in
Y
. By the hypothesis, cl(
f
-1
(
Vc
)) ≤
f
-1
(cl
δ
(
Vc
)). Thus
(3) ⇒ (2). Let
V
be an IF set in
Y
. Then
Vc
is an IF set in
Y
. By the hypothesis,
f
-1
(int
δ
(
Vc
)) ≤ int(
f
-1
>(
Vc
)). 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 δ-neighborhood 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 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 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 q-neighborhood of
f(x(α,β))
. Also, int(cl(
V
)) is an IF δ-neighborhood 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 θ-continuous
function if for each fuzzy point
xα
in
X
and each fuzzy open
q
-neighborhood
V
of
f(xα)
, there exists an fuzzy open
q
-neighborhood
U
of
xα
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 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
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)
≤
Vc
≤ (
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
Bc
is an IF set in
Y
. By (3), cl
θ
(
f
-1
(
Bc
)) ≤
f
-1
(cl
δ
(
Bc
)) for each IF set
B
in
Y
. Therefore
f
-1
(int(B)) = (cl
δ
(
f
-1
(
Bc
)))
c
≥ (
f
-1
(cl
δ
(
Bc
)))
c
= int
θ
(
f
-1
(
B
)).
(4) ⇒ (1): Let
B
be an IF set in
Y
. Then
Bc
is an IF set in
Y
. By (4),
f
-1
(int
δ
(
Bc
)) ≤ int
θ
(
f
-1
(
Bc
)). Thus cl
θ
(
f
-1
(
Bc
)) ≤
f
-1
(cl
δ
(
Bc
)). 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 δ-closed set in
Y
is an IF
θ
-closed set in
X
.
(3) The inverse image of every IF δ-open 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 δ-closed 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 δ-closed set in
X
.
(2) ⇒ (3): Let
B
be an IF δ-open set in
Y
. Then
Bc
is anIF δ-closed set in
Y
. By (4),
f
-1
(
Bc
) = (
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
(
Dc
). 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 q-neighborhood
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 one-to-one,
-
f-1(intδ(f(A))) ≤ intθ(f-1(f(A))) = intθ(A):
Since
f
is onto,
-
intδ(f(A)) =f(f-1(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(f-1(B))) ≤f(intθ(f-1(B))):
Since
f
is one-to-one,
-
f-1(intδ(B)) ≤f-1(f(intθ(f-1(B)))) = intθ(f-1(B)):
By Theorem 3.9,
f
is an IF almost strongly
θ
-continuous function.
4. Conclusion
We characterized the intuitionistic fuzzy δ-continuous functions in terms of IF δ-closure and IF δ-interior, or IF δ-open and IF δ-closed sets, or IF δ-neighborhoods.
Moreover, we characterized the IF weakly δ-continuous, IF almost continuous, and IF almost strongly δ-continuous 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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