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.
Çoker D.
1996
“An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces,”
Journal of Fuzzy Mathematics
4
(4)
749 -
764
Çoker D.
1995
“On intuitionistic fuzzy points,”
Notes on Intuitionistic Fuzzy Sets
1
(2)
79 -
84
Sivaraj D.
1986
“Semiopen set characterizations of almostregular spaces,”
Glasnik Matematicki Series III
21
(2)
437 -
440
G¨urçay H.
,
Çoker D.
,
Es A. H.
1997
“On fuzzy continuity in intuitionistic fuzzy topological spaces,”
Journal of Fuzzy Mathematics
5
(2)
365 -
378
Lee S. J.
,
Lee E. P.
2000
“The category of intuitionistic fuzzy topological spaces,”
Bulletin of the Korean Mathematical Society
37
(1)
63 -
76
Lee S. J.
,
Kim J. T.
2011
“Properties of fuzzy (, s)-semiirresolute mappings in intuitionistic fuzzy topological spaces,”
International Journal of Fuzzy Logic and Intelligent Systems
11
(3)
190 -
196
DOI : 10.5391/IJFIS.2011.11.3.190
Hanafy I. M.
2002
“On fuzzy -open sets and fuzzy -continuity in intuitionistic fuzzy topological spaces,”
Journal of Fuzzy Mathematics
10
(1)
9 -
19
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
Hanafy I. M.
,
Abd El-Aziz
,
Salman T. M.
2006
“Intuitionistic fuzzy -closure operator,”
Boletn de la Asociacin Matemtica Venezolana
13
(1)
27 -
39
Mukherjee M. N.
,
Sinha S. P.
1990
“On some near-fuzzy continuous functions between fuzzy topological spaces,”
Fuzzy Sets and Systems
34
(2)
245 -
254
DOI : 10.1016/0165-0114(90)90163-Z
Mukherjee M. N.
,
Sinha S. P.
1991
“Fuzzy θ-closure operator on fuzzy topological spaces,”
International Journal of Mathematics and Mathematical Sciences
14
(2)
309 -
314
DOI : 10.1155/S0161171291000364
Lee S. J.
,
Eoum Y. S.
2010
“Intuitionistic fuzzy -closure and -interior,”
Communications of the Korean Mathematical Society
25
(2)
273 -
282
DOI : 10.4134/CKMS.2010.25.2.273
Eom Y. S.
,
Lee S. J.
2012
“Delta closure and delta interior in intuitionistic fuzzy topological spaces,”
International Journal of Fuzzy Logic and Intelligent Systems
12
(4)
290 -
295
DOI : 10.5391/IJFIS.2012.12.4.290
Ganguly S.
,
Saha S.
1988
“A note on δ-continuity and δ-connected sets in fuzzy set theory,”
Simon Stevin (A Quarterly Journal of Pure and Applied Mathematics)
62
(2)
127 -
141