In this paper, we obtain two types of adjoint functors between the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense, and the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense. Also, we reveal that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense.
1. Introduction
Chang
[2]
defined fuzzy topological spaces with the concept of fuzzy set introduced by Zadeh
[11]
. After that, many generalizations of the fuzzy topology were studied by several authors like Ŝostak
[10]
, Ramadan
[9]
, and Chattopadhyay and his colleagues
[3]
.
On the other hand, the concept of intuitionistic fuzzy sets was introduced by Atanassov
[1]
as a generalization of fuzzy sets. Çoker
[4]
introduced intuitionistic fuzzy topological spaces by using intuitionistic fuzzy sets. Mondal and Samanta
[7]
introduced the concept of intuitionistic gradation of openness as a generalization of a smooth topology of Ramadan (see
[9]
). Also, using the idea of degree of openness and degree of nonopenness, Çoker and Demirci
[5]
defined intuitionistic fuzzy topological spaces in Ŝostak’s sense as a generalization of smooth topological spaces and intuitionistic fuzzy topological spaces.
Lee and Lee
[6]
revealed that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Çoker’s sense. Also, Park and his colleagues
[8]
showed that the category of intuitionistic fuzzy topological spaces in Çoker’s sense is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense.
The aim of this paper is to continue this investigation of categorical relationships between those categories. We obtain two types of adjoint functors between the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense, and the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense. Also, we reveal that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense.
2. Preliminaries
We will denote the unit interval [0, 1] of the real line by
I
. A member
μ
of
I
^{X}
is called a
fuzzy set
in
X
. By
and
we denote the constant fuzzy sets in
X
with value 0 and 1, respectively. For any
μ
∈
I
^{X}
,
μ
^{c}
denotes the complement
.
Let
X
be a nonempty set. An
intuitionistic fuzzy set A
is an ordered pair
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. By
and
we denote the constant intuitionistic fuzzy sets with value (0, 1) and (1, 0), respectively. Obviously every fuzzy set
μ
in
X
is an intuitionistic fuzzy set of the form (
μ
,
).
Let
f
be a mapping from a set
X
to a set
Y
. Let
A
= (µ
_{A}
, γ
_{A}
) be an intuitionistic fuzzy set in
X
and
B
= (
µ
_{B}
,
γ
_{B}
) an intuitionistic fuzzy set in
Y
. Then
(1) The image of
A
under
f
, denoted by
f
(
A
), is an intuitionistic fuzzy set in
Y
defined by
(2) The inverse image of
B
under
f
, denoted by
f
^{−1}
(
B
), is an intuitionistic fuzzy set in
X
defined by
All other notations are standard notations of fuzzy set theory.
Definition 2.1.
(
[2]
) A
Chang’s fuzzy topology
on
X
is a family
T
of fuzzy sets in
X
which satisfies the following properties:
The pair (
X
,
T
) is called a
fuzzy topological space
.
Definition 2.2.
(
[9]
) A
smooth topology
on
X
is a mapping
T
:
I
^{X}
→
I
which satisfies the following properties:
The pair (
X
,
T
) is called a
smooth topological space
.
Definition 2.3.
(
[4]
) An
intuitionistic fuzzy topology
on
X
is a family
T
of intuitionistic fuzzy sets in
X
which satisfies the following properties:
The pair (
X
,
T
) is called an
intuitionistic fuzzy topological space.
Let
I
(
X
) be a family of all intuitionistic fuzzy sets in
X
and let
I
⊗
I
be the set of the pair (
r
,
s
) such that
r
,
s
∈
I
and
r
+
s
≤ 1.
Definition 2.4.
(
[5]
) Let
X
be a nonempty set. An
intuitionistic fuzzy topology in Ŝostak’s sense
(SoIFT for short) 𝒯 = (𝒯
_{1}
, 𝒯
_{2}
) on
X
is a mapping 𝒯 :
I
(
X
) →
I
⊗
I
which satisfies the following properties:
Then (
X
, 𝒯) = (
X
, 𝒯
_{1}
, 𝒯
_{2}
) is said to be an
intuitionistic fuzzy topological space in Ŝostak’s sense
(SoIFTS for short). Also, we call 𝒯
_{1}
(
A
) the
gradation of openness
of
A
and 𝒯
_{2}
(
A
) the
gradation of nonopenness
of
A
.
Definition 2.5.
(
[5]
) Let
f
: (
X
, 𝒯
_{1}
, 𝒯
_{2}
) → (
Y
, 𝒰
_{1}
, 𝒰
_{2}
) be a mapping from a SoIFTS
X
to a SoIFTS
Y
. Then
f
is said to be
SoIF continuous
if 𝒯
_{1}
(
f
^{−1}
(
B
)) ≥ 𝒯
_{1}
(
B
) and 𝒯
_{2}
(
f
^{−1}
(
B
)) ≤ 𝒯
_{2}
(
B
) for each
B
∈
I
(
Y
).
Let (
X
, 𝒯) be a SoIFTS. Then for each (
r
,
s
) ∈
I
⊗
I
, the family 𝒯
_{(r,s)}
defined by
is an intuitionistic fuzzy topology on
X
. In this case, 𝒯
_{(r,s)}
is called the (
r
,
s
)
level intuitionistic fuzzy topology
on
X
.
Let (
X
,
T
) be an intuitionistic fuzzy topological space. Then for each (
r
,
s
) ∈
I
⊗
I
, a SoIFT
T
^{(r,s)}
:
I
(
X
) →
I
⊗
I
defined by
In this case,
T
^{(r,s)}
is called an (
r
,
s
)
th graded SoIFT
on
X
and (
X
,
T
^{(r,s)}
) is called an (
r
,
s
)
th graded SoIFTS
on
X
.
Definition 2.6.
(
[7]
) Let
X
be a nonempty set. An
intuitionistic fuzzy topology in Mondal and Samanta’s sense
(MSIFT for short)
T
= (
T
_{1}
,
T
_{2}
) on
X
is a mapping
T
:
I^{X}
→
I
⊗
I
which satisfy the following properties:
Then (
X
,
T
) is said to be an
intuitionistic fuzzy topological space in Mondal and Samanta’s sense
(MSIFTS for short).
T
_{1}
and
T
_{2}
may be interpreted as
gradation of openness
and
gradation of nonopenness
, respectively.
Definition 2.7.
(
[7]
) Let
f
: (
X
,
T
_{1}
,
T
_{2}
) → (
Y
,
U
_{1}
,
U
_{2}
) be a mapping. Then
f
is said to be
MSIF contiunous
if
T
_{1}
(
f
^{−1}
(
η
)) ≥
U
_{1}
(
η
) and
T
_{2}
(
f
^{−1}
(
η
)) ≤
U
_{2}
(
η
) for each
η
∈
I
^{Y}
.
Let (
X
,
T
) be a MSIFTS. Then for each (
r
,
s
) ∈
I
⊗
I
, the family
T
_{(r,s)}
defined by
is a Chang’s fuzzy topology on
X
. In this case,
T
_{(r,s)}
is called the (
r
,
s
)
level Chang’s fuzzy topology
on
X
.
Let (
X
,
T
) be a Chang’s fuzzy topological spaces. Then for each (
r
,
s
) ∈
I
⊗
I
, a MSIFT
T
^{(r,s)}
:
I
^{X}
→
I
⊗
I
is defined by
In this case,
T
^{(r,s)}
is called an (
r
,
s
)
th graded MSIFT
on
X
and (
X
,
T
^{(r,s)}
) is called an (
r
,
s
)
th graded MSIFTS
on
X
.
3. The categorical relationships between MSIFTop and SoIFTop
Let
MSIFTop
be the category of all intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense and MSIF continuous mappings, and let
SoIFTop
be the category of all intuitionistic fuzzy topological spaces in Ŝostak’s sense and SoIF continuous mappings.
Theorem 3.1.
Define a functor
F
:
SoIFTop
→
MSIFTop
by
F
(
X
, 𝒯) = (
X
,
F
(𝒯)) and
F
(
f
) =
f
, where
F
(𝒯)(
η
) = (
F
(𝒯)
_{1}
(
η
),
F
(𝒯)
_{2}
(
η
)),
F
(𝒯)
_{1}
(
η
) = ∨ {𝒯
1
(
A
) 
µ_{A}
=
η
},
F
(𝒯)
_{2}
(
η
) = ⋀ {𝒯
_{2}
(
A
) 
µ_{A}
=
η
}. Then
F
is a functor.
Proof
. First, we show that
F
(𝒯) is a MSIFT.
Clearly,
F
(𝒯)(
η
) =
F
(𝒯)
_{1}
(
η
) +
F
(𝒯)
_{2}
(
η
) ≤ 1 for any
η
∈
I
^{X}
.
,
,
, and
.
(2) Suppose that
F
(𝒯)
_{1}
(
η
∧ λ) <
F
(𝒯)
_{1}
(
η
) ∧
F
(𝒯)
_{1}
(λ). Then there is a
t
∈
I
such that
F
(𝒯)
_{1}
(
η
∧λ) <
t
<
F
(𝒯)
_{1}
(
η
) ∧
F
(𝒯)
_{1}
(
λ
). Since
t
<
F
(𝒯)
_{1}
(
η
) = ∨ {
T
_{1}
(
C
) 
µ
_{C}
=
η
}, there is an
A
∈
I
(
X
) such that
t
< 𝒯
_{1}
(
A
) and
µ
_{A}
=
η
. There is a
B
∈
I
(
X
) such that
t
< 𝒯
_{1}
(
B
) and
µ
_{B}
= λ, because
t
<
F
(𝒯)
_{1}
(λ) = ∨{𝒯
_{1}
(
C
) 
µ
_{C}
= λ}. Thus
t
< 𝒯
_{1}
(
A
) ∧ 𝒯
_{1}
(
B
) and
µ
_{A∩B}
=
µ
_{A}
∧
µ
_{B}
=
η
∧ λ. Since 𝒯 is a SoIFT, we obtain
Hence
This is a contradiction. Thus
F
(𝒯)
_{1}
(
η
∧ λ) ≥
F
(𝒯)
_{1}
(
η
) ∧
F
(𝒯)
_{2}
(λ).
Next, assume that
F
(𝒯)
_{2}
(
η
∧ λ) >
F
(𝒯)
_{2}
(
η
) ∨
F
(𝒯)
_{2}
(λ). Then there is an
s
∈
I
such that
Since
s
>
F
(𝒯)
_{2}
(
η
) = ⋀{𝒯
_{2}
(
C
) 
µ
_{C}
=
η
}, there is an
A
∈
I
(
X
) such that
s
> 𝒯
_{2}
(
A
) and
µ
_{A}
=
η
. As
s
>
F
(𝒯)
_{2}
(λ) = ⋀{𝒯
_{2}
(
C
) 
µ
_{C}
= λ}, there is a
B
∈
I
(
X
) such that
s
> 𝒯
_{2}
(
B
) and
µ
_{B}
= λ. So
s
> 𝒯
_{2}
(
A
) ∨ 𝒯
_{2}
(
B
) and
µ
_{A∩B}
=
µ
_{A}
∧
µ
_{B}
=
η
∧ λ. Since 𝒯 is a SoIFT, we have
s
> 𝒯
_{2}
(
A
) ∨ 𝒯
_{2}
(
B
) ≥ 𝒯
_{2}
(
A
∩
B
). Thus
This is a contradiction. Hence
F
(𝒯)
_{2}
(
η
∧ λ) ≤
F
(𝒯)
_{2}
(
η
) ∨
F
(𝒯)
_{2}
(λ).
(3) Suppose that
F
(𝒯)
_{1}
(∨
η_{i}
) < ⋀
F
(𝒯)
_{1}
(
η_{i}
). Then there is a
t
∈
I
such that
F
(𝒯)
_{1}
(∨
η_{i}
) <
t
< ⋀
F
(𝒯)
_{1}
(
η_{i}
). Since
t
<
F
(𝒯)
_{1}
(
η_{i}
) = ∨ {
T
_{1}
(
C
) 
µ_{C}
=
η_{i}
} for each
i
, there is an
A
_{i}
∈
I
(
X
) such that
t
< 𝒯
_{1}
(
A
_{i}
) and
µ_{Ai}
=
η_{i}
. Thus
t
≤ ⋀ 𝒯
_{1}
(
A
_{i}
) and
µ
∪
A_{i}
= ∨
µ_{Ai}
= ∨
η_{i}
. As 𝒯 is a SoIFT, we obtain 𝒯
_{1}
(∪
A
_{i}
) ≥ ⋀ 𝒯
_{1}
(
A
_{i}
). Hence
This is a contradiction. Thus
F
(𝒯)
_{1}
(∨
η_{i}
) ≥ ⋀
F
(𝒯)
_{1}
(
η_{i}
).
Next, assume that
F
(𝒯)
_{2}
(∨
η_{i}
) > ∨
F
(𝒯)
_{2}
(
η_{i}
). Then there is an
s
∈
I
such that
Since
s
>
F
(𝒯)
_{2}
(
η_{i}
) = ⋀{𝒯
_{2}
(
C
) 
µ_{C}
=
η_{i}
} for each
i
, there is a
B
_{i}
∈
I
(
X
) such that
s
> 𝒯
_{2}
(
B
_{i}
) and
µ_{Bi}
=
η_{i}
. Hence
s
≥ ∨𝒯
_{2}
(
B
_{i}
) and
µ
∪
B
_{i}
= ∨
µ_{Bi}
= ∨
η_{i}
. Since 𝒯 is a SoIFT, we have 𝒯
_{2}
(∪
B
_{i}
) ≤ ∨𝒯
_{2}
(
B
_{i}
). Thus
This is a contradiction. Hence
F
(𝒯)
_{2}
( ∨
η_{i}
) ≤ ∨
F(𝒯)
_{2}
(
η_{i}
). Therefore (
X
,
F
(𝒯)) is a MSIFTS.
Finally, we show that if
f
: (
X
, 𝒯) → (
Y
, 𝒰) is SoIF continuous, then
f
: (
X
,
F
(𝒯)) → (
Y
,
F
(𝒰)) is MSIF continuous. Let
F
(𝒯) = (
F
(𝒯)
_{1}
,
F
(𝒯)
_{2}
),
F
(𝒰) = (
F
(𝒰)
_{1}
,
F
(𝒰)
_{2}
), and λ ∈
I
^{Y}
. Then
and
Therefore
F
is a functor.
Theorem 3.2.
Define a functor
G
:
MSIFTop
→
SoIFTop
by
G
(
X
,
T
) = (
X
,
G
(
T
)) and
G
(
f
) =
f
, where
G
(
T
)(
A
) = (
G
(
T
)
_{1}
(
A
),
G
(
T
)
_{2}
(
A
)),
G
(
T
)
_{1}
(
A
) =
T
_{1}
(
µ_{A}
), and
G
(
T
)
_{2}
(
A
) =
T
_{2}
(
µ_{A}
). Then
G
is a functor.
Proof
. First, we show that
G
(
T
) is a SoIFT.
Clearly,
G
(
T
)
_{1}
(
A
) +
G
(
T
)
_{2}
(
A
) =
T
_{1}
(
µ_{A}
) +
T
_{2}
(
µ_{A}
) ≤ 1 for any
A
∈
I
(
X
).
and
(3) Let
A
_{i}
∈
I
(
X
) for each
i
. Then
and
Hence (
X
,
G
(
T
)) is a SoIFT.
Next, we show that if
f
: (
X
,
T
) → (
Y
,
U
) is MSIF continuous, then
f
: (
X
,
G
(
T
)) → (
Y
,
G
(
U
)) is SoIF continuous. Let
B
= (
µ_{B}
,
γ_{B}
) ∈
I
(
Y
). Then
and
Thus
f
: (
X
,
G
(
T
)) → (
Y
,
G
(
U
)) is SoIF continuous. Consequently,
G
is a functor.
Theorem 3.3.
The functor
G
:
MSIFTop
→
SoIFTop
is a left adjoint of
F
:
SoIFTop
→
MSIFTop
.
Proof.
Let (
X
,
T
) be an object in
MSIFTop
and
η
∈
I
^{X}
. Then
Hence l
_{X}
: (
X
,
T
) →
FG
(
X
,
T
) = (
X
,
T
) is MSIF continuous.
Consider (
Y
, 𝒰) ∈
SoIFTop
and a MSIF continuous mapping
f
: (
X
,
T
) →
F
(
Y
, 𝒰). In order to show that
f
:
G
(
X
,
T
) → (
Y
, 𝒰) is a
SoIF
continuous mapping, let
B
∈
I
(
Y
). Then
and
Hence
f
: (
X
,
G
(
T
)
_{1}
,
G
(
T
)
_{2}
) → (
Y
, 𝒰
_{1}
, 𝒰
_{2}
) is a SoIF continuous mapping. Therefore l
_{X}
is a
G
universal mapping for (
X
,
T
) in
MSIFTop
.
Theorem 3.4.
Define a functor
H
:
SoIFTop
→
MSIFTop
by
H
(
X
, 𝒯) = (
X
,
H
(𝒯)) and
H
(
f
) =
f
, where
, and
. Then
H
is a functor.
Proof
. First, we show that
H
(𝒯) is a MSIFT. Obviously,
H
(𝒯)(
η
) =
H
(𝒯)
_{1}
(
η
) +
H
(𝒯)
_{2}
(
η
) ≤ 1 for any
η
∈
I
^{X}
.
,
,
, and
.
(2) Assume that
H
(𝒯)
_{1}
(
η
∧ λ) <
H
(𝒯)
_{1}
(
η
) ∧
H
(𝒯)
_{1}
(λ). Then there is a
t
∈
I
such that
As
, there is an
A
∈
I
(
X
) such that
. Since
, there is a
B
∈
I
(
X
) such that
and
Since 𝒯 is a SoIFT,
t
< 𝒯
_{1}
(
A
) ∧ 𝒯
_{1}
(
B
) ≤ 𝒯
_{1}
(
A
∩
B
). Thus
This is a contradiction. Hence
H
(𝒯)
_{1}
(
η
∧ λ) ≥
H
(𝒯)
_{1}
(
η
) ∧
H
(𝒯)
_{1}
(λ).
Suppose that
H
(𝒯)
_{2}
(
η
∧λ) >
H
(𝒯)
_{2}
(
η
)∨
H
(𝒯)
_{2}
(λ). Then there is an
s
∈
I
such that
Since
, there is an
A
∈
I
(
X
) such that
s
> 𝒯
_{2}
(
A
) and
. As
, there is a
B
∈
I
(
X
) such that
s
> 𝒯
_{2}
(
B
) and
. So
s
> 𝒯
_{2}
(
A
) ∨ 𝒯
_{2}
(
B
) and
Since 𝒯 is a SoIFT, we obtain
s
> 𝒯
_{2}
(
A
)∨𝒯
_{2}
(
B
) ≥ 𝒯
_{2}
(
A
∩
B
). Hence
This is a contradiction. Thus
H
(𝒯)
_{2}
(
η
∧ λ) ≤
H
(𝒯)
_{2}
(
η
) ∨
H
(𝒯)
_{2}
(λ).
(3) Assume that
H
(𝒯)
_{1}
(∨
η_{i}
) < ⋀
H
(𝒯)
_{1}
(
η_{i}
). Then there is a
t
∈
I
such that
As
for each
i
, there is an
A
_{i}
∈
I
(
X
) such that
t
< 𝒯
_{1}
(
A
_{i}
) and
. Hence
t
≤ ⋀𝒯
_{1}
(
A
_{i}
) and
Since 𝒯 is a SoIFT, we have 𝒯
_{1}
(∪
A
_{i}
) ≥ ⋀𝒯
_{1}
(
A
_{i}
). Thus
This is a contradiction. Hence
H
(𝒯)
_{1}
(∨
η_{i}
) ≥ ⋀
H
(𝒯)
_{1}
(
η_{i}
).
Suppose that
H
(𝒯)
_{2}
(∨
η_{i}
) > ∨
H
(𝒯)
_{2}
(
η_{i}
). Then there is an
s
∈
_{I}
such that
H
(𝒯)
_{2}
(∨
η_{i}
) >
s
> ∨
H
(𝒯)
_{2}
(
η_{i}
). Since
for each
i
, there is a
B
_{i}
∈
I
(
X
) such that
s
> 𝒯
_{2}
(
B
_{i}
) and
. Hence
s
≥ ∨𝒯
_{2}
(
B
_{i}
) and
We have 𝒯
_{2}
(∪
B
_{i}
) ≤ ∨𝒯
_{2}
(
B
_{i}
) because 𝒯 is a SoIFT. Thus
This is a contradiction. Hence
H
(𝒯)
_{2}
( ∨
η_{i}
) ≤ ∨
H
(𝒯)
_{2}
(
η_{i}
). Therefore (
X
,
H
(𝒯)) is a MSIFTS.
Next, we show that if
f
: (
X
, 𝒯) → (
Y
, 𝒰) is SoIF continuous, then
f
: (
X
,
H
(𝒯)) → (
Y
,
H
(𝒰)) is MSIF continuous. Let
H
(𝒯) = (
H
(𝒯)
_{1}
,
H
(𝒯)
_{2}
),
H
(𝒰) = (
H
(𝒰)
_{1}
,
H
(𝒰)
_{2}
), and
η
∈
I
^{X}
. Then
and
Therefore
H
is a functor.
Theorem 3.5.
Define a functor
K
:
MSIFTop
→
SoIFTop
by
K
(
X
,
T
) = (
X
,
K
(
T
)) and
K
(
f
) =
f
, where
K
(
T
) = (
K
(
T
)
_{1}
,
K
(
T
)
_{2}
),
, and
. Then
K
is a functor.
Proof
. First, we show that
K
(
T
) is a SoIFT. Clearly,
for any
A
∈
I
(
X
).
,
, and
.
(2) Let
A
,
B
∈
I
(
X
). Then
and
(3) Let
A
_{i }
∈
I
(
X
) for each
i
. Then
and
Thus (
X
,
K
(
T
)) is a SoIFTS.
Finally, we show that if
f
: (
X
,
T
) → (
Y
,
U
) is MSIF continuous, then
f
: (
X
,
K
(
T
)) → (
Y
,
K
(
U
)) is SoIF continuous. Let
B
= (
µ_{B}
,
γ_{B}
) ∈
I
(
Y
). Then
and
Hence
f
: (
X
,
K
(
T
)) → (
Y
,
K
(
U
)) is SoIF continuous. Consequently,
K
is a functor.
Theorem 3.6.
The functor
K
:
MSIFTop
→
SoIFTop
is a left adjoint of
H
:
SoIFTop
→
MSIFTop
.
Proof.
For any (
X
,
T
) in
MSIFTop
and
η
∈
I
^{X}
,
Hence l
_{X}
: (
X
,
T
) →
H
K
(
X
,
T
) = (
X
,
T
) is a MSIF continuous mapping. Consider (
Y
, 𝒰) ∈
SoIFTop
and a MSIF continuous mapping
f
: (
X
,
T
) →
H
(
Y
, 𝒰). In order to show that
f
:
K
(
X
,
T
) → (
Y
, 𝒰) is a SoIF continuous mapping, let
B
∈
I
(
Y
). Then
and
Thus
f
: (
X
,
K
(
T
)) → (
Y
, 𝒰) is SoIF continuous. Hence l
_{X}
is a
K
universal mapping for (
X
,
T
) in
MSIFTop
.
Let (
r
,
s
)
gMSIFTop
be the category of all (
r
,
s
)th graded intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense and MSIF continuous mappings, and let
CFTop
be the category of all Chang’s fuzzy topological spaces and fuzzy continuous mappings.
Theorem 3.7.
Two categories
CFTop
and (
r
,
s
)
gMSIFTop
are isomorphic.
Proof.
Define
F
:
CFTop
→ (
r
,
s
)
gMSIFTop
by
F
(
X
,
T
) = (
X
,
F
(
T
)) and
F
(
f
) =
f
, where
Define
G
: (
r
,
s
)
gMSIFTop
→
CFTop
by
G
(
X
, 𝒯) = (
X
,
G
(𝒯)) and
G
(
f
) =
f
, where

G(𝒯) = 𝒯(r,s)= {η∈IX 𝒯1(η) ≥rand 𝒯2(η) ≤s}.
Then
F
and
G
are functors. Obviously,
GF
(
T
) =
G
(
T
^{(r,s)}
) = (
T
^{(r,s)}
)
_{(r,s)}
=
T
and
FG
(
T
) =
F
(𝒯
_{(r,s)}
) = (𝒯
_{(r,s)}
)
^{(r,s)}
= 𝒯. Hence
CFTop
and (
r
,
s
)
gMSIFTop
are isomorphic.
Theorem 3.8.
The category (
r
,
s
)
gMSIFTop
is a bireflective full subcategory of
MSIFTop.
Proof.
Obviously, (
r
,
s
)
gMSIFTop
is a full subcategory of
MSIFTop
. Let (
X
,
T
) be an object of
MSIFTop
. Then for each (
r
,
s
) ∈
I
⊗
I
, (
X
, (
T
(
_{r,s}
))
^{(r,s)}
) is an object of (
r
,
s
)
gMSIFTop
and l
_{X}
: (
X
,
T
) → (
X
, (
T
_{(r,s)}
)
^{(r,s)}
) is a MSIF continuous mapping. Let (
Y
,
U
) be an object of the category (
r
,
s
)
gMSIFTop
and
f
: (
X
,
T
) → (
Y
,
U
) a MSIF continuous mapping. we need only to check that
f
: (
X
, (
T
_{(r,s)}
)
^{(r,s)}
) → (
Y
,
U
) is a MSIF continuous mapping. Since (
Y
,
U
) ∈ (
r
,
s
)
gMSIFTop
,
U
(
η
) = (1, 0), (
r
,
s
), or (0, 1). Let
U
(
η
) = (1, 0). Then
or
. In fact,
and
In case
U
(
η
) = (0, 1), clearly
U
(
η
) ≤ (
T
_{(r,s)}
)
^{(r,s)}
(
f
^{−1}
(
η
)). Let
U
(
η
) = (
r
,
s
). Since
f
: (
X
,
T
) → (
Y
,
U
) is MSIF continuous,
T
(
f
^{−1}
(
η
)) ≥
U
(
η
) = (
r
,
s
). Thus
f
^{−1}
(
η
) ∈
T
(
r
,
s
), and hence (
T
_{(r,s)}
)
^{(r,s)}
(
f
^{−1}
(
η
)) = (
r
,
s
) =
U
(
η
). Therefore
f
: (
X
, (
T
_{(r,s)}
)(
^{r,s)}
) → (
Y
,
U
) is a MSIF continuous mapping.
From the above theorems, we have the follwing main result.
Theorem 3.9.
The category
CFTop
is a bireflective full subcategory of
MSIFTop
.
4. Conclusion
We obtained two types of adjoint functors between the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense, and the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense. Also, we revealed that the category of Chang’s fuzzy topological spaces is a bireflective full subcategory of the category of intuitionistic fuzzy topological spaces in Mondal and Samanta’s sense.
In further research, we will investigate other properties of the category of intuitionistic fuzzy topological spaces in Ŝostak’s sense.
Acknowledgements
This work was supported by the research grant of Chungbuk National University in 2013.
BIO
Jin Tae Kim received the Ph. D. degree from Chungbuk National University in 2012. His research interests include general topology and fuzzy topology. He is a member of KIIS and KMS.
Email: kjtmath@hanmail.net
Seok Jong Lee received the M. S. and Ph. D. degrees from Yonsei University in 1986 and 1990, respectively. He is a professor at the Department of Mathematics, Chungbuk National University since 1989. He was a visiting scholar in Carleton University from 1995 to 1996, and Wayne State University from 2003 to 2004. His research interests include general topology and fuzzy topology. He is a member of KIIS, KMS, and CMS.
Email: sjl@cbnu.ac.kr
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