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Fuzzy Sets and Systems 190 (2012) 82 – 93
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Lowen LM-fuzzy topological spaces夡
Hai-Yang Lia,b,∗ , Ji-Gen Penga
a Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, PR China
b School of Science, Xi’an Polytechnic University, Xi’an 710048, PR China
Received in revised form 9 July 2011; accepted 9 July 2011
Available online 21 July 2011
Abstract
The aim of this paper is to define and study Lowen LM-fuzzy topological spaces. We discuss the basic properties of Lowen
LM-fuzzy topological spaces, and introduce the notions of interior Lowen topology and exterior Lowen topology of an LM-fuzzy
topology and prove that LLM-FTop (the category of Lowen LM-fuzzy topological spaces) is isomorphism-closed and simultaneously
bireflective and bicoreflective in SLM-FTop (the category of stratified LM-fuzzy topological spaces). Moreover, we also prove that
an LM-fuzzy topological space is an induced LM-fuzzy topological space iff it is a Lowen LM-fuzzy topological space and an
(IC)LM-fuzzy topological space.
© 2011 Elsevier B.V. All rights reserved.
Keywords: Topology; Category; Lowen LM-fuzzy topological spaces
1. Introduction
Since Chang [2] introduced fuzzy set theory into topology, many authors have discussed various aspects of fuzzy
topology. However, in a completely different direction, Höhle [6] created the notion of a topology being viewed as an
L-subset of a powerset (in his case, 2 X ). Then Kubiak [8] and Šostak [20] independently extended Höhle’s notion to
L-subsets of L X . Subsequently, Kubiak and Šostak [9] further extended this notion to M-subsets of L X , thereby creating
LM-fuzzy topologies and the category LM-FTop of LM-fuzzy topological spaces and fuzzy continuous mappings
(L and M are distinct complete lattices).
In 2007, Yue [22] studied stratified, induced and weakly induced LM-fuzzy topological spaces, and proved that
WILM-FTop (the category of weakly induced LM-fuzzy topological spaces and fuzzy continuous mappings) is
a simultaneously reflective and coreflective full subcategory of LM-FTop. Recently, Li and Peng [12] have studied the properties of (IC)LM-topological spaces, which is the generalization of (weakly) induced LM-topological
spaces, and introduced the notions of interior (IC)-fication and exterior (IC)-fication of LM-fuzzy topologies and
showed thatICLM-FTop (the category of (IC)LM-fuzzy topological spaces and fuzzy continuous mappings) is an
夡 This work was supported by the Natural Science Foundation of China (11026201,11071151), the Foundation of Shaanxi Province (2009JK452)
and the Foundation of XPU (09XG10).
∗ Corresponding author at: School of Science, Xi’an Polytechnic University, Xi’an 710048, PR China.
E-mail address: [email protected] (H.-Y. Li).
0165-0114/$ - see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2011.07.005
H.-Y. Li, J.-G. Peng / Fuzzy Sets and Systems 190 (2012) 82 – 93
83
isomorphism-closed full proper subcategory of LM-FTop and ICLM-FTop is a simultaneously bireflective and
bicoreflective full subcategory of LM-FTop.
In 1982, Lowen [14] introduced a very important class of fuzzy topological spaces, called fuzzy neighborhood
spaces. Lowen’s notion has been extended to the L-fuzzy setting (called Lowen space) by Liu and Zhang [16] for L, an
arbitrary completely distributive lattice. This kind of L-topological space has received wide attention for categorical
properties [13,15,16,21,24]. The aim of the present paper is to define and study Lowen LM-fuzzy topological spaces.
We discuss the basic properties of Lowen LM-fuzzy topological spaces, and introduce the notions of interior Lowen
topology and exterior Lowen topology of LM-fuzzy topology and prove that LLM-FTop (the category of Lowen
LM-fuzzy topological spaces and fuzzy continuous mappings) is an isomorphism-closed full proper subcategory of
SLM-FTop (the category of stratified LM-fuzzy topological spaces and fuzzy continuous mappings) and LLM-FTop
is a simultaneously bireflective and bicoreflective full subcategory of SLM-FTop. Moreover, we prove that (X, ) is an
induced LM-fuzzy topological space iff (X, ) is a Lowen LM-fuzzy topological space and an (IC)LM-fuzzy topological
space.
For needed categorical notions, please refer to [1,7,17,19].
2. Preliminaries
We now give some definitions and results to be used in this paper. Let L be a complete
lattice. An element r ∈ L − {0}
is called a coprime element if, for any finite subset K ⊂ L satisfying r ≤
K (the supremum of K), there exists
a k ∈ K such that r ≤ k. An element s ∈ L − {1} is called a prime element if it is a coprime element of L op (the
opposite lattice [4] of L). The set of all coprime elements (resp., prime elements) of L will be written as Copr (L) (resp.,
Pr (L)). We
say that a is a way-below (wedge below) b, in symbols, a>b (ab), if for every directed (arbitrary) subset
D ⊂ L,
D ≥ b implies a ≤ d for some d ∈ D. From [5], we know that Copr (L) is a join-generating set of L if L is
a completely distributive lattice. Hence every element in L is also the supremum of all the coprimes wedge below it. By
the definition
of completely distributive lattice it is easy to see that a complete lattice L is completely distributive iff the
operator : Low(L) → L taking every lower set to its supremum has a left adjoint , and in the case, (a) = {b|ba}
for all a ∈ L. Hence, the wedge below relation has the interpolation property in a completely distributive lattice, that
is, ab implies that there is some c ∈ L such that acb.
The way-below relation on a completely distributive lattice L is called locally multiplicative [23] if for every coprime
a ∈ Copr (L), a>b and a>c imply a>b ∧ c for all b, c ∈ L. If a is a coprime, then a>b if and only if ab (see
[5,23]). Hence L is locally multiplicative if for every coprime a, ab and ac imply ab ∧ c for all b, c ∈ L. Clearly,
[0,1] and power set lattice 2 X are locally multiplicative.
Throughout this paper, L always stands for a completely distributive lattice with locally multiplicative property and
M is a completely distributive lattice. Apparently, the powerset L X (i.e. the set of all L-subset) with the point-wise order
is also a completely distributive lattice with locally multiplicative property, in which the least element and the greatest
element of L X will be written as 0 X and 1 X . Let 1U denote the characteristic function of U ∈ 2 X , and [a] denote the
L-subset taking constant value a, where 2 X is the power set of X and a ∈ L. An L-subset with form [a] ∧ 1U is called
one-step L-subset. Let S denote the set of all the one-step L-subsets of L X . Each mapping f : X −→ Y induces a
mapping f L→ : L X −→ L Y (called L-forward powerset operator, cf. [18]), defined by
{A(x)| f (x) = y} (∀A ∈ L X , ∀y ∈ Y ).
f L→ (A)(y) =
The right adjoint to f L→ (called L-backward powerset operator, cf. [18]) is denoted as f L← and given by
{A ∈ L X | f L→ (A) ≤ B} = B ◦ f (∀B ∈ L Y ).
f L← (B) =
It is known that f L→ preserves arbitrary unions and that f L← preserves arbitrary unions, arbitrary intersections, and
complements when they exist (canonical examples of such morphisms are given in [18]).
For A ∈ L X and p ∈ L, the mapping ˆ p : L X −→ 2 X is defined by ˆ p (A) = {x|A(x) p}, called the strong p-cut of
A. If L is a locally multiplicative completely distributive lattice, then it is easy to verify the following
Lemma 2.1 (Li [10], Yue [22]).
(1) For each p ∈ L , ˆ p : L X −→ 2 X preserves arbitrary suprema and finite meets.
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(2) For each p ∈ L , A ∈ L X , B ∈ L Y and any mapping f : X → Y , we have ˆ p ( f L→ (A)) = f (ˆ p (A)), ˆ p ( f L← (B)) =
f (ˆ p (B)).
(3) A = p∈L ([ p] ∧ 1ˆ p (A) ) = p∈Copr (L) ([ p] ∧ 1ˆ p (A) ).
An LM-fuzzy topology [9] on a set X is defined to be a mapping : L X −→ M satisfying:
(FT1) (1 X ) = (0 X ) = 1;
X (A ∧ B) ≥ (A) ∧ (B);
(FT2) ∀A,
B∈L ,
(FT3) ( t∈T At ) ≥ t∈T (At ), for every family {At |t ∈ T } ⊂ L X .
The value (A) can be interpreted as the degree of openness of A. A fuzzy continuous mapping between LM-fuzzy
topological spaces is a mapping f : (X, ) −→ (Y, ) such that (B) ≤ ( f L← (B)) for all B ∈ L Y . When L = {0, 1},
this definition reduced to that of M-fuzzying topology.
In the following of this section, we give some definitions and results about LM-fuzzy topology.
Definition 2.2 (Fang and Yue [3]). Let be an LM-fuzzy topology on X.
(1) B : L X −→ M is called a base of if B satisfies the following condition:
∀A ∈ L X , (A) =
∈
where the expression
B(B ),
B =A ∈
∈
B =A
∈ B(B )
()
(2) : L X −→ M is called a subbase of if () (A) =
(B )
will be denoted by B () (A).
: L X −→ M is a base of , where
()∈J B =A ∈J
for all A ∈ L X with () standing for “finite intersection”.
Lemma 2.3 (Fang and Yue [3]). Let be an LM-fuzzy topology on X. Then : L X −→ M is the subbase of if and
only if () (1 X ) = 1.
Definition 2.4 (Yue [22]). Let (X, ) be an LM-fuzzy topological space. If ([a]) = 1 for all a ∈ L, then (X, ) is
called a stratified LM-fuzzy topological space.
It is easy to verify that (X, ) is a stratified LM-fuzzy topological space iff ([a]) = 1 for all a ∈ Copr (L). Let
SLM-FTop denote the category of stratified LM-fuzzy topological spaces and fuzzy continuous mappings.
Lemma 2.5. Let (X, ) be a stratified LM-fuzzy topological space. Then ([a] ∧ 1U ) ≤ ([b] ∧ 1U ) (a, b ∈ L , ba
and U ⊂ X ).
Proof. Since (X, ) is stratified, we have
([b] ∧ 1U ) = ([b] ∧ ([a] ∧ 1U )) ≥ ([b]) ∧ ([a] ∧ 1U ) = ([a] ∧ 1U ).
Definition 2.6 (Yue [22]). Let (X, ) be an LM-fuzzy topological space. If (A) =
A ∈ L X , then (X, ) is called an induced LM-fuzzy topological space.
r ∈Copr (L) (1ˆr (A) )
holds for all
Definition 2.7 (Yue [22]). Let (X, ) be an LM-fuzzy topological space. If (A) ≤ r ∈Copr (L) (1ˆr (A) ) holds for all
A ∈ L X , then (X, ) is called a weakly induced LM-fuzzy topological space. Let WILM-FTop denote the category of
weakly induced LM-fuzzy topological spaces and fuzzy continuous mappings.
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85
Definition 2.8 (Li and Peng [12]). Let (X, ) be an LM-fuzzy topological space. If (A) ≤ (1ˆ0 (A) ) holds for all
A ∈ L X , then (X, ) is called an (IC)LM-fuzzy topological space. Let ICLM-FTop denote the category of
(IC)LM-fuzzy topological spaces and fuzzy continuous mappings.
It is easy to verify that induced and weakly induced LM-fuzzy topological spaces are (IC)LM-fuzzy topological
space.
Definition 2.9 (Fang and Yue [3]). Let {(X t , t )}t∈T be a family
of LM-fuzzy topological spaces and pt :
X t −→ X t be the projection. Then the LM-fuzzy topology on t∈T X t whose subbase is defined by
∀A ∈ L
t∈T
Xt
, (A) =
t∈T
t (B)
t∈T ( pt )←
L (B)=A
is called the product LM-fuzzy topology of {t }t∈T , denoted by
space of {(X t , t )}t∈T .
t∈T
t , and (
t∈T
Xt ,
t∈T
t ) is called the product
Definition
2.10 (Yue [22]). Let {(X t , t )}t∈T be a family of LM-fuzzy topological spaces, different X t s be disjoint and
X = t∈T X t , and let : L X −→ M be defined as follows:
∀A ∈ L X , (A) =
t (A|X t ).
t∈T
Then it is easy
an LM-fuzzy topology on X, and is called the sum LM-fuzzy topology of {t }t∈T ,
that is to verify
denoted by t∈T t . ( t∈T X t , t∈T t ) is called the sum space of {(X t , t )}t∈T .
Definition 2.11 (Yue [22]). Let (X, ) be an LM-fuzzy topological space and f : X −→ Y be a surjective mapping. It
is easy to verify that / f L→ : L Y −→ M is an LM-fuzzy topology on Y, where / f L→ is defined by
∀A ∈ L Y , / f L→ (A) = ( f L← (A)).
/ f L→ is called the LM-fuzzy quotient topology of with respect to f, and (Y, / f L→ ) is called the LM-fuzzy quotient
space of (X, ) with respect to f.
Definition 2.12 (Rodabaugh [18]). Let (X, ) be an LM-fuzzy topological space and Y ⊂ X . We call (Y, |Y ) a
subspace of (X, ), where |Y : L Y −→ M is defined by
∀B ∈ L Y , |Y (B) =
{(A)|A ∈ L X , A|Y = B}.
Lemma 2.13 (Yue [22]). Let f : (X, ) −→ (Y, ) be a mapping and be a subbase of . If ( f L← (B)) ≥ (B) for all
B ∈ L Y , then f is fuzzy continuous.
3. Lowen LM-fuzzy topological spaces
In the section, we mainly introduce the definition of a Lowen LM-fuzzy topological space and discuss its basic
properties such as the properties that the product space and the sum space of Lowen LM-fuzzy topological spaces are
also Lowen LM-fuzzy topological spaces. Moreover, we show that (X, ) is an induced LM-fuzzy topological space iff
(X, ) is a Lowen LM-fuzzy topological space and an (IC)LM-fuzzy topological space.
Definition 3.1. An
LM-fuzzy topological space (X, ) is said to be a Lowen LM-fuzzy topological space iff for all
A ∈ L X , (A) = a∈Copr (L) ([a] ∧ 1ˆa (A) ). Let LLM-FTop denote the category of Lowen LM-fuzzy topological
spaces and fuzzy continuous mappings.
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H.-Y. Li, J.-G. Peng / Fuzzy Sets and Systems 190 (2012) 82 – 93
Example 3.2. Let L = M = 2 X . Then L is not only a complete Boolean algebra but a locally multiplicative completely
distributive lattice. We consider the map : L X −→ M defined by
A(x) ∨
A(x) → 0 .
(A) =
x∈X
x∈X
Then is a stratified LM-fuzzy topology on X (See [7, Example 4.2.1(b)]). We will show that is a Lowen LM-fuzzy
topology on X in the following.
For any A ∈ L X and any t ∈ Copr (L), if ˆt (A) = X or , then
([t] ∧ 1ˆt (A) ) = 1 ≥ (A).
If ˆt (A) X and ˆt (A) , then there exist x1 , x2 ∈ X such that tA(x1 ) and tA(x2 ), which imply that A (x1 ) ≤ t and A(x2 ) ≤ t , and hence
(A) =
A(x) ∨
A(x) → 0 ≤ t .
x∈X
x∈X
Note
([t] ∧ 1ˆt (A) ) =
([t] ∧ 1ˆt (A) )(x) ∨
x∈X
([t] ∧ 1ˆt (A) )(x) → 0 = t .
x∈X
Therefore for any A ∈ L X and any t ∈ Copr (L), we have (A) ≤ ([t] ∧ 1ˆt (A) ), which implies that is the Lowen
LM-fuzzy topology on X by Theorem 3.4(2).
Lemma 3.3. (1) A Lowen LM-fuzzy topological space (X, ) is a stratified LM-fuzzy topological space.
(2) An induced LM-fuzzy topological space (X, ) is a Lowen LM-fuzzy topological space.
Proof.
(1) Since (X, ) is a Lowen LM-fuzzy topological space, we have
(1 X ) ≤
([a] ∧ 1ˆa (1 X ) ) =
([a]),
a∈Copr (L)
a∈Copr (L)
and hence ([a]) = 1 (∀a ∈ Copr (L)), which implies that (X, ) is a stratified LM-fuzzy topological space.
(2) Since (X, ) is an induced LM-fuzzy topological space, for all A ∈ L X , and a ∈ Copr (L), we have
(1ˆa (A) ) = ([a]) ∧ (1ˆa (A) ) ≤ ([a] ∧ 1ˆa (A) ),
which implies that
(A) =
(1ˆa (A) ) ≤
a∈Copr (L)
([a] ∧ 1ˆa (A) ).
a∈Copr (L)
Hence (X, ) is a Lowen LM-fuzzy topological space by Theorem 3.4(2). Theorem 3.4. Let (X, ) be an LM-fuzzy topological space and S be the set of all the one-step L-subset of L X . Then
the following are equivalent.
(1) (X, ) is a Lowen LM-fuzzy topological space.
(2) For all A ∈ L X and a ∈ Copr (L), (A) ≤ ([a] ∧ 1ˆa (A) ).
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87
(3) There exists a base B of satisfying B([a]) = 1 (∀a ∈ L) and
⎧
⎫
⎬
⎨
B(B ) B = A and B ∈ S .
∀A ∈ L X , (A) =
⎩
⎭
∈
∈
(4) There exists a subbase of satisfying ([a]) = 1 (∀a ∈ L) and
⎧
⎨
∀A ∈ L X , () (A) =
(B )|()∈J B = A and B ∈
⎩
⎫
⎬
S
∈J
⎭
is a base of .
Proof. (1) ⇐⇒ (2) follows from the definition of LM-fuzzy topology.
(2) ⇒ (3): Let B : S −→ M defined by B(A) = (A) (∀A ∈ S). Then B([a]) = 1 (∀a ∈ L) by Lemma 3.3 (1).
We will show B is a base of in the following. By (2) and the definition of LM-fuzzy topology, we have
(A) ≤
([a] ∧ 1ˆa (A) ) =
B([a] ∧ 1ˆa (A) ) ≤
B(B ) (B ∈ S)
a∈Copr (L)
≤
∈
a∈Copr (L)
∈
B =A ∈
(A) = (A),
B =A
which implies that B is a base of .
(3) ⇒ (4): It is easily concluded by taking = B.
(4) ⇒ (2): For all a ∈ Copr (L), A ∈ L X , we have
([a] ∧ 1ˆa (A) ) =
∈
B =[a]∧1ˆa (A) ∈ ()∈ C =B ∈
∈
D =A ∈ ()∈ E =[a]∧1ˆa (D ) ∈
∈
D =A ∈ ()∈ F =D ∈
∈
D =A ∈ ()∈ F =D ∈
≥
≥
≥
(C ) (C ∈ S)
(E ) (E ∈ S)
([a] ∧ 1ˆa (F ) ) (F ∈ S)
(F ) ≥ (A).
Theorem 3.5. An LM-fuzzy topological space (X, ) is an induced LM-fuzzy topological space iff (X, ) is a Lowen
LM-fuzzy topological space and an (IC)LM-fuzzy topological space.
Proof. It suffices to show the sufficiency. Since (X, ) is a Lowen LM-fuzzy topological space, we have, for all A ∈ L X ,
(A) ≤
([a] ∧ 1ˆa (A) ).
a∈Copr (L)
By the definition of (IC)LM-fuzzy topology,
([a] ∧ 1ˆa (A) ) ≤ (1ˆa (A) ))
and hence
(A) ≤
((1ˆa (A) )).
a∈Copr (L)
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H.-Y. Li, J.-G. Peng / Fuzzy Sets and Systems 190 (2012) 82 – 93
By Lemma 3.3(1) and the definition of Lowen LM-fuzzy topology, we have
(A) ≥
([a] ∧ 1ˆa (A) ) ≥
(1ˆa (A) ).
a∈Copr (L)
Hence
(A) =
a∈Copr (L)
((1ˆa (A) )),
a∈Copr (L)
which implies that (X, ) is an induced LM-fuzzy topological space. Theorem 3.6. Let (X, ) be a Lowen LM-fuzzy topological space and Y ⊂ X . Then the subspace (Y, |Y ) of (X, ) is
also a Lowen LM-fuzzy topological space.
Proof. Since (X, ) is a Lowen LM-fuzzy topological space, for all A ∈ L Y and a ∈ Copr (L), we have
|Y (A) = {(B)|B ∈ L X , B|Y = A} ≤
{(B)|B ∈ L X , [a] ∧ 1ˆa (B) |Y = [a] ∧ 1ˆa (A) }
≤ {([a] ∧ 1ˆa (B) )|B ∈ L X , [a] ∧ 1ˆa (B) |Y = [a] ∧ 1ˆa (A) } ≤ |Y ([a] ∧ 1ˆa (A) )
which implies that (Y, |Y ) is a Lowen LM-fuzzy topological space. Theorem 3.7. Let (X, ) be a Lowen LM-fuzzy topological space and f : X −→ Y be a surjective mapping. Then
the LM-fuzzy quotient space (Y, / f L→ ) of (X, ) with respect to f is also a Lowen LM-fuzzy topological space.
Proof. Since (X, ) is a Lowen LM-fuzzy topological space, for all A ∈ L Y and a ∈ Copr (L), we have
/ f L→ (A) = ( f L← (A)) ≤ ([a] ∧ 1ˆa ( f L← (A)) ) = ( f L← ([a] ∧ 1ˆa (A) )) = / f L→ ([a] ∧ 1ˆa (A) )),
which implies that (Y, / f L→ ) is a Lowen LM-fuzzy topological space. Theorem
3.8. Let {(X t , t )}t∈T be a family of Lowen LM-fuzzy topological spaces. Then the product space ( t∈T X t ,
t∈T t ) of {(X t , t )}t∈T is also a Lowen LM-fuzzy topological space.
Proof. Let be the subbase of . Then, for all A ∈ L X and a ∈ Copr (L), we have
(A) =
t (B) ≤
t (B) ≤
t∈T ( pt )←
L (B)=A
t∈T ( pt )←
L ([a]∧1ˆa (B) )=[a]∧1ˆa (A)
t∈T ( pt )←
L ([a]∧1ˆa (B) )=[a]∧1ˆa (A)
= ([a] ∧ 1ˆa (A) ).
Hence, for all A ∈ L X and a ∈ Copr (L), (A) ≤ ([a] ∧ 1ˆa (A) ), which implies that (
LM-fuzzy topological space. t∈T
Xt ,
t ([a] ∧ 1ˆa (B) )
t∈T
t ) is a Lowen
)}t∈T be a family of Lowen LM-fuzzy topological spaces, different X t ’s be disjoint. Then
Theorem 3.9. Let
{(X t , t
the sum space ( t∈T X t , t∈T t ) of {(X t , t )}t∈T is a Lowen LM-fuzzy topological space iff for all t ∈ T, (X t , t )
is a Lowen LM-fuzzy topological space.
Proof. Necessity: For all t ∈ T, At ∈ L X t and a ∈ Copr (L), we have
t (At ) =
t (A∗|X t ) = (A∗) ≤ ([a] ∧ 1ˆa (A∗) ) =
t ([a] ∧ 1ˆa (A∗) |X t ) = t ([a] ∧ 1ˆa (A) ),
t∈T
where
A∗(x) =
t∈T
A(x), x ∈ X t
0,
x∈
/ Xt
Thus for every t ∈ T, (X t , t ) is a Lowen LM-fuzzy topological space.
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89
Sufficiency: For every A ∈ L X and a ∈ Copr (L), we have
t (A|X t ) ≤
t ([a] ∧ 1ˆa (A|X t ) ) =
t ([a] ∧ 1ˆa (A) |X t ) = ([a] ∧ 1ˆa (A) ),
(A) =
t∈T
which implies that (
t∈T
t∈T
Xt ,
t∈T
t∈T
t ) is a Lowen LM-fuzzy topological space. 4. Interior and exterior Lowen topology of an LM-fuzzy topology
In the section, we mainly study some categorical properties of LM-fuzzy topological spaces. Especially, based on
the interior and exterior Lowen topology of an LM-fuzzy topology, we prove that LLM-FTop (the category of Lowen
LM-fuzzy topological spaces) is isomorphism-closed and simultaneously bireflective and bicoreflective in SLM-FTop
(the category of stratified LM-fuzzy topological spaces). For this we first prove Lemmas 4.1 and 4.2.
Lemma 4.1. Let (X, ) be a stratified LM-fuzzy topological space and I L () : L X −→ M be defined by
([a] ∧ 1ˆa (A) ) (∀A ∈ L X ).
I L ()(A) =
a∈Copr (L)
Then I L () is the largest Lowen LM-fuzzy topology on X which is contained in . We call I L () the interior Lowen
topology of .
Proof. Clearly, I L ()(A) ≤ (A) for every A ∈ L X . By Lemma 2.5 and the definition of I L , for every a ∈ Copr (L)
and A ∈ L X , we have
I L ()([a] ∧ 1ˆa (A) ) =
([b] ∧ 1ˆa (A) ) = ([a] ∧ 1ˆa (A) ).
ba,b∈Copr (L)
Hence
I L ()(A) =
a∈Copr (L)
([a] ∧ 1ˆa (A) ) =
I L ()([a] ∧ 1ˆa (A) ),
a∈Copr (L)
which implies that I L () is the Lowen LM-fuzzy topology on X which is contained in . Set ≤ and be a Lowen
LM-fuzzy topology on X. Then, for all A ∈ L X ,
([a] ∧ 1ˆa (A) ) ≤
([a] ∧ 1ˆa (A) ) = I L ()(A).
(A) =
a∈Copr (L)
a∈Copr (L)
Therefore, I L () is the largest Lowen LM-fuzzy topology on X which is contained in . Let (X, ) be an LM-fuzzy topological space and : L X −→ M be defined by
a r ( {(B)|ˆr (B) = U }), A = [a] ∧ 1U (∀a ∈ L , r ∈ Copr (L) and U ⊂ X )
(A) =
(A)
others
It is easy to verify that is a subbase of one LM-fuzzy topology on X. We denote this LM-fuzzy topology by E L ()
and call it the exterior Lowen topology of (see Lemma 4.2).
Lemma 4.2. Let (X, ) be an LM-fuzzy topological space. Then E L () is the smallest Lowen LM-fuzzy topology on X
which contains .
Proof. Firstly, we show that E L () is the Lowen LM-fuzzy topology on X. In fact, for every A ∈ L X and a ∈ Copr (L),
we have
E L ()(A) =
(C ),
∈
B =A ∈ ()∈ C =B ∈
90
H.-Y. Li, J.-G. Peng / Fuzzy Sets and Systems 190 (2012) 82 – 93
and
E L ()([a] ∧ 1ˆa (A) ) =
∈
≥
∈
(C )
B =[a]∧1ˆa (A) ∈ ()∈ C =B ∈
B =A ∈ ()∈ C =B ∈
([a] ∧ 1ˆa (C ) ).
If C is not a one-step L-subset, then
([a] ∧ 1ˆa (C ) ) =
{(B)|ˆr (B) = ˆa (C )} ≥ (C ) = (C ).
a r
If C is a one-step L-subset and let C = [b] ∧ 1V , then
{(B)|ˆr (B) = V }
(C ) =
br
and, for all ab,
([a] ∧ 1ˆa (C ) ) = ([a] ∧ 1V ) =
{(B)|ˆr (B) = V } ≥ (C ).
a r
Hence E L ()(A) ≤ E L ()([a] ∧ 1ˆa (A) ), which implies that E L () is the Lowen LM-fuzzy topology on X which
contains .
Secondly, we show that E L () is the smallest Lowen LM-fuzzy topology on X which contains . Let ≤ and be a Lowen LM-fuzzy topology on X. We need to prove that E L () ≤ . It suffices to show that (A) ≤ (A) for all
A ∈ L X . It is clear that E() ≤ when A is not a one-step L-subset. If A is a one-step L-subset and let A = [b] ∧ 1U ,
then
{(B)|ˆr (B) = U } ≤
{(B)|ˆr (B) = U } ≤
{([r ] ∧ 1U )|ˆr (B) = U }
(A) =
br
≤
br
br
([r ] ∧ 1U ) ≤ ([b] ∧ 1U ) = (A),
br
and thus E() ≤ . Corollary 4.3. (X, ) is a Lowen LM-fuzzy topological space iff any two of E L (), I L (), are equal (equivalently,
all the three are equal).
Lemma 4.4.
(1) Let (X, ) be a stratified LM-fuzzy topological space and (Y, ) be a Lowen LM-fuzzy topological space. Then
f : (X, ) −→ (Y, ) is a fuzzy continuous mapping iff f : (X, I L ()) −→ (Y, I L ()) = (Y, ) is a fuzzy continuous
mapping.
(2) Let (X, ) be an LM-fuzzy topological space and (Y, ) be a Lowen LM-fuzzy topological space. Then f : (Y, ) −→
(X, ) is a fuzzy continuous mapping iff f : (Y, E L ()) = (Y, ) −→ (X, E L ()) is a fuzzy continuous mapping.
Proof.
(1) The sufficiency is obvious and we need to show the necessity. Suppose that f : (X, ) −→ (Y, ) is fuzzy continuous,
i.e., (B) ≤ ( f L← (B)) for every B ∈ L Y . Since (Y, ) is a Lowen LM-fuzzy topological space, we have
([a] ∧ 1ˆa (B) ) ≤
( f L← ([a] ∧ 1ˆa (B) ))
(B) =
a∈Copr (L)
=
a∈Copr (L)
([a] ∧ 1ˆa ( f L← (B)) ) = I L ()( f L← (B)),
a∈Copr (L)
which implies that f : (X, I L ()) −→ (Y, I ()) = (Y, ) is a fuzzy continuous mapping.
H.-Y. Li, J.-G. Peng / Fuzzy Sets and Systems 190 (2012) 82 – 93
91
(2) The sufficiency is obvious and we need to show the necessity. It suffices to show for all A = [a] ∧ 1U (∀a ∈ L and
U ⊂ X ), (A) ≤ ( f L← (A)) by the definition of E L () and Lemma 2.13. Suppose that f : (Y, ) −→ (X, ) is a
fuzzy continuous mapping, i.e., (A) ≤ ( f L← (A)) for all A ∈ L X . Since (Y, ) is a Lowen LM-fuzzy topological
space, we have
(A) =
([a] ∧ 1ˆa (A) ),
a∈Copr (L)
and hence
(A) =
a r
=
a r
{(B)|ˆr (B) = U } ≤
{ f L← (B)|ˆr (B) = U }
a r
([r ] ∧ 1 f −1 (U ) ) ≤ ([a] ∧ 1 f −1 (U ) ) = ( f L← (A)),
which implies that f : (Y, E L ()) = (Y, ) −→ (X, E L ()) is a fuzzy continuous mapping. Let i : LLM-FTop −→ SLM-FTop (LM-FTop) be the inclusion functor. By proof of Lemma 4.4, we may show
the following
Theorem 4.5. (1) IL : SLM-FTop −→ LLM-FTop is functor and IL ⵬i.
(2) EL : LM-FTop −→ LLM-FTop is functor and i⵬EL .
Corollary 4.6. LLM-FTop is an isomorphism-closed full proper subcategory of SLM-FTop which is simultaneously
bireflective and bicoreflective in SLM-FTop, and given a stratified LM-fuzzy topological space (X, ), its reflection
and coreflection are given by id X : (X, ) −→ (X, I L ()) and id X : (X, E L ()) −→ (X, ), respectively, where
id X : X −→ X is the identity mapping.
As every right adjoint preserves limits and every left adjoint preserves colimits, we have the following Corollaries.
Corollary 4.7.
(1) Let {(X i , i )}i∈I be a family of LM-fuzzy topological spaces. Then
EL
Xi ,
i =
Xi ,
E L (i ) .
i∈I
i∈I
i∈I
i∈I
(2) Let {(X i , i )}i∈I be a family of stratified LM-fuzzy topological spaces, different X t s be disjoint. Then
IL
Xi ,
i =
Xi ,
I L (i ) .
i∈I
i∈I
i∈I
i∈I
Corollary 4.8. (1) Let (X, ) be an LM-fuzzy topological space and Y ⊂ X . Then E L (|Y ) = E L ()|Y .
(2) Let (X, ) be a stratified LM-fuzzy topological space and f : X −→ Y be a surjective mapping. (Y, / f L→ ) is the
LM-fuzzy quotient space of (X, ) with respect to f. Then I L (/ f L→ ) = I L ()/ f L→ .
Theorem 4.9. (1) Let (X, ) be a stratified LM-fuzzy topological space and U ⊂ X . Then I L ()|U ≤ I L (|U ).
(2) Let (X, ) be an LM-fuzzy topological space and (Y, / f L→ ) be the LM-fuzzy quotient space of (X, ) with respect
to f. Then E L (/ f L→ ) ≤ E L ()/ f L→ .
Proof.
(1) We have I L ()|U ≤ I L (|U ) by Theorem 3.6 and the definition of I L ().
(2) We have E L (/ f L→ ) ≤ E L ()/ f L→ by Theorem 3.7 and the definition of E L (). 92
H.-Y. Li, J.-G. Peng / Fuzzy Sets and Systems 190 (2012) 82 – 93
Remark 4.10. The following two counterexamples show that the above inequalities in Theorem 4.9 cannot be replaced
by equalities.
(1) Let X = L = M = [0, 1], U = [0, 0.5] and (X, ) be a stratified LM-fuzzy topological space, where ([a] ∧
1[0.25,0.5] ) = (1[0.25,0.5] ∨ ([0.5] ∧ 1[0,0.25] )) = 1 (∀a ∈ L) and for others A ∈ L X , (A) = 0.
It is easy to verify that I L () : L X −→ M is defined by I L ()([a] ∧ 1[0.25,0.5] ) = I L ()([a]) = 1 (∀a ∈ L) and for
others A ∈ L X , I L ()(A) = 0. On the one hand, I L ()|U (1[0.25,0.5] ∨ ([0.5] ∧ 1[0,0.25] )) = 0. On the other hand,
I L (|U )(1[0.25,0.5] ∨ ([0.5] ∧ 1[0,0.25] )) = 1. Therefore, I L ()|U < I L (|U ).
(2) Let X = [−1, 1], Y = L = M = [0, 1], : L X −→ M is defined by (0 X ) = (1 X ) = ([0.5] ∧ 1[0.5,1] ) =
(1[−1,−0.5] ) = 1.
It is easy to verify that / f L→ : L Y −→ M is defined by / f L→ (0Y ) = / f L→ (1Y ) = 1 and for others B ∈
L Y , / f L→ (B) = 0. Hence E L (/ f L→ )([0.5] ∧ 1[0.5,1] ) = 0. On the other hand, E L ()/ f L→ ([0.5] ∧ 1[0.5,1] ) =
E L ()( f L← ([0.5] ∧ 1[0.5,1] )) = E L ()([0.5] ∧ 1[0.5,1]∪[−1,−0.5] )) = 1. Therefore, E(/ f L→ ) < E()/ f L→ .
Moreover, we have
Theorem 4.11.
Let {(X t , t )}t∈T be a family of LM-fuzzy topological spaces, different X t ’s be disjoint. Then
(t ) = E L ( t∈T t ).
t∈T
EL
Proof. Clearly,
E L ( t∈T t ) ≤ t∈T E L (t ) by Theorem 3.9 and the definition of E L (). Conversely, let ∈ Copr (L)
and t∈T E L (t )(A) (∀A ∈ L X ), i.e.,
t
E L (t )(A) =
E L (t )(A|X t ) =
t (E ).
t∈T
t∈T
t∈T
∈t
t =D t
t
Dt =A|X t ∈t ()∈t E ∈
Then for all t ∈ T , there exists {Dt }∈t ⊂ L X t such that
(i) ∈t Dt = A|X t ;
t }
t
t
X t such that ()
(ii) For each ∈ t , there exists {E ∈t ⊂ L
∈t E = D ;
t ).
(iii) For each ∈ t , we have ≤ t (E t )∗ ∈ L X be defined as follows:
Let (Dt )∗ ∈ L X , (E (Dt )∗(x) =
Dt (x), x ∈ X t
0,
x∈
/ Xt
t
(E )∗(x) =
t (x), x ∈ X
E t
0,
x∈
/ Xt
Then we have
t
t
t
(Dt )∗ = A, ()∈t (E )∗ = (Dt )∗ and t (E ) = t∈T t ((E )∗).
t∈T ∈t
Hence ≤ EL
t∈T
t
t )∗). Note that
((E t (A) =
t∈T
We have ≤ E L (
∈
t∈T
t∈T
t
(C ).
B =A ∈ ()∈ C =B ∈
t )(A), and hence
t∈T
E L (t ) ≤ E L (). Remark
4.12.
Let {(X t , t )}t∈T be a family of LM-fuzzy topological spaces. Then I L (
( t∈T X t , t∈T I L (t )), and the inequality cannot be replaced by equality.
t∈T
Xt ,
t∈T
t ) ≥
H.-Y. Li, J.-G. Peng / Fuzzy Sets and Systems 190 (2012) 82 – 93
93
Proof. It is easy to verify that I L
t∈T X t , t∈T t ≥
t∈T X t , t∈T I L (t ) by Theorem 3.8 and the definition
of I L (). The following example shows that the inequality cannot be replaced by equality.
Let X = L = M = [0, 1] and (X, ) be a stratified LM-fuzzy topological space, where ([a] ∧ 1[0.25,0.5] ) =
(1[0.25,0.5] ∨ ([0.5] ∧ 1[0,0.25] )) = 1 (∀a ∈ L) and for others A ∈ L X , (A) = 0. It is easy to verify that I L () :
L X −→ M is defined by I L ()([a] ∧ 1[0.25,0.5] ) = I L ()([a]) = 1 (∀a ∈ L) and for others A ∈ L X , I L ()(A) = 0. On
the one hand I L ()× I L ()([0.5]∧1[0,0.25]×X ∪X ×[0,0.25] ) = 0. On the other hand, ×([0.5]∧1[0,0.25]×X ∪X ×[0,0.25] ) = 1
and I L ( × )([0.5] ∧ 1[0,0.25]×X ∪X ×[0,0.25] ) = 1. Therefore I L () × I L () < I L ( × ). 5. Conclusion
In the paper, we firstly introduce the definition of a Lowen LM-fuzzy topological space and discuss its basic properties
such as the properties that the product space and the sum space of Lowen LM-fuzzy topological spaces are also Lowen
LM-fuzzy topological spaces. Secondly, we study some categorical properties of Lowen LM-fuzzy topological spaces.
For example, based on the interior and exterior Lowen topology of an LM-fuzzy topology, we show that LLM-FTop
(the category of Lowen LM-fuzzy topological spaces) is isomorphism-closed and simultaneously bireflective and
bicoreflective in SLM-FTop (the category of stratified LM-fuzzy topological spaces). Moreover, we also show that
(X, ) is an induced LM-fuzzy topological space iff (X, ) is a Lowen LM-fuzzy topological space and an (IC)LM-fuzzy
topological space. In the future, there are still some categorical properties of Lowen LM-fuzzy topological spaces which
are worth studying. For example, is LLM-FTop Cartesian closed?
Acknowledgments
The authors wish to thank Prof. S.E. Rodabaugh and the anonymous referees for their valuable comments and helpful
suggestions.
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