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On L-fuzzy topological spaces
Fuzzy Sets and Systems 149 (2005) 473 – 484 www.elsevier.com/locate/fss
On L-fuzzy topological spaces夡 Jie Zhanga,∗ , Fu-Gui Shib , Chong-You Zhengc a College of Science, North China University of Technology, Beijing 100041, PR China b Department of Applied Mathematics, Beijing Technology Institute, Beijing 100081, PR China c Department of Mathematics, Capital Normal University, Beijing 100037, PR China
Received 25 May 2000; received in revised form 26 April 2004; accepted 10 May 2004
Abstract In this paper, we study L-fuzzy topological spaces, where L represents a completely distributive lattice. We shall investigate the level decomposition of L-fuzzy topologies on X and the corresponding L-fuzzy continuous maps. In addition, we shall establish the representation theorems of L-fuzzy topologies on X. © 2004 Elsevier B.V. All rights reserved. Keywords: L-fuzzy topology; L-topology; L-fuzzy continuous map; L-continuous map
1. Introduction The concept of [0, 1]-topology, also known as Chang’s fuzzy topology, was first introduced in 1968 by Chang [1] and later extended by other authors. See [5,6,16,20]. In 1980, Höhle [12] suggested that a topology can be viewed as an L-subset of 2X . In 1985, Kubiak [17] and Šostak [27] further extended the idea to the more general setting of L-subsets in LX , where L = [0, 1]. A series of papers by the same authors [18,28,29] further established the theory of L-fuzzy topology on LX . During the 1990s, the topic had been pursued by many other investigators. Those who wish to learn more about the area can find related work in [2,3,7,8,9,10,22,33]. With few exceptions, this early work is focused on the case when L = [0, 1]. 夡
The project is supported by the Science Research Foundation of North China University of Technology, the plan of Science and Technology Developing of Beijing city Education Committee of China, and the National Natural Science Foundation of China (No. 10371079). ∗ Corresponding author. E-mail address: [email protected] (J. Zhang). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.05.007
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In 1995 and 1997, respectively, Höhle and Šostak [14], Kubiak and Šostak [18] pushed the concept of L-fuzzy topological space even further to situations where L is more general than [0, 1]. In 1999, Höhle and Šostak [15] established axiomatic foundations of L-fuzzy topology when L is a complete quasi-monoidal lattice (CQML). Rodabaugh [23] established categorical foundations of variable-basis C-fuzzy topology when C is any subcategory of the category CQML. In the present paper, we study the level decomposition of an L-fuzzy topology and the corresponding Lfuzzy continuous maps. In addition, we also establish some representation theorems for L-fuzzy topologies on X. In fact, Negoita and Ralescu [21] first gave the situation where the lattice F (X, L), i.e., all L-valued subsets of X, can be characterized by a mapping defined on L that takes values in 2X . Wuyts [32] studied the [0,1]-topologies by their level-topologies, whose level topologies are classical ones ({0, 1}-topologies). The main results of this paper are several representation theorems for L-fuzzy topology on X, where L represents a completely distributive lattice. Based on the results of this paper, we have also developed representation theorems for the category L-FTOP which consists of L-fuzzy topological spaces and L-fuzzy continuous maps. Due to page limitation, we shall report this work elsewhere. 2. Preliminaries Throughout this paper, L represents a completely distributive lattice with the smallest element ⊥ and the greatest element , where ⊥ = . We define M(L) to be the set of all non-zero ∨-irreducible (or coprime) elements in L such that a∈M(L) iff a b ∨ c implies a b or a c. Let P (L) be the set of all non-unit prime elements in L such that a∈P (L) iff a b ∧ d implies a b or a c. Finally, let X be a non-empty usual set, and LX be the set of all L-fuzzy sets on X. For each a∈L, let a denote a constant-valued L-fuzzy set with a as its value. Let ⊥ and be the smallest element and the greatest element in LX , respectively. For the empty set ∅ ⊂ L, we define ∧∅ = and ∨∅ = ⊥. Definition 2.1 (Wang [31]). Suppose that a∈L and A ⊆ L. (1) A is called a maximal family of a if (a) inf A = a, (b) ∀B ⊆ L, infB a implies that ∀x∈A there exists y∈B such that y x. (2) A is called a minimal family of a if (a) sup A = a, (b) ∀B ⊆ L, sup B a implies that ∀x∈A there exists y∈B such that y x. Remark 2.2. Hutton [16] proved that if L is a completely distributive lattice and a∈L, then there exists B ⊆ L such that (i) a = B, and (ii) if A ⊆ L and a = A, then for each b∈B there is a c∈A such that b c. However, if ∀a∈L, and if there exists B ⊆ L satisfying (i) and (ii), then in general L is not a completely distributive lattice. To this end, Wang [31] introduced the following modification of condition (ii), (ii ) if A ⊆ L and a A, then for each b∈B there is a c∈A such that b c. Wang proved that a complete lattice L is completely distributive if and only if for each element a in L, there exists B ⊆ L satisfying (i) and (ii ). Such a set B is called a minimal set of a by Wang [31]. The
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concept of maximal family is the dual concept of minimal family, and a complete lattice L is completely distributive if and only if for each element a in L, there exists a maximal family B ⊆ L. Let (a) denote the union of all maximal families of a. Likewise, let (a) denote the union of all minimal sets of a. Finally, let ∗ (a) = (a) ∩ P (L) and ∗ (a) = (a) ∩ M(L). One can easily see that both (a) and ∗ (a) are maximal sets of a. Likewise, both (a) and ∗ (a) are minimal sets of a. Also, we have () = ∅ and (⊥) = ∅. Lemma 2.3 (Shi [24,25,26] and Wang [31]). For a∈L and a map T : LX → L, we define T[a] = {A∈LX | T(A) a}
and
T[a] = {A∈LX |a ∈ / (T(A))}.
Let T be a map from LX to L and a, b∈L. Then (1) a∈(b) ⇒ T[b] ⊆ T[a] ; a∈(b) ⇒ T[a] ⊆ T[b] . ∗ ∗ (2) a ⊆ (b) ⇔ ∗ (a) ⊆∗ (b) ⇔ (b) b ⇔ (a) ⊆ (a) ⇔ (b) ⊆ (a). (3) ( i∈ ai ) = i∈ (ai ) and ( i∈ ai ) = i∈ (ai ) for any sub-family {ai }i∈ ⊆ L. 3. Level decompositions of an L-fuzzy topology Definition 3.1 (Hutton [16]). An L-fuzzy topology on X is a map T : LX → L satisfying the following three axioms: (1) T() = ; X (2) T(A ∧ T(B) for every A, B∈L ; ∧ B) T(A) (3) T( i∈ Ai ) i∈ T(Ai ) for every family {Ai | i∈} ⊆ LX . The pair (X, T) is called an L-fuzzy topological space (L-fts). For every A∈LX , T(A) is called the degree of openness of the fuzzy subset A. Just as an L-topology on X is an ordinary subset of LX , an L-fuzzy topology on X is a fuzzy subset of LX . In many papers a level decomposition of an L-fuzzy topology was considered: T is an L-fuzzy topology on X if and only if ∀a∈L, T[a] is an L-topology on X.
Excepting this, we have the followings level decompositions of an L-fuzzy topology: Theorem 3.2. Let T be a map T : LX → L. Then the following conditions are equivalent: (1) T is an L-fuzzy topology on X. (2) ∀a∈M(L), T[a] is an L-topology on X. (3) ∀a∈L, T[a] is an L-topology on X. (4) ∀a∈P (L), T[a] is an L-topology on X. Proof. (1) ⇒ (2): This part is obvious. (2) ⇒ (1): For each a∈M(L), we have ∈T[a] and T() a. Accordingly, {a | a∈M(L)} = . T()
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Thus, T() = . Next, let A, B∈LX . Clearly, when T(A) ∧ T(B) = ⊥, we have T(A ∧ B) T(A) ∧T(B). Otherwise, if T(A)∧T(B) > ⊥, then for each a T(A)∧T(B), we have T(A) a, T(B) a, or A, B∈T[a] . Consequently, we have A ∧ B∈T[a] , or T(A ∧ B) a. This further implies that {a∈M(L) | a T(A) ∧ T(B)} = T(A) ∧ T(B). T(A ∧ B) Let {Ai | i∈} ⊆ LX . Then for a∈M(L) and a T(Ai ), we have T(Ai ) a and Ai ∈T[a] for i∈ each i∈. The proof follows because i∈ Ai ∈T[a] , T( i∈ Ai ) a and
T Ai T(Ai ) = T(Ai ). a∈M(L) a i∈
i∈
i∈
(1) ⇒ (3): ∀a∈L, since T(⊥) = T() = and () = ∅, we have a ∈ / () = (T(⊥)) = (T()). Thus ⊥, ∈T[a] . Next, let A, B∈T[a] . Then a ∈ / (T(A)) and a ∈ / (T(B)). Hence a∈ / (T(A)) ∪ (T(B)) = (T(A) ∧ T(B)) ⊇ (T(A ∧ B)). Furthermore, since a ∈ / (T(A ∧ B)), we have A ∧ B∈T[a] . Finally, let Ai ∈T[a] . Then a ∈ / (T(Ai )) for each i∈ and a∈ / . (T(Ai )) = T(Ai ) ⊇ T Ai i∈
i∈
i∈
Consequently, we have a ∈ / (T( i∈ Ai )), or i∈ Ai ∈T[a] . (3) ⇒ (4): This part is obvious. ⇒ (1): ∀a∈P (L), it is clear that ∈T[a] . Thus a ∈ / (T()), ∗ (T()) = ∅, and T() = (4) ∗ X (T()) = . Next, let A, B∈L . Then clearly, when T(A) ∧ T(B) = ⊥, we have T(A ∧ B) T(A) ∧ T(B). If T(A) ∧ T(B) > ⊥, then for each a∈P (L) and a ∈ / (T(A) ∧ T(B)) = (T(A)) ∪ (T(B)), we have a ∈ / (T(A)), (T(B)) and a ∈ / (T(A ∧ B)). Accordingly, we have ∗ (T(A) ∧ T(B)) ⊇ ∗ (T(A ∧ B)),
or
T(A ∧ B) T(A) ∧ T(B).
Finally, suppose that {Ai | i∈} ⊆ LX . Then obviously T Ai T(Ai ), if T(Ai ) = ⊥. i∈
i∈
i∈
> ⊥. Then ∀a∈P (L) and a ∈ / ( i∈ T(Ai )) = (T(Ai )). It i∈ follows that a ∈ / (T(Ai )), Ai ∈T[a] for each i∈ and i∈ Ai ∈T[a] . Hence ⊆ ∗ ∗ T Ai T(Ai ) , or T Ai T(Ai ). Suppose now that
i∈
i∈ T(Ai )
i∈
i∈
i∈
We can now state the following decomposition theorem of L-fuzzy topology. The proof is straightforward and therefore omitted.
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Theorem 3.3. Let T be an L-fuzzy topology on X. Then [a] [a] T= (a ∧ T[a] ) = (a ∧ T[a] ) = (a ∨ T ) = (a ∨ T ). a∈L
a∈M(L)
a∈L
a∈P (L)
Corollary 3.4. Let T1 and T2 be L-fuzzy topologies on X, then the following conditions are equivalent: (1) T1 = T2 . (2) ∀a∈L, T1[a] = T2[a] . (3) ∀a∈M(L), T1[a] = T2[a] . (4) ∀a∈L, T1[a] = T2[a] . (5) ∀a∈P (L), T1[a] = T2[a] . Theorem 3.5. Let T be an L-fuzzy topology on X, then (1) ∀a∈L, T[a] = b∈(a) T[b] . (2) ∀a∈M(L), T[a] = b∈∗ (a) T[b] . (3) ∀a∈L, T[a] = a∈(b) T[b] . (4) ∀a∈P (L), T[a] = a∈∗ (b) T[b] . b∈P (L)
Proof. (1) By Lemma 2.3, we have that ∀a∈L, T[a] ⊆ b∈(a) T[b] . To show that T[a] ⊇ b∈(a) T[b] , we take A∈LX and A∈ b∈(a) T[b] . Notice that ∀b∈(a), T(A) b. Hence T(A) {b | b∈(a)} = a, which implies that A∈T[a] . (2) The proof is similar to (1). (3) By Lemma 2.3, we have that ∀a∈L, T[a] ⊆ a∈(b) T[b] . To show that T[a] ⊇ a∈(b) T[b] , we take A∈LX and A∈ a∈(b) T[b] . Notice that ∀b∈L and a∈(b), it follows that b∈ / (T(A)). We prove by contradiction as follows. Suppose that a∈ ( T (A)). Notice that T (A)=∧{b | b∈(T(A))} and (T(A)) = {(b) | b∈(T(A))}. There must exist b∈(T(A)) such that a∈(b). But this is impossible. (4) The proof is similar to (3). Remark 3.6.(1) b∈(a) implies b>a, where > is way-below relation [4], i.e. b>a if and only if for every up-directed set S in L, ∨S a implies that ∃s∈S such that s b; (2) If a∈M(L), then b∈ ∗ (a) if and only if b>a. (3) ∀a∈M(L), T[a] = b∈∗ (a) T[b] ⇔ T[a] = T[b] . b >a b∈M(L)
Proof. (1) Since (a) is a minimal set of a, from Definition 2.1, we have that for every up-directed set S in L, if ∨S a, then ∀b∈(a) there exists s∈S such that s b. It follows that b>a. (2) Let a∈M(L) and b>a. From Theorems 1.3.6 and 1.3.8 in [19] and Definition 2.1, we know that ∗ (a) is both an up-directed set and a lower set, and ∨∗ (a) = a. Hence, ∃b ∈∗ (a) such that a b b. In other words, b∈∗ (a). Conversely, if b∈∗ (a), then since ∗ (a) ⊆ (a), b∈∗ (a) implies b∈(a). It follows that b>a. (3) It is obvious. Theorem 3.7. Let {Ta | a∈M(L)} be a family of L-topologies on X. Then the following conditions are equivalent:
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(1) There exists an L-fuzzy topology T on X such that T[a] = Ta for each a∈M(L). (2) ∀a∈M(L), Ta = b∈∗ (a) Tb . Proof. (1) ⇒ (2): This holds because of Theorem 3.5. (2) ⇒ (1): Let T = a∈M(L) (a ∧ Ta ). Obviously, we have Ta ⊆ T[a] . For any A∈T[a] , we have T(A) a and {b∈M(L) | A∈Tb } a. Next, since ∗ (a) is a minimal family of a, for each b∈∗ (a), there exists b ∈M(L) such that b b and A∈Tb ⊆ Tb . Therefore, A∈ b∈∗ (a) Tb = Ta . Similarly, we can state the following theorems. Theorem 3.8. Let {Ta | a∈P (L)} be a family of L-topologies on X. Then the following conditions are equivalent: [a] (1) There exists an L-fuzzy topology T on X such that T = Ta for each a∈P (L). (2) ∀a∈P (L), Ta = a∈∗ (b) Tb . Theorem 3.9. Let {Ta | a∈L} be a family of L-topologies on X. Then the following conditions are equivalent: (1) There exists an L-fuzzy topology T on X such that T[a] = Ta for each a∈L. (2) ∀a∈L, Ta = b∈(a) Tb . Theorem 3.10. Let {Ta | a∈L} be a family of L-topologies on X. Then the following conditions are equivalent: (1) There exists an L-fuzzy topology T on X such that T[a] = Ta for each a∈L. (2) ∀a∈L, Ta = a∈(b) Tb . 4. Representation theorems of L-fuzzy topology Let LT[X] denote the family of all L-topologies on X. Let LFT[X] denote the family of all L-fuzzy topologies on X. The order relation on LFT[X] is defined as follow: ∀T1 , T2 ∈LFT[X],
T1 4T2 ⇔ ∀A∈LX ,
T1 (A) T2 (A).
Theorem 4.1 (Höhle [15]). (LFT[X], 4) is a complete lattice. In fact, it is a complete sub-meet-semilattice X X of L(L ) , i.e. closed under the ∧ of L(L ) . To facilitate further illustration, let us define the following classes: F (b) , UL [X] = F : L → LT[X] | ∀a∈L, F (a) = a∈(b) UL [X] = F : L → LT[X] | ∀a∈L, F (a) = F (b) , b∈(a)
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F : M(L) → LT[X] | ∀a∈M(L), F (a) = F (b) , b∈∗ (a) F (b) . UP (L) [X] = F : P (L) → LT[X] | ∀a∈P (L), F (a) = ∗ UM(L) [X] =
a∈ (b)
In addition, let us define the following order relations within the classes UL [X], UL [X], UM(L) [X] and UP (L) [X]: L
F1 , F2 ∈UL [X], F1 4 F2 ⇔ ∀a∈L, F1 (a) ⊆ F2 (a), F1 , F2 ∈UL [X], F1 4L F2 ⇔ ∀a∈L, F1 (a) ⊆ F2 (a), F1 , F2 ∈UM(L) [X], F1 4M(L) F2 ⇔ ∀a∈M(L), F1 (a) ⊆ F2 (a), F1 , F2 ∈UP (L) [X], F1 4P (L) F2 ⇔ ∀a∈P (L), F1 (a) ⊆ F2 (a). L Theorem 4.2. (UL [X], 4 ), UL [X], 4L , (UM(L) [X], 4M(L) ) and (UP (L) [X], 4P (L) ) are complete lat L tices. Obviously, UL [X], 4 and UL [X], 4L are complete sub-meet-semilattices of the lattice 4L (LT[X])L (i.e., closed under the ∧ of (LT[X])L , when {Fi | i∈} ⊆ UL [X], F = i∈ Fi be defined 4L as ∀a∈L, F (a) = F (a); when {F | i∈ } ⊆ UL [X], F = i∈ Fi be defined as ∀a∈L, F (a) = i i i∈ F (a)), UM(L) [X], 4M(L) is a complete sub-meet-semilattices of the lattice (LT[X])M(L) , and i∈ i UP (L) [X], 4P (L) is a complete sub-meet-semilattices of the lattice (LT[X])P (L) . Proof. ∀a∈L, let us define F⊥ (a) = {∅, X} and F (a) = LX . Clearly, we have F⊥ , F ∈UL [X], and they L are the smallest element and the greatest element in (UL [X], 4 ), respectively. Next, let {Fi | i∈} ⊆ L 4 UL [X] and F = i∈ Fi . Since F (a) = Fi (a) = Fi (b) = Fi (b) = F (b), i∈
i∈ a∈(b)
a∈(b) i∈
a∈(b)
L
it follows that F ∈UL [X]. Accordingly, (UL [X], 4 ) is a complete lattice. The same argument can be used to prove the rest of the theorem. The following representation theorem of L-fuzzy topology follows naturally. Theorem 4.3. The map f : LFT[X] → UL [X], T −→ FT ( for every a∈L and FT (a) = T[a] ) is an isomorphism in the category of complete meet-semilattices and f −1 : UL [X] → LFT[X], F −→ TF = a∈L (a ∨ F (a)). Proof. For each T∈LFT[X], it is easy to verify that FT (a) = T[a] = T[b] = FT (b). a∈(b)
a∈(b)
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Hence, FT ∈UL [X]. Next, by Theorems 3.2, 3.3 and Corollary 3.4, it suffices to show that f is an injection. Since x ∈( / TF )[] iff ∈(TF (x)) = ((a ∨ F (a))(x)) = {(a) | a∈L, x ∈F / (a)} a∈L
/ (a) iff x ∈ / ∈(a) F (a) = F (), we have FTF () = iff there exists a∈L such that ∈(a) and x ∈F (TF )[] = F (). This shows that FTF = F . It follows that f is a surjection as well as a bijection, and (a ∨ F (a)). f −1 : UL [X] → LFT[X], F −→ TF = a∈L
Next, let T1 , T2 ∈LFT[X], {Ti | i∈} ⊆ LFT[X]. Then it is straightforward to show that f (T1 )4 4L f (T2 ) when T1 4T2 . Hence f ( i∈ Ti ) = i∈ f (Ti ) and the proof is complete.
L
The following Theorem follows directly from the above proof. Theorem 4.4. The map f : LFT[X] → UP (L) [X], T −→ FT ( for every a∈P (L), FT (a) = T[a] ) −1 : U is an isomorphism P (L) [X] → LFT[X], in the category of complete meet-semilattices and f F −→ TF = a∈P (L) (a ∨ F (a)). Theorem 4.5. The map f : LFT[X] → UL [X], T −→ FT ( for every a∈L, FT (a) = T[a] ) is an isomorphism in the category of complete meet-semilattices and f −1 : UL [X] → LFT[X], F −→ TF = a∈L (a ∨ F (a)). Proof. For each T∈LFT[X], it is easy to verify that FT (a) = T[a] = T[b] = FT (b). b∈(a)
b∈(a)
Hence FT ∈UL [X]. Next, by Theorem 3.3 and Corollary 3.4, it suffices to show that f is an injection. It is proved easily that x∈(TF )[] iff TF (x) = (a ∧ F (a))(x) = {a | x∈F (a)} a∈L
iff (because of Lemma 2.3) (a) = (∨{a | x∈F (a)}) ⊇ (). x∈F (a)
On the other hand, we can prove x∈F () = F (a) iff ∀a∈(a), x∈F (a) iff a∈()
x∈F (a)
(a) ⊇ ().
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(In fact, obviously ∀a∈(), x∈F (a) implies (a) ⊇ (a) = (). Conversely, for x∈F (a) a∈() each c∈() ⊆ x∈F (a) (a), then there exists a∈L such that c∈(a) and x∈F (a) = b∈(a) F (b). It show that x∈F (c)). So, we conclude that x∈(TF )[] iff x∈F (), i.e., FTF () = (TF )[] = F (). This shows that FTF = F . It follows that f is a surjection as well as a bijection, and (a ∧ F (a)). f −1 : UL [X] → LFT[X], F −→ TF = a∈L
Next, let T1 , T2 ∈LFT[X], {Ti | i∈} ⊆ LFT[X]. Then it is straightforward to show that f (T1 )4L 4L f (T2 ) when T1 4T2 . Hence f ( i∈ Ti ) = i∈ f (Ti ) and the proof is complete. The following Theorem follows directly from the above proof. Theorem 4.6. The map f : LFT[X] → UM(L) [X], T −→ FT ( for every a∈M(L), FT (a) = T[a] ) −1 : U is an isomorphism M(L) [X] → LFT[X], in the category of complete meet-semilattices and f F −→ TF = a∈M(L) (a ∧ F (a)). 5. L-fuzzy continuous maps Definition 5.1 (Höhle [15]). Let (X, T1 ) and (Y, T2 ) be two L-topological spaces. Let f : X → Y be a map. f : (X, T1 ) → (Y, T2 ) is called L-fuzzy continuous if for every A∈LY we have T1 (f ← (A)) T2 (A), where f ← (A) = A ◦ f. From Definition 5.1, obviously, f : (X, T1 ) → (Y, T2 ) is an L-fuzzy continuous map if and only if ∀a∈M(L), f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. Excepting this, we have the followings equivalent conditions: Theorem 5.2. Let (X, T1 ) and (Y, T2 ) be L-fuzzy topological spaces and f : X → Y be a map. Then the following conditions are equivalent: (1) f : (X, T1 ) → (Y, T2 ) is an L-fuzzy continuous map. (2) ∀a∈M(L), f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. (3) ∀a∈L, f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. (4) ∀a∈P (L), f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. Proof. (1) ⇒ (2): This part is obvious. (2) ⇒ (1): ∀A∈LY , a∈M(L) such that a T2 (A), we have A∈T2[a] and f ← (A)∈T1 [a] by the continuity of f : (X, T1[a] ) → (Y, T2[a] ). Accordingly, T1 (f ← (A)) a for each a∈M(L) M(T2 (A)), ← where M(T2 (A)) = {a∈M(L) | a T2 (A)}. It follows that T1 (f (A)) M(T2 (A)) = T2 (A). (1) ⇒ (3): ∀A∈LY , since T2 (A) T1 (f ← (A)), it follows from Lemma 2.3 that a ∈ / (T1 (f ← (A))) [a] [a] ← when ∀a∈L, if a ∈ / (T2 (A)). In other words, if A∈T1 , then f (A)∈T1 . Thus f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map.
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(3) ⇒ (4): This is obvious. (4) ⇒ (1): For ∀a∈P (L) and A∈LY , if a ∈ / (T2 (A)), then A∈T2[a] . Thus f ← (A)∈T1[a] by the continu/ (T1 (f ← (A))) and ∗ (T2 (A)) ity of f : (X, T1[a] ) → (Y, T2[a] ). In other words, a ∈ ∗ ← ⊇ (T1 (f (A))). It follows from Lemma 2.3 that T2 (A) T1 (f ← (A)). Hence the proof is completed. Definition 5.3. Let (X, T1 ) and (Y, T2 ) be L-fuzzy topological spaces. A map f : (X, T1 ) → (Y, T2 ) is called an L-fuzzy open map if for each A∈LX we have T1 (A) T2 (f → (A)). Theorem 5.4. Let (X, T1 ) and (Y, T2 ) be L-fuzzy topological spaces, and f : X → Y be a map. Then the following conditions are equivalent: (1) f : (X, T1 ) → (Y, T2 ) is an L-fuzzy open map. (2) f : (X, T1[a] ) → (Y, T2[a] ) is an L-fuzzy open map for each a∈M(L). (3) f : (X, T1[a] ) → (Y, T2[a] ) is an L-open map for each a∈L. (4) f : (X, T1[a] ) → (Y, T2[a] ) is an L-open map for each a∈P (L). Proof. (1) ⇒ (2): This is obvious by Definition 5.3. X (2) ⇒ (1): For a given clearly, T1 (A) T2 (f → (A)). If T1 (A) > ⊥, A∈L , if T1 (A) = ⊥, then → then since T1 (A) = M(T1 (A)), we have T2 (f (A)) a for each a∈M(T1 (A)). Hence T2 (f → (A)) {a | a∈M(T1 (A))} = T1 (A). (1) ⇒ (3): ∀A∈LX . From part (1) of the theorem and Lemma 2.3, we have (T1 (A)) ⊇ (T2 (f → (A))). It follows that a ∈ / (T2 (f → (A))) if a ∈ / (T1 (A)) for each a∈L. In other words, f → (A)∈ [a] [a] T2 if A∈ T1 for each a∈L. Hence statement (3) holds. (3) ⇒ (4): This part is obvious. (4) ⇒ (1): ∀a∈P (L) and A∈LX , from part (4) of the theorem, if a ∈ / (T1 (A)), then a ∈ / (T2 → ∗ ∗ → → (f (A)). Thus, (T1 (A)) ⊇ (T2 (f (A))). We have from Lemma 2.3 that T1 (A) T2 (f (A)). Hence the proof is completed. Definition 5.5. Let (X, T1 ) and (Y, T2 ) be two L-fuzzy topological spaces. Let f : X → Y be a map. f : (X, T1 ) → (Y, T2 ) is called an L-fuzzy homeomorphism if f is bijective and f and f ← are L-fuzzy continuous maps. Theorem 5.6. Let (X, T1 ) and (X, T2 ) be two L-fuzzy topological spaces, and f : X → Y be a bijective map. The following conditions are equivalent: (1) f : (X, T1 ) → (Y, T2 ) is an L-fuzzy homeomorphism. (2) f : (X, T1[a] ) → (Y, T2[a] ) is a homeomorphism for each a∈M(L). (3) f : (X, T1[a] ) → (Y, T2[a] ) is a homeomorphism for each a∈L. (4) f : (X, T1[a] ) → (Y, T2[a] ) is a homeomorphism for each a∈P (L). Proof. It follows from Definitions 5.1, 5.3, 5.5 and Theorems 5.2 and 5.4.
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Definition 5.7 (He [11]). Let X and Y be two non-empty sets. Let L and L be two completely distributive lattices. Let g : L → L and f : X → Y be two maps. The right adjoint map of g, denoted by g ∗ : L → L, is defined as g ∗ (b) = {a∈L | g(a) b}, for b∈L. The map H = g f : LX → LY is defined as H (A)(y) = {g(A(x)) | f (x) = y, x∈X},
for A∈LX and y∈Y.
The map H is called a bi-induced map of g and f. The inverse of H, denoted by H ← : LY → LX , is defined as H ← (B) = g ∗ ◦ B ◦ f , where B∈LY . Definition 5.8 (He [11]). Let H = g f : (LX , T) → (LY , R) be a bi-induced map. H is called fuzzy continuous, if for each B∈LY , T(H ← (B)) g ∗ (R(B)). Theorem 5.9. Let H = g f : (LX , T) → (LY , R) be a bi-induced map. Let g be a join preserving order homomorphism. Then the following conditions are equivalent: (1) H is a fuzzy continuous map. (2) For each a∈L, H : (LX , T[g ∗ (a)] ) → (LY , R[a] ) is a continuous map. (3) For each a∈M(L), H : (LX , T[g ∗ (a)] ) → (LY , R[a] ) is a continuous map. Proof. (1) ⇒ (2): Since R(B) a, ∀B∈R[a] , we have T(H ← (B)) g ∗ (R(B)) g ∗ (a). This implies that H ← (B)∈T[g ∗ (a)] . (2) ⇒ (3): This is obvious. (3) ⇒ (1): ∀A∈LY , if R(B) = ⊥, then T(H ← (B)) g ∗ (R(B)). Likewise, if R(B) > ⊥, then foreach a∈M(R(B)), we have R (B) a. Hence T(H ← (B)) g ∗ (a). This implies that T(H ← (B)) ∗ ∗ {g (a) | a∈M(R(B))} = g ( {a | a∈M(R(B))}) = g ∗ (R(B)). Definition 5.10 (He [11]). Let H = g f : (LX , T) → (LY , R) be a bi-induced map. H is called a fuzzy open map if for each A∈LX , R(H (A)) g(T(A)). Theorem 5.11. Let H = g f : (LX , T) → (LY , R) be a bi-induced map. Let g be a join preserving order homomorphism. Then the following conditions are equivalent: (1) H is a fuzzy open map. (2) For each a∈L, H : (LX , T[a] ) → (LY , R[g(a)] ) is an open map. (3) For each a∈M(L), H : (LX , T[a] ) → (LY , R[g(a)] ) is an open map. Proof. (1) ⇒ (2): ∀a∈L and A∈LX , if A∈T[a] , then T(A) a. Hence, R(H (A)) g(T(A)) g(a). It follows that H (A)∈R[g(a)] . (2) ⇒ (3): This is obvious. (3) ⇒ (1): Clearly, ∀A∈LX , if T(A) = ⊥, then R(H (A)) g(T(A)) = g(⊥) = ⊥. Likewise, if T(A) > ⊥, then by (3), for each a∈M(T(A)), we have R(H (A)) g(a). Hence, R(H (A)) {g(a) | a∈M(T(A))} = g( {a | a∈M(T(A))}) = g(T(A)).
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Acknowledgements The authors are deeply grateful to the referees and the Area Editor for the valuable suggestions. References [1] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182–190. [2] K.C. Chattopadyay, R.N. Hazra, S.K. Samanta, Gradation of openness: fuzzy topology, Fuzzy Sets and Systems 49 (1992) 237–242. [3] K.C. Chattopadyay, S.K. Samanta, Fuzzy topology: fuzzy closure operator, fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Systems 54 (1993) 207–212. [4] G. Gierz et al., A Compendium of Continuous Lattices, Springer, Berlin, 1980. [5] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174. [6] J.A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973) 737–742. [7] V. Gregori, A. Vidal, Fuzziness in Chang’s fuzzy topological spaces, Rend. Inst. Mat. Univ. Trieste 30 (Suppl.) (1999) 111–121. [8] V. Gregori, A. Vidal, Gradation of openness and Chang’s fuzzy topologies, Fuzzy Sets and Systems 109 (2000) 223–244. [9] J. Gutierrez Garcia, M.A. De Prada Vicente, A. Šostak, Even fuzzy topological and related structures, Quaestiones Math. 20 (1997) 327–351. [10] R.N. Hazra, S.K. Samanta, K.C. Chattopadyay, Fuzzy topology redefined, Fuzzy Sets and Systems 45 (1992) 79–80. [11] M. He, Bi-induced map on L-fuzzy sets, Kexue tongbao 6 (1986) 475 (in Chinese). [12] U. Höhle, Uppersemicontinuous fuzzy sets and applications, J. Math. Anal. Appl. 78 (1980) 659–673. [13] U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Dordrecht, Boston, 1999. [14] U. Höhle, A.P. Šostak, A general theory of fuzzy topological spaces, Fuzzy Sets and Systems 73 (1995) 131–149. [15] U. Höhle, A.P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Dordrecht, Boston, 1999, pp. 123–272. [16] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975) 74–79. [17] T. Kubiak, On fuzzy topologies, Ph. D. Thesis, Adam Mickiewicz University, Pozna´n, Poland, 1985. [18] T. Kubiak, A.P. Šostak, Lower set-valued fuzzy topologies, Quaestiones Math. 20 (1997) 423–429. [19] Y.M. Liu, M.K. Luo, Fuzzy Topology, World Scientific Publishing, Singapore, 1998. [20] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621–633. [21] C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Birkhäuser Verlag, Basel, Stuttgart, 1975. [22] W. Peeters, Subspaces of smooth fuzzy topologies and initial smooth fuzzy structures, Fuzzy Sets and Systems 104 (1999) 423–433. [23] S.E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Dordrecht, Boston, 1999, pp. 273–388. [24] F.G. Shi, Theory of L -nested sets and L -nested sets and its applications, Fuzzy Systems Math. 4 (1995) 63–72 (in Chinese). [25] F.G. Shi, L-fuzzy sets and prime element nested sets, J. Math. Res. Exposition 16 (1996) 398–402 (in Chinese). [26] F.G. Shi, Theory of molecular nested sets and its applications, J. Yantai Teachers Univ. 1 (1996) 33–36 (in Chinese). [27] A.P. Šostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo Ser. II 11 (Suppl.) (1985) 98–103. [28] A.P. Šostak, Two decades of fuzzy topology: basic ideas, notations and results, Russian Math. Surveys 44 (1989) 125–186. [29] A.P. Šostak, On some modifications of fuzzy topologies, Mat. Vesnik 41 (1989) 20–37. [30] A.P. Šostak, Basic structures of fuzzy topology, J. Math. Sci. 78 (6) (1996) 662–701. [31] G.J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems 47 (1992) 351–376. [32] P. Wuyts, On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies, Fuzzy Sets and Systems 12 (1984) 71–85. [33] M.S. Ying, A new approach for fuzzy topology (I), Fuzzy Sets and Systems 39 (1991) 303–321.
On L-fuzzy topological spaces夡 Jie Zhanga,∗ , Fu-Gui Shib , Chong-You Zhengc a College of Science, North China University of Technology, Beijing 100041, PR China b Department of Applied Mathematics, Beijing Technology Institute, Beijing 100081, PR China c Department of Mathematics, Capital Normal University, Beijing 100037, PR China
Received 25 May 2000; received in revised form 26 April 2004; accepted 10 May 2004
Abstract In this paper, we study L-fuzzy topological spaces, where L represents a completely distributive lattice. We shall investigate the level decomposition of L-fuzzy topologies on X and the corresponding L-fuzzy continuous maps. In addition, we shall establish the representation theorems of L-fuzzy topologies on X. © 2004 Elsevier B.V. All rights reserved. Keywords: L-fuzzy topology; L-topology; L-fuzzy continuous map; L-continuous map
1. Introduction The concept of [0, 1]-topology, also known as Chang’s fuzzy topology, was first introduced in 1968 by Chang [1] and later extended by other authors. See [5,6,16,20]. In 1980, Höhle [12] suggested that a topology can be viewed as an L-subset of 2X . In 1985, Kubiak [17] and Šostak [27] further extended the idea to the more general setting of L-subsets in LX , where L = [0, 1]. A series of papers by the same authors [18,28,29] further established the theory of L-fuzzy topology on LX . During the 1990s, the topic had been pursued by many other investigators. Those who wish to learn more about the area can find related work in [2,3,7,8,9,10,22,33]. With few exceptions, this early work is focused on the case when L = [0, 1]. 夡
The project is supported by the Science Research Foundation of North China University of Technology, the plan of Science and Technology Developing of Beijing city Education Committee of China, and the National Natural Science Foundation of China (No. 10371079). ∗ Corresponding author. E-mail address: [email protected] (J. Zhang). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.05.007
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In 1995 and 1997, respectively, Höhle and Šostak [14], Kubiak and Šostak [18] pushed the concept of L-fuzzy topological space even further to situations where L is more general than [0, 1]. In 1999, Höhle and Šostak [15] established axiomatic foundations of L-fuzzy topology when L is a complete quasi-monoidal lattice (CQML). Rodabaugh [23] established categorical foundations of variable-basis C-fuzzy topology when C is any subcategory of the category CQML. In the present paper, we study the level decomposition of an L-fuzzy topology and the corresponding Lfuzzy continuous maps. In addition, we also establish some representation theorems for L-fuzzy topologies on X. In fact, Negoita and Ralescu [21] first gave the situation where the lattice F (X, L), i.e., all L-valued subsets of X, can be characterized by a mapping defined on L that takes values in 2X . Wuyts [32] studied the [0,1]-topologies by their level-topologies, whose level topologies are classical ones ({0, 1}-topologies). The main results of this paper are several representation theorems for L-fuzzy topology on X, where L represents a completely distributive lattice. Based on the results of this paper, we have also developed representation theorems for the category L-FTOP which consists of L-fuzzy topological spaces and L-fuzzy continuous maps. Due to page limitation, we shall report this work elsewhere. 2. Preliminaries Throughout this paper, L represents a completely distributive lattice with the smallest element ⊥ and the greatest element , where ⊥ = . We define M(L) to be the set of all non-zero ∨-irreducible (or coprime) elements in L such that a∈M(L) iff a b ∨ c implies a b or a c. Let P (L) be the set of all non-unit prime elements in L such that a∈P (L) iff a b ∧ d implies a b or a c. Finally, let X be a non-empty usual set, and LX be the set of all L-fuzzy sets on X. For each a∈L, let a denote a constant-valued L-fuzzy set with a as its value. Let ⊥ and be the smallest element and the greatest element in LX , respectively. For the empty set ∅ ⊂ L, we define ∧∅ = and ∨∅ = ⊥. Definition 2.1 (Wang [31]). Suppose that a∈L and A ⊆ L. (1) A is called a maximal family of a if (a) inf A = a, (b) ∀B ⊆ L, infB a implies that ∀x∈A there exists y∈B such that y x. (2) A is called a minimal family of a if (a) sup A = a, (b) ∀B ⊆ L, sup B a implies that ∀x∈A there exists y∈B such that y x. Remark 2.2. Hutton [16] proved that if L is a completely distributive lattice and a∈L, then there exists B ⊆ L such that (i) a = B, and (ii) if A ⊆ L and a = A, then for each b∈B there is a c∈A such that b c. However, if ∀a∈L, and if there exists B ⊆ L satisfying (i) and (ii), then in general L is not a completely distributive lattice. To this end, Wang [31] introduced the following modification of condition (ii), (ii ) if A ⊆ L and a A, then for each b∈B there is a c∈A such that b c. Wang proved that a complete lattice L is completely distributive if and only if for each element a in L, there exists B ⊆ L satisfying (i) and (ii ). Such a set B is called a minimal set of a by Wang [31]. The
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concept of maximal family is the dual concept of minimal family, and a complete lattice L is completely distributive if and only if for each element a in L, there exists a maximal family B ⊆ L. Let (a) denote the union of all maximal families of a. Likewise, let (a) denote the union of all minimal sets of a. Finally, let ∗ (a) = (a) ∩ P (L) and ∗ (a) = (a) ∩ M(L). One can easily see that both (a) and ∗ (a) are maximal sets of a. Likewise, both (a) and ∗ (a) are minimal sets of a. Also, we have () = ∅ and (⊥) = ∅. Lemma 2.3 (Shi [24,25,26] and Wang [31]). For a∈L and a map T : LX → L, we define T[a] = {A∈LX | T(A) a}
and
T[a] = {A∈LX |a ∈ / (T(A))}.
Let T be a map from LX to L and a, b∈L. Then (1) a∈(b) ⇒ T[b] ⊆ T[a] ; a∈(b) ⇒ T[a] ⊆ T[b] . ∗ ∗ (2) a ⊆ (b) ⇔ ∗ (a) ⊆∗ (b) ⇔ (b) b ⇔ (a) ⊆ (a) ⇔ (b) ⊆ (a). (3) ( i∈ ai ) = i∈ (ai ) and ( i∈ ai ) = i∈ (ai ) for any sub-family {ai }i∈ ⊆ L. 3. Level decompositions of an L-fuzzy topology Definition 3.1 (Hutton [16]). An L-fuzzy topology on X is a map T : LX → L satisfying the following three axioms: (1) T() = ; X (2) T(A ∧ T(B) for every A, B∈L ; ∧ B) T(A) (3) T( i∈ Ai ) i∈ T(Ai ) for every family {Ai | i∈} ⊆ LX . The pair (X, T) is called an L-fuzzy topological space (L-fts). For every A∈LX , T(A) is called the degree of openness of the fuzzy subset A. Just as an L-topology on X is an ordinary subset of LX , an L-fuzzy topology on X is a fuzzy subset of LX . In many papers a level decomposition of an L-fuzzy topology was considered: T is an L-fuzzy topology on X if and only if ∀a∈L, T[a] is an L-topology on X.
Excepting this, we have the followings level decompositions of an L-fuzzy topology: Theorem 3.2. Let T be a map T : LX → L. Then the following conditions are equivalent: (1) T is an L-fuzzy topology on X. (2) ∀a∈M(L), T[a] is an L-topology on X. (3) ∀a∈L, T[a] is an L-topology on X. (4) ∀a∈P (L), T[a] is an L-topology on X. Proof. (1) ⇒ (2): This part is obvious. (2) ⇒ (1): For each a∈M(L), we have ∈T[a] and T() a. Accordingly, {a | a∈M(L)} = . T()
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Thus, T() = . Next, let A, B∈LX . Clearly, when T(A) ∧ T(B) = ⊥, we have T(A ∧ B) T(A) ∧T(B). Otherwise, if T(A)∧T(B) > ⊥, then for each a T(A)∧T(B), we have T(A) a, T(B) a, or A, B∈T[a] . Consequently, we have A ∧ B∈T[a] , or T(A ∧ B) a. This further implies that {a∈M(L) | a T(A) ∧ T(B)} = T(A) ∧ T(B). T(A ∧ B) Let {Ai | i∈} ⊆ LX . Then for a∈M(L) and a T(Ai ), we have T(Ai ) a and Ai ∈T[a] for i∈ each i∈. The proof follows because i∈ Ai ∈T[a] , T( i∈ Ai ) a and
T Ai T(Ai ) = T(Ai ). a∈M(L) a i∈
i∈
i∈
(1) ⇒ (3): ∀a∈L, since T(⊥) = T() = and () = ∅, we have a ∈ / () = (T(⊥)) = (T()). Thus ⊥, ∈T[a] . Next, let A, B∈T[a] . Then a ∈ / (T(A)) and a ∈ / (T(B)). Hence a∈ / (T(A)) ∪ (T(B)) = (T(A) ∧ T(B)) ⊇ (T(A ∧ B)). Furthermore, since a ∈ / (T(A ∧ B)), we have A ∧ B∈T[a] . Finally, let Ai ∈T[a] . Then a ∈ / (T(Ai )) for each i∈ and a∈ / . (T(Ai )) = T(Ai ) ⊇ T Ai i∈
i∈
i∈
Consequently, we have a ∈ / (T( i∈ Ai )), or i∈ Ai ∈T[a] . (3) ⇒ (4): This part is obvious. ⇒ (1): ∀a∈P (L), it is clear that ∈T[a] . Thus a ∈ / (T()), ∗ (T()) = ∅, and T() = (4) ∗ X (T()) = . Next, let A, B∈L . Then clearly, when T(A) ∧ T(B) = ⊥, we have T(A ∧ B) T(A) ∧ T(B). If T(A) ∧ T(B) > ⊥, then for each a∈P (L) and a ∈ / (T(A) ∧ T(B)) = (T(A)) ∪ (T(B)), we have a ∈ / (T(A)), (T(B)) and a ∈ / (T(A ∧ B)). Accordingly, we have ∗ (T(A) ∧ T(B)) ⊇ ∗ (T(A ∧ B)),
or
T(A ∧ B) T(A) ∧ T(B).
Finally, suppose that {Ai | i∈} ⊆ LX . Then obviously T Ai T(Ai ), if T(Ai ) = ⊥. i∈
i∈
i∈
> ⊥. Then ∀a∈P (L) and a ∈ / ( i∈ T(Ai )) = (T(Ai )). It i∈ follows that a ∈ / (T(Ai )), Ai ∈T[a] for each i∈ and i∈ Ai ∈T[a] . Hence ⊆ ∗ ∗ T Ai T(Ai ) , or T Ai T(Ai ). Suppose now that
i∈
i∈ T(Ai )
i∈
i∈
i∈
We can now state the following decomposition theorem of L-fuzzy topology. The proof is straightforward and therefore omitted.
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Theorem 3.3. Let T be an L-fuzzy topology on X. Then [a] [a] T= (a ∧ T[a] ) = (a ∧ T[a] ) = (a ∨ T ) = (a ∨ T ). a∈L
a∈M(L)
a∈L
a∈P (L)
Corollary 3.4. Let T1 and T2 be L-fuzzy topologies on X, then the following conditions are equivalent: (1) T1 = T2 . (2) ∀a∈L, T1[a] = T2[a] . (3) ∀a∈M(L), T1[a] = T2[a] . (4) ∀a∈L, T1[a] = T2[a] . (5) ∀a∈P (L), T1[a] = T2[a] . Theorem 3.5. Let T be an L-fuzzy topology on X, then (1) ∀a∈L, T[a] = b∈(a) T[b] . (2) ∀a∈M(L), T[a] = b∈∗ (a) T[b] . (3) ∀a∈L, T[a] = a∈(b) T[b] . (4) ∀a∈P (L), T[a] = a∈∗ (b) T[b] . b∈P (L)
Proof. (1) By Lemma 2.3, we have that ∀a∈L, T[a] ⊆ b∈(a) T[b] . To show that T[a] ⊇ b∈(a) T[b] , we take A∈LX and A∈ b∈(a) T[b] . Notice that ∀b∈(a), T(A) b. Hence T(A) {b | b∈(a)} = a, which implies that A∈T[a] . (2) The proof is similar to (1). (3) By Lemma 2.3, we have that ∀a∈L, T[a] ⊆ a∈(b) T[b] . To show that T[a] ⊇ a∈(b) T[b] , we take A∈LX and A∈ a∈(b) T[b] . Notice that ∀b∈L and a∈(b), it follows that b∈ / (T(A)). We prove by contradiction as follows. Suppose that a∈ ( T (A)). Notice that T (A)=∧{b | b∈(T(A))} and (T(A)) = {(b) | b∈(T(A))}. There must exist b∈(T(A)) such that a∈(b). But this is impossible. (4) The proof is similar to (3). Remark 3.6.(1) b∈(a) implies b>a, where > is way-below relation [4], i.e. b>a if and only if for every up-directed set S in L, ∨S a implies that ∃s∈S such that s b; (2) If a∈M(L), then b∈ ∗ (a) if and only if b>a. (3) ∀a∈M(L), T[a] = b∈∗ (a) T[b] ⇔ T[a] = T[b] . b >a b∈M(L)
Proof. (1) Since (a) is a minimal set of a, from Definition 2.1, we have that for every up-directed set S in L, if ∨S a, then ∀b∈(a) there exists s∈S such that s b. It follows that b>a. (2) Let a∈M(L) and b>a. From Theorems 1.3.6 and 1.3.8 in [19] and Definition 2.1, we know that ∗ (a) is both an up-directed set and a lower set, and ∨∗ (a) = a. Hence, ∃b ∈∗ (a) such that a b b. In other words, b∈∗ (a). Conversely, if b∈∗ (a), then since ∗ (a) ⊆ (a), b∈∗ (a) implies b∈(a). It follows that b>a. (3) It is obvious. Theorem 3.7. Let {Ta | a∈M(L)} be a family of L-topologies on X. Then the following conditions are equivalent:
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(1) There exists an L-fuzzy topology T on X such that T[a] = Ta for each a∈M(L). (2) ∀a∈M(L), Ta = b∈∗ (a) Tb . Proof. (1) ⇒ (2): This holds because of Theorem 3.5. (2) ⇒ (1): Let T = a∈M(L) (a ∧ Ta ). Obviously, we have Ta ⊆ T[a] . For any A∈T[a] , we have T(A) a and {b∈M(L) | A∈Tb } a. Next, since ∗ (a) is a minimal family of a, for each b∈∗ (a), there exists b ∈M(L) such that b b and A∈Tb ⊆ Tb . Therefore, A∈ b∈∗ (a) Tb = Ta . Similarly, we can state the following theorems. Theorem 3.8. Let {Ta | a∈P (L)} be a family of L-topologies on X. Then the following conditions are equivalent: [a] (1) There exists an L-fuzzy topology T on X such that T = Ta for each a∈P (L). (2) ∀a∈P (L), Ta = a∈∗ (b) Tb . Theorem 3.9. Let {Ta | a∈L} be a family of L-topologies on X. Then the following conditions are equivalent: (1) There exists an L-fuzzy topology T on X such that T[a] = Ta for each a∈L. (2) ∀a∈L, Ta = b∈(a) Tb . Theorem 3.10. Let {Ta | a∈L} be a family of L-topologies on X. Then the following conditions are equivalent: (1) There exists an L-fuzzy topology T on X such that T[a] = Ta for each a∈L. (2) ∀a∈L, Ta = a∈(b) Tb . 4. Representation theorems of L-fuzzy topology Let LT[X] denote the family of all L-topologies on X. Let LFT[X] denote the family of all L-fuzzy topologies on X. The order relation on LFT[X] is defined as follow: ∀T1 , T2 ∈LFT[X],
T1 4T2 ⇔ ∀A∈LX ,
T1 (A) T2 (A).
Theorem 4.1 (Höhle [15]). (LFT[X], 4) is a complete lattice. In fact, it is a complete sub-meet-semilattice X X of L(L ) , i.e. closed under the ∧ of L(L ) . To facilitate further illustration, let us define the following classes: F (b) , UL [X] = F : L → LT[X] | ∀a∈L, F (a) = a∈(b) UL [X] = F : L → LT[X] | ∀a∈L, F (a) = F (b) , b∈(a)
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F : M(L) → LT[X] | ∀a∈M(L), F (a) = F (b) , b∈∗ (a) F (b) . UP (L) [X] = F : P (L) → LT[X] | ∀a∈P (L), F (a) = ∗ UM(L) [X] =
a∈ (b)
In addition, let us define the following order relations within the classes UL [X], UL [X], UM(L) [X] and UP (L) [X]: L
F1 , F2 ∈UL [X], F1 4 F2 ⇔ ∀a∈L, F1 (a) ⊆ F2 (a), F1 , F2 ∈UL [X], F1 4L F2 ⇔ ∀a∈L, F1 (a) ⊆ F2 (a), F1 , F2 ∈UM(L) [X], F1 4M(L) F2 ⇔ ∀a∈M(L), F1 (a) ⊆ F2 (a), F1 , F2 ∈UP (L) [X], F1 4P (L) F2 ⇔ ∀a∈P (L), F1 (a) ⊆ F2 (a). L Theorem 4.2. (UL [X], 4 ), UL [X], 4L , (UM(L) [X], 4M(L) ) and (UP (L) [X], 4P (L) ) are complete lat L tices. Obviously, UL [X], 4 and UL [X], 4L are complete sub-meet-semilattices of the lattice 4L (LT[X])L (i.e., closed under the ∧ of (LT[X])L , when {Fi | i∈} ⊆ UL [X], F = i∈ Fi be defined 4L as ∀a∈L, F (a) = F (a); when {F | i∈ } ⊆ UL [X], F = i∈ Fi be defined as ∀a∈L, F (a) = i i i∈ F (a)), UM(L) [X], 4M(L) is a complete sub-meet-semilattices of the lattice (LT[X])M(L) , and i∈ i UP (L) [X], 4P (L) is a complete sub-meet-semilattices of the lattice (LT[X])P (L) . Proof. ∀a∈L, let us define F⊥ (a) = {∅, X} and F (a) = LX . Clearly, we have F⊥ , F ∈UL [X], and they L are the smallest element and the greatest element in (UL [X], 4 ), respectively. Next, let {Fi | i∈} ⊆ L 4 UL [X] and F = i∈ Fi . Since F (a) = Fi (a) = Fi (b) = Fi (b) = F (b), i∈
i∈ a∈(b)
a∈(b) i∈
a∈(b)
L
it follows that F ∈UL [X]. Accordingly, (UL [X], 4 ) is a complete lattice. The same argument can be used to prove the rest of the theorem. The following representation theorem of L-fuzzy topology follows naturally. Theorem 4.3. The map f : LFT[X] → UL [X], T −→ FT ( for every a∈L and FT (a) = T[a] ) is an isomorphism in the category of complete meet-semilattices and f −1 : UL [X] → LFT[X], F −→ TF = a∈L (a ∨ F (a)). Proof. For each T∈LFT[X], it is easy to verify that FT (a) = T[a] = T[b] = FT (b). a∈(b)
a∈(b)
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Hence, FT ∈UL [X]. Next, by Theorems 3.2, 3.3 and Corollary 3.4, it suffices to show that f is an injection. Since x ∈( / TF )[] iff ∈(TF (x)) = ((a ∨ F (a))(x)) = {(a) | a∈L, x ∈F / (a)} a∈L
/ (a) iff x ∈ / ∈(a) F (a) = F (), we have FTF () = iff there exists a∈L such that ∈(a) and x ∈F (TF )[] = F (). This shows that FTF = F . It follows that f is a surjection as well as a bijection, and (a ∨ F (a)). f −1 : UL [X] → LFT[X], F −→ TF = a∈L
Next, let T1 , T2 ∈LFT[X], {Ti | i∈} ⊆ LFT[X]. Then it is straightforward to show that f (T1 )4 4L f (T2 ) when T1 4T2 . Hence f ( i∈ Ti ) = i∈ f (Ti ) and the proof is complete.
L
The following Theorem follows directly from the above proof. Theorem 4.4. The map f : LFT[X] → UP (L) [X], T −→ FT ( for every a∈P (L), FT (a) = T[a] ) −1 : U is an isomorphism P (L) [X] → LFT[X], in the category of complete meet-semilattices and f F −→ TF = a∈P (L) (a ∨ F (a)). Theorem 4.5. The map f : LFT[X] → UL [X], T −→ FT ( for every a∈L, FT (a) = T[a] ) is an isomorphism in the category of complete meet-semilattices and f −1 : UL [X] → LFT[X], F −→ TF = a∈L (a ∨ F (a)). Proof. For each T∈LFT[X], it is easy to verify that FT (a) = T[a] = T[b] = FT (b). b∈(a)
b∈(a)
Hence FT ∈UL [X]. Next, by Theorem 3.3 and Corollary 3.4, it suffices to show that f is an injection. It is proved easily that x∈(TF )[] iff TF (x) = (a ∧ F (a))(x) = {a | x∈F (a)} a∈L
iff (because of Lemma 2.3) (a) = (∨{a | x∈F (a)}) ⊇ (). x∈F (a)
On the other hand, we can prove x∈F () = F (a) iff ∀a∈(a), x∈F (a) iff a∈()
x∈F (a)
(a) ⊇ ().
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(In fact, obviously ∀a∈(), x∈F (a) implies (a) ⊇ (a) = (). Conversely, for x∈F (a) a∈() each c∈() ⊆ x∈F (a) (a), then there exists a∈L such that c∈(a) and x∈F (a) = b∈(a) F (b). It show that x∈F (c)). So, we conclude that x∈(TF )[] iff x∈F (), i.e., FTF () = (TF )[] = F (). This shows that FTF = F . It follows that f is a surjection as well as a bijection, and (a ∧ F (a)). f −1 : UL [X] → LFT[X], F −→ TF = a∈L
Next, let T1 , T2 ∈LFT[X], {Ti | i∈} ⊆ LFT[X]. Then it is straightforward to show that f (T1 )4L 4L f (T2 ) when T1 4T2 . Hence f ( i∈ Ti ) = i∈ f (Ti ) and the proof is complete. The following Theorem follows directly from the above proof. Theorem 4.6. The map f : LFT[X] → UM(L) [X], T −→ FT ( for every a∈M(L), FT (a) = T[a] ) −1 : U is an isomorphism M(L) [X] → LFT[X], in the category of complete meet-semilattices and f F −→ TF = a∈M(L) (a ∧ F (a)). 5. L-fuzzy continuous maps Definition 5.1 (Höhle [15]). Let (X, T1 ) and (Y, T2 ) be two L-topological spaces. Let f : X → Y be a map. f : (X, T1 ) → (Y, T2 ) is called L-fuzzy continuous if for every A∈LY we have T1 (f ← (A)) T2 (A), where f ← (A) = A ◦ f. From Definition 5.1, obviously, f : (X, T1 ) → (Y, T2 ) is an L-fuzzy continuous map if and only if ∀a∈M(L), f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. Excepting this, we have the followings equivalent conditions: Theorem 5.2. Let (X, T1 ) and (Y, T2 ) be L-fuzzy topological spaces and f : X → Y be a map. Then the following conditions are equivalent: (1) f : (X, T1 ) → (Y, T2 ) is an L-fuzzy continuous map. (2) ∀a∈M(L), f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. (3) ∀a∈L, f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. (4) ∀a∈P (L), f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map. Proof. (1) ⇒ (2): This part is obvious. (2) ⇒ (1): ∀A∈LY , a∈M(L) such that a T2 (A), we have A∈T2[a] and f ← (A)∈T1 [a] by the continuity of f : (X, T1[a] ) → (Y, T2[a] ). Accordingly, T1 (f ← (A)) a for each a∈M(L) M(T2 (A)), ← where M(T2 (A)) = {a∈M(L) | a T2 (A)}. It follows that T1 (f (A)) M(T2 (A)) = T2 (A). (1) ⇒ (3): ∀A∈LY , since T2 (A) T1 (f ← (A)), it follows from Lemma 2.3 that a ∈ / (T1 (f ← (A))) [a] [a] ← when ∀a∈L, if a ∈ / (T2 (A)). In other words, if A∈T1 , then f (A)∈T1 . Thus f : (X, T1[a] ) → (Y, T2[a] ) is an L-continuous map.
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(3) ⇒ (4): This is obvious. (4) ⇒ (1): For ∀a∈P (L) and A∈LY , if a ∈ / (T2 (A)), then A∈T2[a] . Thus f ← (A)∈T1[a] by the continu/ (T1 (f ← (A))) and ∗ (T2 (A)) ity of f : (X, T1[a] ) → (Y, T2[a] ). In other words, a ∈ ∗ ← ⊇ (T1 (f (A))). It follows from Lemma 2.3 that T2 (A) T1 (f ← (A)). Hence the proof is completed. Definition 5.3. Let (X, T1 ) and (Y, T2 ) be L-fuzzy topological spaces. A map f : (X, T1 ) → (Y, T2 ) is called an L-fuzzy open map if for each A∈LX we have T1 (A) T2 (f → (A)). Theorem 5.4. Let (X, T1 ) and (Y, T2 ) be L-fuzzy topological spaces, and f : X → Y be a map. Then the following conditions are equivalent: (1) f : (X, T1 ) → (Y, T2 ) is an L-fuzzy open map. (2) f : (X, T1[a] ) → (Y, T2[a] ) is an L-fuzzy open map for each a∈M(L). (3) f : (X, T1[a] ) → (Y, T2[a] ) is an L-open map for each a∈L. (4) f : (X, T1[a] ) → (Y, T2[a] ) is an L-open map for each a∈P (L). Proof. (1) ⇒ (2): This is obvious by Definition 5.3. X (2) ⇒ (1): For a given clearly, T1 (A) T2 (f → (A)). If T1 (A) > ⊥, A∈L , if T1 (A) = ⊥, then → then since T1 (A) = M(T1 (A)), we have T2 (f (A)) a for each a∈M(T1 (A)). Hence T2 (f → (A)) {a | a∈M(T1 (A))} = T1 (A). (1) ⇒ (3): ∀A∈LX . From part (1) of the theorem and Lemma 2.3, we have (T1 (A)) ⊇ (T2 (f → (A))). It follows that a ∈ / (T2 (f → (A))) if a ∈ / (T1 (A)) for each a∈L. In other words, f → (A)∈ [a] [a] T2 if A∈ T1 for each a∈L. Hence statement (3) holds. (3) ⇒ (4): This part is obvious. (4) ⇒ (1): ∀a∈P (L) and A∈LX , from part (4) of the theorem, if a ∈ / (T1 (A)), then a ∈ / (T2 → ∗ ∗ → → (f (A)). Thus, (T1 (A)) ⊇ (T2 (f (A))). We have from Lemma 2.3 that T1 (A) T2 (f (A)). Hence the proof is completed. Definition 5.5. Let (X, T1 ) and (Y, T2 ) be two L-fuzzy topological spaces. Let f : X → Y be a map. f : (X, T1 ) → (Y, T2 ) is called an L-fuzzy homeomorphism if f is bijective and f and f ← are L-fuzzy continuous maps. Theorem 5.6. Let (X, T1 ) and (X, T2 ) be two L-fuzzy topological spaces, and f : X → Y be a bijective map. The following conditions are equivalent: (1) f : (X, T1 ) → (Y, T2 ) is an L-fuzzy homeomorphism. (2) f : (X, T1[a] ) → (Y, T2[a] ) is a homeomorphism for each a∈M(L). (3) f : (X, T1[a] ) → (Y, T2[a] ) is a homeomorphism for each a∈L. (4) f : (X, T1[a] ) → (Y, T2[a] ) is a homeomorphism for each a∈P (L). Proof. It follows from Definitions 5.1, 5.3, 5.5 and Theorems 5.2 and 5.4.
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Definition 5.7 (He [11]). Let X and Y be two non-empty sets. Let L and L be two completely distributive lattices. Let g : L → L and f : X → Y be two maps. The right adjoint map of g, denoted by g ∗ : L → L, is defined as g ∗ (b) = {a∈L | g(a) b}, for b∈L. The map H = g f : LX → LY is defined as H (A)(y) = {g(A(x)) | f (x) = y, x∈X},
for A∈LX and y∈Y.
The map H is called a bi-induced map of g and f. The inverse of H, denoted by H ← : LY → LX , is defined as H ← (B) = g ∗ ◦ B ◦ f , where B∈LY . Definition 5.8 (He [11]). Let H = g f : (LX , T) → (LY , R) be a bi-induced map. H is called fuzzy continuous, if for each B∈LY , T(H ← (B)) g ∗ (R(B)). Theorem 5.9. Let H = g f : (LX , T) → (LY , R) be a bi-induced map. Let g be a join preserving order homomorphism. Then the following conditions are equivalent: (1) H is a fuzzy continuous map. (2) For each a∈L, H : (LX , T[g ∗ (a)] ) → (LY , R[a] ) is a continuous map. (3) For each a∈M(L), H : (LX , T[g ∗ (a)] ) → (LY , R[a] ) is a continuous map. Proof. (1) ⇒ (2): Since R(B) a, ∀B∈R[a] , we have T(H ← (B)) g ∗ (R(B)) g ∗ (a). This implies that H ← (B)∈T[g ∗ (a)] . (2) ⇒ (3): This is obvious. (3) ⇒ (1): ∀A∈LY , if R(B) = ⊥, then T(H ← (B)) g ∗ (R(B)). Likewise, if R(B) > ⊥, then foreach a∈M(R(B)), we have R (B) a. Hence T(H ← (B)) g ∗ (a). This implies that T(H ← (B)) ∗ ∗ {g (a) | a∈M(R(B))} = g ( {a | a∈M(R(B))}) = g ∗ (R(B)). Definition 5.10 (He [11]). Let H = g f : (LX , T) → (LY , R) be a bi-induced map. H is called a fuzzy open map if for each A∈LX , R(H (A)) g(T(A)). Theorem 5.11. Let H = g f : (LX , T) → (LY , R) be a bi-induced map. Let g be a join preserving order homomorphism. Then the following conditions are equivalent: (1) H is a fuzzy open map. (2) For each a∈L, H : (LX , T[a] ) → (LY , R[g(a)] ) is an open map. (3) For each a∈M(L), H : (LX , T[a] ) → (LY , R[g(a)] ) is an open map. Proof. (1) ⇒ (2): ∀a∈L and A∈LX , if A∈T[a] , then T(A) a. Hence, R(H (A)) g(T(A)) g(a). It follows that H (A)∈R[g(a)] . (2) ⇒ (3): This is obvious. (3) ⇒ (1): Clearly, ∀A∈LX , if T(A) = ⊥, then R(H (A)) g(T(A)) = g(⊥) = ⊥. Likewise, if T(A) > ⊥, then by (3), for each a∈M(T(A)), we have R(H (A)) g(a). Hence, R(H (A)) {g(a) | a∈M(T(A))} = g( {a | a∈M(T(A))}) = g(T(A)).
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