L-FUZZY UNIFORM SPACES Reprinted from the Journal of the

J. Korean Math. Soc. 44 (2007), No. 6, pp. 1383–1396

L-FUZZY UNIFORM SPACES Yue-li Yue and Fu-gui Shi

Reprinted from the Journal of the Korean Mathematical Society Vol. 44, No. 6, November 2007 c °2007 The Korean Mathematical Society

J. Korean Math. Soc. 44 (2007), No. 6, pp. 1383–1396

L-FUZZY UNIFORM SPACES Yue-li Yue and Fu-gui Shi Abstract. The aim of this paper is to study L-fuzzy uniformizable spaces. A new kind of topological fuzzy remote neighborhood system is defined and used for investigating the relationship between L-fuzzy co-topology and L-fuzzy (quasi-)uniformity. It is showed that this fuzzy remote neighborhood system is different from that in [23] when U is an Lfuzzy quasi-uniformity and they will be coincident when U is an L-fuzzy uniformity. It is also showed that each L-fuzzy co-topological space is L-fuzzy quasi-uniformizable.

1. Introduction It is well-known that uniformity is a very important concept close to topology and a convenient tool for investigating topology. Fuzzy versions of (quasi-) uniformity theory were established by B. Hutton [8], R. Lowen [13], U. H¨ohle [5] and Shi [18–19], etc. Fuzzy (quasi-)uniformity in Hutton’s sense has been accepted by many authors and has attracted wide attention in the literature. Up till now there are many spectacular and creative works about the theory of Hutton uniformities (See [2, 4, 8, 10, 14, 17, 21, 23–25]) and Zhang [25] gave a comparison of various uniformities in fuzzy topology. Extension of Hutton’s quasi-uniformities—I-fuzzy uniformity—was considered in [2]. Later, in [21] fuzzy uniformities for lattices more general than I, namely, so called (L, K)-fuzzy uniformities were considered. Finally, in [4], the paper specially devoted to the analysis of different approaches to the theory of fuzzy uniformities in the context of fuzzy sets, an essentially more general concept of an L-valued uniformity was studied by using a filter approach. Further, in [17], there is a significant extension of Hutton for quasi-uniformities without using filters explicitly and without any distributivity and with general tensor products generating the intersection axiom. In [23], fuzzy remote neighborhood system was used for studying L-fuzzy quasi-uniformity. The aim of this paper is to define another topological fuzzy remote neighborhood system from a given L-fuzzy quasi-uniformity and use it to study the Received March 28, 2006. 2000 Mathematics Subject Classification. 54E15, 54A40. Key words and phrases. L-fuzzy co-topology, fuzzy remote neighborhood system, L-fuzzy (quasi-)uniformity. c °2007 The Korean Mathematical Society

1383

1384

YUE-LI YUE AND FU-GUI SHI

relationship between L-fuzzy co-topology and L-fuzzy quasi-uniformity. We show that this fuzzy remote neighborhood system is different from that in [23] when U is an L-fuzzy quasi-uniformity. However, when U is an L-fuzzy uniformity, they will be coincident. We prove that each L-fuzzy co-topological space is L-fuzzy quasi-uniformizable. In this article, all lattices are assumed to be complete. 2. Preliminaries Let a, b be elements in L. An element a ∈ L is said to be coprime if a ≤ b ∨ c implies that a ≤ b or a ≤ c. The set of all coprimes of L is denoted by c(L). We say a is way below (wedge below) b, in symbols, a W ¿ b (a C b) or b À a (b B a), if for every directed (arbitrary) subset D ⊆ L, D ≥ b implies a ≤ d for some d ∈ D. Clearly if a ∈ L is a coprime, then a ¿ b if and only if a C b. A complete lattice L is said to be continuous (completely distributive) if every element in L is the supremum of all the elements way below (wedge below) it. Proposition 2.1 ([3]). Let L be a complete lattice. The following conditions are equivalent: (1) L is completely distributive; (2) L is distributive W continuous lattice with enough coprimes; (3) The operator : Low(L) → L sending every lower set to its supremum has a left adjoint β, and in this case β(a) = {b| b C a}. From (3) in the above proposition it is easy to see that the wedge below relation has the interpolation property in a completely distributive lattice, this is to say, a C b implies there is some c ∈ L such that a C c C b. Throughout this paper, L and M are two completely distributive lattices and there is an order reversing involution 0 on L. LX is the set of all L-fuzzy sets on X. A0 ∈ LX defined by A0 (x) = (A(x))0 . The set of all coprimes of LX is denoted by c(LX ). Let e|A denote the set {B ∈ LX |e 6≤ B ≥ A} for e ∈ c(LX ) and A ∈ LX . Let F : X → Y be an ordinary mapping, define L-fuzzy mapping FL→ : LXW → LY and its L-fuzzy reverse mapping FL← : LY → LX by FL→ (A)(y) = {A(x)| x ∈ X, f (x) = y} for A ∈ LX and y ∈ Y , and FL← (B)(x) = B(f (x)) for B ∈ LY and x ∈ X (following the notation in [15]), respectively. Definition 2.2 ([6, 7, 11, 16, 20]). An L-fuzzy co-topology is a mapping η : LX → M such that (FCT1) η(1X ) = η(0X ) = 1; X (FCT2) η(AV∨ B) ≥ η(A) V ∧ η(B) for all A, B ∈ L ; (FCT3) η( Aj ) ≥ η(Aj ) for every family {Aj |j ∈ J} ⊆ LX . j∈J

j∈J

The pair (LX , η) is called an L-fuzzy co-topological space. A mapping F : (LX , η) → (LY , η1 ) is said to be fuzzy continuous with respect to η and η1 if η(FL← (B)) ≥ η1 (B) for all B ∈ LY . Let L-FCTOP denote the category of L-fuzzy co-topological spaces and fuzzy continuous mappings.

L-FUZZY UNIFORM SPACES

1385

If η is an L-fuzzy co-topology on X, then τ is an L-fuzzy topology on X, where τ : LX → M is defined by τ (A) = η(A0 ). The converse is also true. For convenience, in this paper, we use L-fuzzy co-topology. For undefined notions about category, please refer to [1]. Definition 2.3 ([22]). A topological fuzzy remote neighborhood system is a set R = {Re |e ∈ c(LX )} of mappings Re : LX → M such that (FRN1) Re (1X ) = 0, Re (0X ) = 1; (FRN2) Re (A) > 0 ⇒ e 6≤ A; (FRN3) Re (A ∨ B) W = Re (A) V ∧ Re (B); (FRN4) Re (A) = B∈e|A a6≤B Ra (B). Lemma 2.4 ([22]). Let η : LX → M be an L-fuzzy co-topology. Then we have (1) Rη = {Reη |e ∈ c(LX )} is a topological fuzzy remote neighborhood system, where Reη is defined by ½ W B∈e|A η(B), e 6≤ A, Reη (A) = 0, e≤A for e ∈ c(LX ) and A ∈ LX . (2) If η and ζ are two L-fuzzy co-topologies which determine the same topological fuzzy remote neighborhood system, then η = ζ. Lemma 2.5 ([22]). Let R = {Re |e ∈ c(LX )} be a topological V fuzzy remote neighborhood system and η : LX → M be defined by η(A) = e6≤A Re (A) for all A ∈ LX . Then η is an L-fuzzy co-topology. Furthermore, if R and ∫ are two topological fuzzy remote neighborhood systems which determine the same L-fuzzy co-topology, then R = ∫ . Lemma 2.6 ([23]). Let R = {Re |e ∈ c(LX )} be a set satisfying (FRN1)– (FRN3). Then theWfollowing V two statements are equivalent (FRN4) Re (A) = B∈e|A a6≤B Ra (B); V W (FRN4∗ ) Re (A) = B∈e|A (Re (B) ∧ a6≤B Ra (A)). 3. L-fuzzy uniform spaces In this section, a new kind of topological fuzzy remote neighborhood system is defined by a given L-fuzzy quasi-uniformity. We will use this kind of remote neighborhood system to study the relationship between L-fuzzy quasiuniformity and L-fuzzy co-topology. Let H(LX ) denote the family of all mappings f : LX → LX such that: X (1) A W ≤ f (A) for all WA∈L ; (2) f ( j∈J Aj ) = j∈J f (Aj ) for {Aj }j∈J ⊆ LX . f1 denotes the biggest element of H(LX ), i.e., f1 (A) = 0X when A = 0X and f1 (A) = 1X otherwise. For f, g ∈ H(LX ), we have that f ∧ g ∈ H(LX ) and

1386


f ◦ g ∈ H(LX ), where f ∧ g(A) =

^

f (B) ∨ g(C) and f ◦ g(A) = f (g(A)).

B∨C=A

For each f ∈ H(LX ), let f C (B) = following proposition.

V

{C ∈ LX |f (C 0 ) ≤ B 0 ), then we have the

Proposition 3.1 ([8, 12]). (1) f C ∈ H(LX ); (2) f ≤ g implies f C ≤ g C ; (3) (f C )C = f ; (4) (f ◦ g)C = g C ◦ f C ; (5) (fW∧ g)C = f CW∧ g C ; (6) ( t∈T ft )C = t∈T ftC . Suppose F : X → Y is a mapping, f ∈ H(LY ), define F ⇐ (f ) : LX → LX by F ⇐ (f )(A) = FL← ◦ f ◦ F → (A) for all A ∈ LX , then we have Proposition 3.2 ([8, 12, 14, 24]). (1) F ⇐ (f ) ∈ H(LX ); (2) f ≤ g implies F ⇐ (f ) ≤ F ⇐ (g); (3) F ⇐ (f C ) = (F ⇐ (f ))C ; (4) F ⇐ (f ◦ g) ≤ F ⇐ (f ) ◦ F ⇐ (g). Definition 3.3 ([2, 4, 21]). An L-fuzzy quasi-uniformity is a mapping U : H(LX ) → M such that (FQU1) U(f1 ) = 1; (FQU2) U(f ∧ g)W= U(f ) ∧ U(g) for all f, g ∈ H(LX ); (FQU3) U(f ) = g◦g≤f U (g) for all f ∈ H(LX ). The pair (LX , U) is called an L-fuzzy quasi-uniform space. An L-fuzzy quasiuniformity is called an L-fuzzy uniformity if it also satisfies the following condition: (FQU4) U(f ) = U (f C ) for all f ∈ H(LX ); A mapping F : (LX , U) → (LY , U1 ) is called fuzzy (quasi-)uniformly continuous if U(F ⇐ (g)) ≥ U1 (g) for all g ∈ H(LY ). The category of L-fuzzy quasi-uniform spaces and continuous mappings is denoted by L-HuQUnif. (U )

Lemma 3.4 ([23]). Let (LX , U) be an L-fuzzy quasi-uniform space and Re : W (U ) LX → M be defined by Re (A) = e6≤f (A) U(f ) for all A ∈ LX . Then R(U ) = (U )

{Re |e ∈ c(LX )} is a topological fuzzy remote neighborhood system. In L-topology, given an L-quasi-uniformity U, we know that the mapping i : LX → LX defined by _ i(A) = {C ∈ LX | ∃f ∈ U, f (C) ≤ A} is an interior operator and c : LX → LX defined by ^ c(A) = {f (A)| f ∈ U}


1387

is a closure operator. Hence, i and c generate two L-topologies δi and δc , respectively. For L-quasi-uniform space, δi is not necessary coincident with δc . But δi = δc is valid when U is an L-uniformity. In Lemma 3.4, one kind of topological fuzzy remote neighborhood system is generated by L-fuzzy quasiuniformity. In fact, the idea of this topological remote neighborhood system is due to the definition of the closure operator above. In the following discussion, we will define another topological remote neighborhood system on account of the interior operator. We show that the two topological remote neighborhood systems may not be coincident when U is an L-fuzzy quasi-uniformity and they will be coincident when U is an L-fuzzy uniformity. Theorem 3.5. Let (LX , U) be an L-fuzzy quasi-uniform space and ReU : LX → M be defined by _ _ ∀A ∈ LX , ReU (A) = U(f ). e6≤C f (C 0 )≤A0

Then RU = {ReU |e ∈ c(LX )} is a topological fuzzy remote neighborhood system. Proof. We need to check (FRN1)–(FRN4). (FRN1), (FRN2) and (FRN3) are straightforward, what remains is to prove (FRN4). From Lemma W 2.6, we know that it is equivalent to check (FRN4∗ ). Since ReU (A) ≥ B∈e|A (ReU (B) ∧ W V U U U B∈e|A (Re (B) ∧ a6≤B Ra (A)) is obvious, it suffices to show that Re (A) ≤ V U U a6≤B Ra (A)). Let α C Re (A), i.e., _ _ _ _ _ U(f ) = U(g). α C ReU (A) = e6≤C f (C 0 )≤A0

e6≤C f (C 0 )≤A0 g◦g≤f

Then there exist C ∈ LX , f ∈ H(LX ) and g ∈ H(LX ) such that e 6≤ C ≥ (g(C 0 ))0 ≥ (g ◦ g(C 0 ))0 ≥ f (C 0 )0 ≥ A and α ≤ U(g). Let B = (g(C 0 ))0 . Then B ∈ e|A. Furthermore, we have _ _ _ ReU (B) = U(h) ≥ U(h) ≥ U (g) ≥ α e6≤D h(D 0 )≤B 0

and ^ a6≤B

RaU (A) =

^ _

_

a6≤B a6≤D h(D 0 )≤A0

Then α ≤ ReU (B) ∧

V

h(C 0 )≤B 0

U(h) ≥

^

_

U (h) ≥

a6≤B h(B 0 )≤A0

^

U (g) ≥ α.

a6≤B

RaU (A). Therefore, _ ^ RaU (A)). α≤ (ReU (B) ∧

a6≤B

B∈e|A

a6≤B

From the arbitrariness of α, we have ReU (A) ≤ Thus the conclusion holds.

W

U B∈e|A (Re (B) ∧

V a6≤B

RaU (A)). ¤

1388


Theorem 3.6. Let (LX , U ) be an L-fuzzy quasi-uniform space. Then ReU can also be written as W follows: W (1) ReU (A) = e6≤C f ◦f (C 0 )≤A0 U(f ); W W (2) ReU (A) = C∈LX e6≤f (C 0 )0 ≥(f ◦f (C 0 ))0 ≥A U (f ) ; W (3) ReU (A) = e6≤f C (A) U (f ). Proof. (1) and (2) are trivial. (3) can be obtained by the definition of f C . ¤ Remark 3.7. When U is an L-fuzzy quasi-uniformity, RU is not necessary coincident with R(U ) . The following example can show this (also see it in [12]). Example 3.8. Let X = [0, 1], L = M = {0, 1}, D = {(x, y) ∈ X × X| 0 ≤ x ≤ y ≤ 1} and define fD : 2X → 2X by fD (U ) = {y ∈ X| ∃x ∈ U, s.t., (x, y) ∈ D}. Now define U : H(2X ) → {0, 1} as follows: ½ 1, f ≥ fD , U(f ) = 0, others. Then it is easy to verify that U is an L-fuzzy quasi-uniformity (in fact, it is a crisp quasi-uniformity). Since fD ([0, 21 ]) = [0, 1], we have _ 1 Re(U ) ([0, ]) = U (f ) = 0 2 1 e6∈f ([0, 2 ])

for all e ∈ X. But 1 ReU ([0, ]) = 2

_ e6∈f C ([0, 12 ])= (U)

for all e 6∈ [0, 21 ]. Hence Re generated are not coincident.

T {C|f (C 0 )⊆( 21 ,1]}

U(f ) = 1

6= ReU . Therefore, the two topologies they

However, for L-fuzzy uniform spaces, the two topological remote neighborhood systems are the same, just as the following theorem shows. Theorem 3.9. Let U be an L-fuzzy uniformity, then RU = R(U ) . Proof. Since U is an L-fuzzy uniformity, we have U(f ) = U(f C ) for all f ∈ H(LX ). Hence, _ _ _ U(f ) = U(f C ) ≤ U(g). e6≤f C (A)

e6≤f C (A)

e6≤g(A)

(U )

This is to say ReU (A) ≤ Re (A). Conversely, by (f C )C = f , we have _ _ U(g) = U(g C ) ≤ e6≤(g C )C (A)

e6≤g(A) (U )

_

U (f ).

e6≤f C (A) (U)

This is to say ReU (A) ≥ Re (A). Therefore, ReU (A) = Re (A).

¤


1389

Let (LX , U) be an L-fuzzy quasi-uniform space and ηU : LX → M be defined by ηU (A) =

^

ReU (A) =

e6≤A

^

_

U(f )

e6≤A e6≤f C (A)

for all A ∈ LX . From Lemma 2.5, we know that ηU is an L-fuzzy co-topology on X and call it the generated L-fuzzy co-topology by U . Theorem 3.10. Let (LX , η) be an L-fuzzy co-topological space. Then there is one L-fuzzy quasi-uniformity Uη on X such that the generated L-fuzzy cotopology by Uη is just η, i.e., η = ηUη . This is to say that each L-fuzzy cotopological space is L-fuzzy quasi-uniformizable. Proof. Let U ∈ LX and fU : LX → LX be defined as follows:  A 6≤ U,  1X , U, 0X 6= A ≤ U, fU (A) =  0X , A = 0X . Then fU ∈ H(LX ) and fU ◦ fU = fU . Define Uη : H(LX ) → M by _ i=n Uη (f ) = {∧i=n i=1 η(Ui )| f ≥ ∧i=1 fUi0 , n ∈ N }. It is easy to verify that Uη is an L-fuzzy quasi-uniformity on X. Now we prove that η = ηUη . Noting that fAC0 (A) = A, from the definition of ηUη , we have ^ _ _ ^ i=n ηUη (A) = {∧i=n η(A) = η(A). i=1 η(Ui )|f ≥ ∧i=1 fUi0 , n ∈ N } ≥ e6≤A e6≤f C (A)

e6≤A

This is to say ηUη ≥ η. On the other hand, we have ^ _ _ i=n ηUη (A) = {∧i=n i=1 η(Ui )| f ≥ ∧i=1 fUi0 , n ∈ N } e6≤A e6≤f C (A)

≤

^

_

e6≤A e6≤f C (A)

≤ ≤

^

_

e6≤A

e6≤f C (A)

_

C i=n η(Ui )| f C ≥ ∧i=n {∧i=1 i=1 fU 0 , n ∈ N } i

_

C i=n {∧i=n i=1 η(Ui )| f (A) ≥ ∧i=1 fUi (A), n ∈ N }

^ _ j=m {η(∧j=1 Uj )| e 6≤ ∧j=m j=1 Uj ≥ A, m ∈ N }

e6≤A

≤

^ _ {η(B)| e 6≤ B ≥ A}

e6≤A

= η(A). This concludes the proof.

¤

1390


Example 3.11. We consider the L-fuzzy unit interval I(L) in L-topological spaces. Let X be the L-fuzzy unit interval I(L) and τ (I(L)) be the usual L-topology on X = I(L). For more detail about L-fuzzy unit interval, please refer to [11, 12 , 14]. Define η : LX → [0, 1] by ½ 1 A0 ∈ τ (I(L)), η(A) = 0 others. In fact, η is just the characteristic function of usual co-topology on I(L). From [12, 14], we know that I(L) is (quasi-)unformizable. Hence it is also L-fuzzy (quasi-)uniformizable. Theorem 3.12. If F : (LX , U ) → (LY , U1 ) is fuzzy quasi-uniformly continuous, then F : (LX , ηU ) → (LY , ηU1 ) is fuzzy continuous. Proof. Let B ∈ LY and α C ηU1 (B). Since F : (LX , U) → (LY , U1 ) is fuzzy quasi-uniformly continuous, we have U1 (f ) ≤ U (F ⇐ (f )) for all f ∈ H(LY ). Hence, _ ^ _ ^ U1 (f ) ≤ U(F ⇐ (f )). α C ηU1 (B) = e6≤B e6≤f C (B)

Noting that FL→ (a) such that FL→ (a) 6≤ Then h(a) ∈ H(LX ) α≤

^

← (B) a6≤FL

e6≤B e6≤f C (B)

FL← (B), ⇐

6 B when a 6≤ ≤ we can find some f(a) ∈ H(LY ) C f(a) (B) and α ≤ U(F (f(a) )). Now let h(a) = F ⇐ (f(a) ). ← and a 6≤ hC (a) (FL (B)). Hence, ^ _ U (h(a) ) ≤ U(h) = ηU (FL← (B)). ← (B) a6≤hC (F ← (B)) a6≤FL L

Therefore, ηU1 (B) ≤ ηU (FL← (B)) from the arbitrariness of α. So F : (LX , ηU ) → (LY , ηU1 ) is fuzzy continuous. ¤ Theorem 3.13. If F : (LX , η) → (LY , η1 ) is fuzzy continuous, then F : (LX , Uη ) → (LY , Uη1 ) is fuzzy quasi-uniformly continuous. Proof. Let F : (LX , η) → (LY , η1 ) be fuzzy continuous. From the definition of Uη1 , we know that _ i=n Uη1 (f ) = {∧i=n i=1 η1 (Ui )|f ≥ ∧i=1 fUi0 , n ∈ N }. i=n Moreover, if f ≥ ∧i=1 fUi0 , then we have i=n ⇐ i=n ← (U )0 F ⇐ (f ) ≥ F ⇐ (∧i=n i=1 fUi0 ) = ∧i=1 (F (fUi0 )) = ∧i=1 fFL i

Since F : (LX , η) → (LY , η1 ) is fuzzy continuous, we have ∧i=n i=1 η1 (Ui ) ≤ ← ⇐ η(F (U )). Hence, U (f ) ≤ U (F (f )). Therefore, F : (LX , Uη ) → ∧i=n i η1 η i=1 L Y (L , Uη1 ) is fuzzy quasi-uniformly continuous. ¤ Theorem 3.14. Let G : L-FCTOP → L-HuQUnif be defined by G((LX , τ )) = (LX , Uτ ). Then G is an embedding functor from L-FCTOP to L-HuQUnif.


1391

Remark 3.15. In [9], Kim studied the relationship between L-fuzzy quasiuniformities and L-fuzzy topologies and showed that each L-fuzzy topological space is L-fuzzy quasi-uniformizable if L is an order dense chain. From Theorem 3.10, we know that each L-fuzzy co-topological space is L-fuzzy quasiuniformizable and the condition that L is an order dense chain is not needed. Kim’s approach is different from ours. The following example shows this. Example 3.16. Let X be a single-point set and let L = {0, 1, a, b} be the diamond-type lattice, that is, a ∨ b = 1, a ∧ b = 0, a0 = b and b0 = a. Hence, we know that c(L) = {a, b} and we do not distinguish L and LX . Define τ : L → M as follows:   1, λ ∈ {0, 1}, b, λ = a, τ (λ) =  a, λ = b. Then τ is an L-fuzzy topology and its corresponding L-fuzzy co-topology is   1, λ ∈ {0, 1}, a, λ = a, η(λ) = τ (λ0 ) =  b, λ = b. It is easy to verify that H(LX ) is the set {f1 , f 2 , f 3 , f 4 }, where f1 , f 2 , f 3 , f 4 are defined as follows:   0, λ = 0, 0, λ = 0,       1, λ = 1, 1, λ = 1, 2 f1 (λ) = f (λ) = 1, λ = a, a, λ = a,       1, λ = b. b, λ = b. and

 0,    1, f 3 (λ) =  a,   1,

λ = 0, λ = 1, λ = a, λ = b.

 0, λ = 0,    1, λ = 1, f 4 (λ) =  1, λ = a,   b, λ = b.

It is easy to check that f1C = f1 , (f 2 )C = f 2 , (f 3 )C = f 4 and (f 4 )C = f 3 . Furthermore,   λ = 0, λ = 0,  0,  0, 3 1, λ ∈ {1, b}, 1, λ ∈ {1, a}, = f 4 fa (λ) = =f , fb (λ) =   b, λ = b. a, λ = a. Hence from Theorem 4.2 in  0,    b, Uη (f ) = a,    1,

[5] and Theorem 3.10 in this paper, we have  f = f 2, 0, f = f 2 ,    3 f =f , b, f = f 3 , Uτ (f ) = 4 f =f , a, f = f 4 ,    f = f1 . 1, f = f1 .

1392


By Theorem 3.5 in [9], we have ½ 1, λ = 1, IUτ (λ, a) = 0, others.

½ IUτ (λ, b) =

1, 0,

λ = 1, others.

Therefore, from Theorem 3.6 in [9], we can get that ½ 1, λ ∈ {0, 1}, τUτ (λ) = 0, others. Then τUτ 6= τ . According to our approach in this paper, we have    0, λ ∈ {1, a},  0, λ ∈ {1, b}, U 1, λ = 0, 1, λ = 0, RaUη (λ) = Rb η (λ) =   b, λ = b. a, λ = a. Hence

  1, λ ∈ {0, 1}, a, λ = a, ηUη (λ) =  b, λ = b.

Therefore, ηUη = η. Question 3.17. In L-topology, we know that an L-topological space (LX , δ) is L-uniformizable if and only if it is Hutton completely regular. We do not know how to define completely regular separation axiom in L-fuzzy topological spaces so that it can be compatible with L-uniformizable space. 4. L-fuzzy pointwise uniformities Shi [18, 19] studied pointwise L-(quasi-)uniformities. One remarkable advantage of Shi’s (quasi-)uniformity is that it can directly reflect the characteristics of pointwise L-topology, i.e., the relations between a point and its quasi-coincident neighborhood or remote neighborhood. In [23], we studied ˇ the extension of Shi’s quasi-uniformity in a Kubiak-Sostak sense. Similar to those in section 3, the purpose of this section is to define another topological remote neighborhood system by a pointwise L-fuzzy quasi-uniformity. First, we recall some notions and results in [18, 19, 23]. Let D(LX ) denote the set of all mappings d : c(LX ) → LX such that e 6≤ d(e) for all e ∈ c(LX ). d0 is the smallest element of D(LX ), i.e., d0 (e) = 0 for all e ∈ c(LX ). For d, g ∈ D(LX ), we define (1) d ≤ g if and only if d(e) ≤ g(e) for all e ∈ c(LX ), (2) (d ∨ g)(e) = V d(e) ∨ g(e) for all e ∈ c(LX ) (3) (d ¦ g)(e) = {d(a)| a ∈ c(LX ), a 6≤ g(e)} for all e ∈ c(LX ). Then d ∨ g ∈ D(LX ), d ¦ g ∈ D(LX ), d ¦ g ≤ d, d ¦ g ≤ g and the operations ∨ and ¦ satisfy associative law. Definition 4.1 ([18, 19]). An order-preserving mapping d ∈ D(LX ) is said to be symmetric, if for all λ, µ ∈ c(LX ), there exists a ∈ c(LX ) such that a 6≤ λ0 and µ 6≤ d(a) implies that there exists b ∈ c(LX ) such that b 6≤ µ0 and λ 6≤ d(b).


1393

Let Ds (LX ) denote the set of all symmetric mappings in D(LX ). Definition 4.2 ([23]). A pointwise L-fuzzy quasi-uniformity is a mapping U : D(LX ) → M such that (FQU1) U(d0 ) = 1; (FQU2) U(d ∨ g)W= U (d) ∧ U(g) for all d, g ∈ D(LX ); (FQU3) U(d) = g¦g≥d U(g) for all d ∈ D(LX ). If U is a pointwise L-fuzzy quasi-uniformity on X, the pair (LX , U) is called a pointwise L-fuzzy quasi-uniform space. Definition 4.3 ([22]). A mapping B : D(LX ) → M is called a base of one pointwise L-fuzzy quasi-uniformity if it satisfies: (FB1) B(d0 ) = 1; (FB2) B(d ∨ g)W≥ B(d) ∧ B(g) for all d, g ∈ D(LX ); (FB3) B(d) ≤ g¦g≥d B(g) for all d ∈ D(LX ). Definition 4.4. Let U be a pointwise L-fuzzy quasi-uniformity on X. U is called a pointwise L-fuzzy uniformity if there exists a mapping B : D(LX ) → M with B(d) = 0 for all d ∈ D(LX ) −WDs (LX ) such that B is a base of U, i.e., B satisfies (FB1)–(FB3) and U(d) = g≥d B(g). Lemma 4.5 ([23]). Let (LX , U ) be a pointwise L-fuzzy quasi-uniform space (U ) and Re : LX → M be defined by _ U(d). ∀A ∈ LX , Re(U ) (A) = A≤d(e) (U )

Then R(U ) = {Re |e ∈ c(LX )} is a topological fuzzy remote neighborhood system. Theorem 4.6. Let (LX , U ) be a pointwise L-fuzzy quasi-uniform space and ReU : LX → M be defined by _ ∀A ∈ LX , ReU (A) = U(d). e6≤

W

d(a)0

a6≤A0

Then RU = {ReU |e ∈ c(LX )} is a topological fuzzy remote neighborhood system. Proof. We only check (FRN4∗ ). Since _ ^ ReU (A) ≥ (ReU (B) ∧ RaU (A)) a6≤B

B∈e|A

is obvious, it suffices to show that ReU (A) ≤ Let _ α C ReU (A) = U(d) = e6≤

W

a6≤A0

d(a)0

W

e6≤

U B∈e|A (Re (B)

_ W a6≤A0

∧

_ d(a)0 g¦g≥d

V a6≤B

U (g).

RaU (A)).

1394


W Then there exist d, g ∈ D(LX ) with e 6≤ a6≤A0 d(a)0 such that g ¦ g ≥ d and W α ≤ U (g). Let B = b6≤A0 g(b)0 . Then B ∈ e|A. Since _

g(c)0 =

c6≤B 0

we have e 6≤

_ V c6≤ b6≤A0 g(b)

W

_

g(c)0 =

_

b6≤A0

_

g(c)0 =

(g ¦ g(b))0 ≤

b6≤A0

c6≤g(b)

_

(d(b))0 ,

b6≤A0

g(c)0 . Hence,

c6≤B 0

_

ReU (A) = e6≤

W a6≤B 0

Furthermore, we have ^ ^ RaU (A) =

_

W a6≤B a6≤ b6≤A0 d(b)0

a6≤B

Then α ≤ ReU (B) ∧

U(d) ≥ U(g) ≥ α. d(a)0

V a6≤B

U(d) ≥

^

U(g) ≥ α.

a6≤B

RaU (A). Therefore,

α≤

_

(ReU (B) ∧

^

RaU (A)).

a6≤B

B∈e|A

From the arbitrariness of α, we have ^ _ RaU (A)). (ReU (B) ∧ ReU (A) ≤ a6≤B

B∈e|A

Thus the conclusion holds.

¤

Theorem 4.7. If (LX , U) is a pointwise L-fuzzy uniform space, then ReU ≤ (U ) Re for all e ∈ c(LX ). Proof. Since U is a pointwise L-fuzzy uniformity, there exists a base B of U such that B(d) = 0 for all d ∈ D(LX ) − Ds (LX ). Let _ _ _ α C ReU (A) = U (d) = B(d∗ ). W e6≤ a6≤A0 d(a)0

e6≤

W

a6≤A0

d(a)0 d∗ ≥d

W Then there exist d ∈ D(LX ) and d∗ ∈ Ds (LX ) such that e 6≤ a6≤A0 d(a)0 , W d∗ ≥ d and α ≤ B(d∗ ). Hence, e 6≤ a6≤A0 d∗ (a)0 . Since d∗ is symmetric, we W (U ) have d∗ (e) ≥ A. Therefore, α ≤ B(d∗ ) ≤ A≤d(e) U(d) = Re (A). From the (U )

(U )

arbitrariness of α, we have ReU (A) ≤ Re (A), i.e., ReU ≤ Re . (U )

Question 4.8. Is ReU = Re space?

¤

valid when (LX , U) is a pointwise L-fuzzy uniform


1395

References [1] J. Ad´ amek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories, J. Wiley & Sons, New York, 1990. ˇ [2] D. Dzhajanbajev and A. P. Sostak, On a fuzzy uniform structure, Tartu Ul. Toimetised No. 940 (1992), 31–36. [3] G. Gierz, K. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott, A compendium of continuous lattice, Springer, Berlin, 1980. ˇ errez Garcia, M. A. de Prada Vicente, and A. P. Sostak, A unified approach to [4] J. Guti´

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

the concept of fuzzy L-uniform space, Trends Log. Stud. Log. Libr., 20, Kluwer Acad. Publ., Dordrecht, 2003. U. H¨ ohle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math. 38 (1982), no. 3, 289–323. , Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl. 78 (1980), no. 2, 659–673. ˇ U. H¨ ohle and A. P. Sostak, Axiomatic foundations of fixed-basis fuzzy topology, Handb. Fuzzy Sets Ser., 3, Kluwer Acad. Publ., Boston, MA, 1999. B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. 58 (1977), no 3, 557–571. C. Y. Kim, L-fuzzy quasi-uniformizable spaces, Indian J. Pure Appl. Math. 35 (2004), no. 5, 599–619. W. Kotz´ e, Uniform spaces, Handb. Fuzzy Sets Ser., 3, Kluwer Acad. Publ., Boston, MA, 1999. T. Kubiak, On fuzzy topologies (PhD Thesis, Adam Mickiewicz, Poznan (Poland), 1985. Y. M. Liu and M. K. Luo, Fuzzy Topology, World Scientific, Singapore, 1997. R. Lowen, Fuzzy uniform spaces, J. Math. Anal. Appl. 82 (1981), no. 2, 370–385. S. E. Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl. 129 (1988), no. 1, 37–70. , Powerset operator foundations for poslat fuzzy set theories and topologies, Handb. Fuzzy Sets Ser., 3, Kluwer Acad. Publ., Boston, MA, 1999. , Categorical foundations of variable-basis fuzzy topology, Handb. Fuzzy Sets Ser., 3, Kluwer Acad. Publ., Boston, MA, 1999. , Axiomatic foundations for uniform operator quasi-uniformities, Trends Log. Stud. Log. Libr., 20, Kluwer Acad. Publ., Dordrecht, 2003. F. G. Shi, Pointwise uniformities in fuzzy set theory, Fuzzy Sets and Systems 98 (1998), no. 1, 141–146. F. G. Shi, J. Zhang, and C. Y. Zheng, L-proximities and totally bounded pointwise L-uniformities, Fuzzy Sets and Systems 133 (2003), no. 3, 321–331. ˇ A. P. Sostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo (2) Suppl. No. 11 (1985), 89–103. , Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), no. 6, 662–701. ˇ Y. Yue and J. Fang, Categories isomorphic to the Kubiak-Sostak extension of TML, Fuzzy Sets and Systems 157 (2006), no. 6, 832–842. ˇ , Extension of Shi’s quasi-uniformities in a Kubiak-Sostak sense, Fuzzy Sets and Systems 157 (2006), no. 14, 1956–1969. D. Zhang, Stratified Hutton uniform spaces, Fuzzy Sets and Systems 130 (2002), no. 3, 337–346. , A comparison of various uniformities in fuzzy topology, Fuzzy Sets and Systems 140 (2003), no. 3, 399–422.

1396


Yue-Li Yue Department of Mathematics Ocean University of China Qingdao 266-071, P. R. China E-mail address: [email protected] Fu-gui Shi Department of Mathematics Beijing Institute of Technology Beijing 100-081, P.R. China E-mail address: [email protected]