O-convergence of fuzzy nets and its applications
Fuzzy Sets and Systems 140 (2003) 499 – 507 www.elsevier.com/locate/fss
O-convergence of fuzzy nets and its applications Fu-Gui Shia;∗ , Chong-You Zhengb;1 a
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China b Department of Mathematics, Capital Normal University, Beijing 100037, People’s Republic of China Received 19 September 2000; received in revised form 25 November 2002; accepted 16 January 2003
Abstract In this paper, an O-convergence theory of fuzzy nets is built in L-topological spaces by means of neighborhoods of fuzzy points. It has many nice properties. It can be used to characterize the closed set, open set, T2 separation axiom and fuzzy compactness. c 2003 Elsevier B.V. All rights reserved. Keywords: L-topology; O-cluster point; O-limit point; O-convergence; T2 separation axiom; Fuzzy compactness
1. Introduction Pu and Liu [7] introduced the concept of Q-neighborhoods of a fuzzy point and built successfully a theory of Moore–Smith convergence of fuzzy nets. Adherence points and accumulation points of fuzzy sets can be characterized by means of fuzzy nets. In [5] it was pointed out that L-convergence classes and L-topologies completely determined each other. In this paper, O-convergence theory of fuzzy nets is presented in terms of neighborhoods of fuzzy points. Although it is not so ideal as the convergence theory in [5], it still has many nice properties. In particular we can characterize closed sets, open sets, T2 separation axiom and fuzzy compactness by means of O-convergence and O-cluster points of fuzzy nets. Hence we overcome the di?culty which the neighborhood method meets. In 1976, the concept of fuzzy compactness was introduced in [0,1]-topological spaces by R. Lowen [6]. Subsequently its characterization was given by G.J. Wang in terms of -net in [10] and by Chadwick in term of Clter in [1]. In 1988, Wang extended Lowen’s fuzzy compactness into L-topology in terms of -nets in [12], where is a nonzero co-prime element in lattice value ∗
Corresponding author. Tel.: +86-1-068940246. E-mail addresses: [email protected], [email protected] (F.-G. Shi). 1 The Project is supported by the National Natural Science Foundation of China.
c 2003 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(03)00019-8
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L of LX . Wang’s deCnition of fuzzy compactness is not equivalent to KubiMak’s deCnition of fuzzy compactness in [4]. In fact, KubiMak’s deCnition of fuzzy compactness implies Wang’s deCnition of fuzzy compactness, but the inverse is not true. Up till now we have not known whether KubiMak’s deCnition of fuzzy compactness can be characterized by means of fuzzy nets. Since the set of all nonzero co-prime elements in a completely distributive lattice is -generating, based on the action of -nets ( is a nonzero co-prime element), we need a completely distributive lattice L in LX . This assumption is not too strong. In fact if L is a continuous lattice with a quasi-complementation, then Lop is also a continuous lattice. Thus our assumption of completely distributivity is only added a distributive law, i.e. a ∧ (b ∨ c)= (a ∧ b) ∨ (a ∧ c). As pointed out by KubiMak [4], for a continuous lattice L and a topological space (X; T ), T = L !L (T ) is not true in general, but by Proposition 3.5 in [4] we know that one su?cient condition of T = L !L (T ) is that L is completely distributive.
2. Preliminaries Throughout this paper (L; ; ; ) is a completely distributive lattice with an order-reversing involution “ ”. X is a nonempty set. The smallest element and the largest element in L are denoted by 0 and 1. LX is the set of all L-fuzzy sets (or L-set for short) on X . The smallest element and the largest element in LX are denoted by 0 and 1. An element a is called co-prime [2] if a6b ∨ c implies a6b or a6c. The set of nonzero co-prime elements in L is denoted by M (L). Note that the set of all nonzero co-prime elements in LX are exactly the set of all L-fuzzy points x ( ∈ M (L)) deCned by x (x) = and x (y) = 0 otherwise. We shall not distinguish a crisp set and its character function. An L-topological space (L-ts) is a pair (LX ; ), where (⊆ LX ) contains 0 and 1 and is closed for any suprema and Cnite inCma. Members of are called open, and their quasi-complements are called closed. A closed L-set P is called a closed remote-neighborhood(or closed R-neighborhood) of x if x P. An open L-set Q is called an open Q-neighborhood of x if Q is a closed Rneighborhood of x . An open L-set U is called an open neighborhood of x ∈ M (LX ) if x 6U . All closed R-neighborhoods of x are denoted by − (x ). All open Q-neighborhoods of x are denoted by Qo (x ). All open neighborhoods of x are denoted by No (x ). x ∈ M (LX ) is said to be quasi-coincident with B ∈ LX if x B . As proved by Hutton [3], if L is a completely distributive lattice and a ∈ L, then there exists B ⊆ L such that: (1) a = B, and (2) If A ⊆ L and a = A, then for each b ∈ B there is a c ∈ A such that b6c. But for a complete lattice L, if ∀a ∈ L, there exists B ⊆ L satisfying (1) and (2), then in general L is not a completely distributive lattice. To turn L into a completely distributive lattice, Wang revised condition (2) as follows. (2 ) If A ⊆ L and a6 A, then for each b ∈ B there is a c ∈ A such that b6c. 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 (1) and (2 ). Such a set B is called a minimal set of a by Wang [11]. Let (a) denote the union of all minimal set of a and ∗ (a) = (a) ∩ M (L). One easily sees that both (a) and ∗ (a) are minimal set of a (see [5,11]).
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Proposition 2.1 (Gierz et al. [2], Liu and Luo [5], Wang [11]). If L is a completely distributive lattice, then M (L) is a join-generating set of L. In a completely distributive lattice L, it is easy to see that b ∈ ∗ (a) ⇔ b a, where is the way below relation [2] in L. Denition 2.2 (Shi [9]). Let A ∈ LX and a ∈ L, we deCne A(a) = {x ∈ X | a ∈ (A(x))}; A(a) = {x ∈ X | A(x) a}: It is easy to prove that A(0) = A(0) ; A(a) ⊂ A[a] ; A(a) = ba A[b] = ba A(b) . A[a] = {x ∈ X | A(x) ¿ a};
Denition 2.3 (Liu and Luo [5]). If D is a directed set, then every mapping S : D→M (LX ) is called a net in LX . A net T : E→M (LX ) is called a subnet of S : D→M (LX ) if there exists a mapping N : E→D such that (i) T = S ◦ N ; (ii) For each n0 ∈ D, there exists m0 ∈ E such that N (m)¿n0 for m¿m0 . Since for a co-prime element , ∗ () is a directed set, we know that S = {xr | r ∈ ∗ ()} is a net in LX . Denition 2.4 (Liu and Luo [5]). An L-ts (LX ; ) is called a T2 space (or HausdorR space) if for any x ; y ∈ M (LX ) with x = y, there exist U ∈ Qo (x ); V ∈ Qo (y ) such that U ∧ V = 0. Denition 2.5 (Wang [12], Zhao [14]). Let (LX ; ) be an L-ts and B ∈ LX . A ⊆ is called an -Rneighborhood family of B, brieSy -RF of B, if for each point x 6B, there exists A ∈ A such that x A. A is called an − -R-neighborhood family of B, brieSy − -RF of B, if there exists ! ∈ ∗ () such that A is a !-RF of B, where ∈ M (L). In 1983, Wang presented a characterization of Lowen fuzzy compactness by means of fuzzy nets. In 1988, he extended it into L-topology as follows. Denition 2.6 (Wang [12]). Let (LX ; ) be an L-ts and D ∈ LX . D is called fuzzy compact if for all ∈ M (L) and for all ! ∈ ∗ (), every constant -net in D has a cluster point x! 6D. The following theorem gives two characterizations of DeCnition 2.6. Theorem 2.7 (Shi [8]). Let (LX ; ) be an L-ts and D ∈ LX . Then the following statements are equivalent. (1) D is fuzzy compact. (2) For all ∈ M (L), each − -RF of D has a =nite subfamily which is an -RF of D. (3) For all ∈ M (L), each − -RF of D has a =nite subfamily which is an − -RF of D. Condition (3) in Theorem 2.7 is exactly the deCnition in [13].
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KubiMak also extended Lowen fuzzy compactness into L-topology, but Kubiak’s deCnition is not equivalent to DeCnition 2.6. Moreover we have not known whether KubiMak’s fuzzy compactness can be characterized by means of fuzzy nets.
3. O-convergence of nets Denition 3.1. A net S with index set D is also denoted by {S(n) | n ∈ D} or {S(n)}n∈D . For A ∈ LX , a net S is said to quasi-coincide with A if ∀n ∈ D; S(n)A . Denition 3.2. Let {S(n) | n ∈ D} be a net in (LX ; ), x ∈ M (LX ). S eventually possesses the property P, if there exists n0 ∈ D such that ∀n¿n0 , S(n) always possesses the property P. S frequently possesses the property P, if for every n ∈ D, there always exists n0 ∈ D such that n0 ¿n and S(n0 ) possesses the property P. x is an O-cluster point of S, if ∀U ∈ No (x ), S is frequently in U . x is an O-limit point of S, if ∀U ∈ No (x ), S is eventually in U , in this case we also say that S O-converges to x , denoted O by S → x . It is easy to prove the following theorem. Theorem 3.3. Let S be a net in (LX ; ), T a subnet of S and x ; x ∈ M (LX ). Then (1) S = {xr | r ∈ ∗ ( )} O-converges to x . O (2) S → x implies that x is an O-cluster point of S. (3) If x 6x and x is an O-cluster point of S, then x is also an O-cluster point of S. O
O
(4) S → x 6x ⇒ S → x . O
O
(5) S → x ⇒ T → x . (6) x is an O-cluster point of T ⇒ x is an O-cluster point of S. O (7) x is an O-cluster point of S if and only if S has a subnet R such that R → x . Proof. It is simple and is omitted. The following theorem characterizes the closure of an L-set. Theorem 3.4. Let x ∈ M (LX ); B ∈ LX . Then x quasi-coincides with B− if and only if there exists O a net S quasi-coinciding with B such that S → x . Proof. Necessity. Suppose that x quasi-coincides with B− . Then ∀U ∈ No (x ); U (B− ) . Further B− U . Hence BU . This implies U B . Take S(U ) ∈ M (LX ) such that S(U )6U; S(U )B . We obtain a net {S(U ) | U ∈ No (x )} O-converging to x and it quasi-coincides with B. O Su?ciency. Let {S(n)} be a net quasi-coinciding with B and S → x . If x 6(B− ) , then ∃n0 ∈ D such that ∀n¿n0 ; S(n)6(B− ) 6B , contradicts that S quasi-coincides with B.
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The following two corollaries are obvious. Corollary 3.5. Let (LX ; ) be an L-ts and A ∈ LX . Then the following conditions are equivalent: (1) A is closed. (2) For each net S quasi-coinciding with A, if x is an O-cluster point of S, then x A . O (3) For each net S quasi-coinciding with A, if S → x , then x A . Corollary 3.6. Let (LX ; ) be an L-ts and A ∈ LX . Then the following conditions are equivalent: (1) A is open. O (2) ∀x 6A, S → x implies S is eventually in A. (3) ∀x 6A, if x is O-cluster point of S, then S is frequently in A. Moreover we can easily prove the following result. Theorem 3.7. Let f : (LX ; )→(LY ; %) be an L-value Zadeh’s type mapping. Then the following conditions are equivalent. (1) f is continuous. (2) For any net S in LX , if x is an O-cluster point of S, then f(x ) is an O-cluster point of f(S). O O (3) For any net S in LX , if S → x , then f(S) → f(x ). The following theorem gives a characterization of a T2 L-ts. Theorem 3.8. An L-ts (LX ; ) is T2 if and only if for any x ; y ∈ M (LX ) with x = y, there exist U ∈ No (x ); V ∈ No (y ) such that U ∧ V = 0. Proof. (⇒) Suppose that (LX ; ) is T2 and x ; y ∈ M (LX ) with x = y. Then for any co-prime element a and any co-prime element b , there exist U (a) ∈ Qo (xa ); V (b) ∈ Qo (yb ) such that U ∧ V = 0. Let U = a U (a) and V = b V (b), then U ∈ No (x ); V ∈ No (y ) and U ∧ V = 0. (⇐) Suppose that for any xa ; yb ∈ M (LX ) with x = y, there exist U ∈ No (xa ); V ∈ No (yb ) such that U ∧ V = 0. To prove that (LX ; ) is T2 , suppose that x ; y ∈ M (LX ) with x = y. Then for any o co-prime element a and anyco-prime element b No (yb ) , there exist U (a) ∈ No (xa ); V (b) ∈ such that U ∧ V = 0. Let U = a U (a) and V = b V (b), then U ∈ Q (x ); V ∈ Qo (y ) and U ∧ V = 0. This shows that (LX ; ) is T2 . A T2 L-ts can be characterized by O-convergence of nets. Theorem 3.9. An L-ts (LX ; ) is T2 if and only if for each net S in LX having O-limit points, support points of O-limit points of S is unique.
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Proof. Suppose that (LX ; ) is T2 and x ; y are two O-limit points of a net S(n). If x = y, then there exists U ∈ No (x ); V ∈ No (y ) such that U ∧ V = 0. Since x ; y are two O-limit points of S(n), we can take n0 such that when n¿n0 , S(n)6U and S(n)6V , further S(n)6U ∧ V . But U ∧ V = 0. This is a contradiction. Therefore it follows that x = y. Conversely suppose that support points of O-limit points of each net in LX having O-limit points is unique. We prove that (LX ; ) is T2 . Suppose that it is not T2 . Then there exists x ; y ∈ M (LX ) with x = y such that ∀U ∈ No (x ); ∀V ∈ No (y ), U ∧ V = 0. ∀U ∈ No (x ); ∀V ∈ No (y ), take S(UV ) ∈ M (LX ) such that S(UV )6U ∧ V and let S = {S(UV ) | U ∈ No (x ); V ∈ No (y )}: ∀(U1 ; V1 ); (U2 ; V2 ) ∈ No (x ) × No (y ) we deCne (U1 ; V1 )6(U2 ; V2 ) ⇔ U1 ¿U2 ; V1 ¿V2 . Then No (x ) × No (y ) is a directed set. It is easy to see that S is a net O-converging to x and y . Thus we obtain that x = y, this contradicts x = y. The proof is obtained.
4. Characterizations of fuzzy compactness KubiMak pointed out that his deCnition of fuzzy compactness cannot be restated in terms of open L-sets by simply applying the order-reversing involution on LX , since this involution need not be order-reversing with respect to the way-below relation. In this section, we shall present some characterizations of Wang’s deCnition of fuzzy compactness by means of open L-sets and O-cluster points. We Crst prove the following Lemma. Lemma 4.1. For each a ∈ L − {0}, let Q(a) = {b ∈ L | ba }; Q∗ (a) = Q(a) ∩ M (L). Then the following conditions are true. ∗ () ⇒ ∗ ( Q(!)) ∩ Q ∗ () = ∅. (1) For each ∈ L, ! ∈ (2) b ∈ ∗ ( Q(a)) ⇒ Q(b)¿a. Proof. (1) For ∈ L, ! ∈ ∗ (), we suppose that ∗ ( Q(!)) ∩ Q∗ () = ∅. Then ∀e ∈ ∗ ( Q(!)); e 6 . Hence Q(!)6 , i.e. 6 {c | c ∈ Q(!)}. Since ! ∈ ∗ (), there exists a c ∈ Q(!) such that !6c . This is a contradiction. (2) Suppose that b ∈ ∗ ( Q(a)). Then ∀ca , b6c. Hence ∀d ∈ Q(b); dc . This implies that a6d. Further we obtain that Q(b)¿a. To characterize fuzzy compactness, the following deCnition is useful. Denition 4.2. Let (LX ; ) be an L-ts, G∈ LX and ∈ M (L). ) ⊆ is called a Q -open cover of G, if for each x G , it follows that x 6 ) = {A | A ∈ )}. It is obvious that ) is a Q -open cover of 1 if and only if ) is an open cover of constant fuzzy set .) is a Q -open cover of G if and only if ) is an open cover of ∧ G ( ) if and only if G ∨( ))¿.
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Theorem 4.3. Let (LX ; ) be an L-ts and G ∈ LX . Then G is fuzzy compact if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each Q -open cover ) of G has a =nite subfamily B such that B is a Q! -open cover of G. Proof. (⇒) Suppose that G is fuzzy compact, ∈ M (L) and ) is a Q -open cover of G.For each ! ∈ ∗ (), take an a ∈ ∗ () such that ! ∈ ∗ (a). By Lemma 4.1(1) we can take b ∈ ∗ ( Q(!)) ∩ Q∗ (a). Take c ∈ ∗ (b) such that c ∈ Q∗ (a). Now we prove that ) is a c-RF of G. Infact, suppose that ) is not a c-RF of G. Then there exists a ∩ xc 6G such that xc 6 ) , i.e., c6 {A (x) | A ∈ )}. Hence a {A(x) | A ∈ )} and aG(x) . This shows that ) is not a Qa -open cover of G. This contradicts that ) is a Q -open cover of G. Thus ) is a b− -RF of G. Hence by fuzzy compactness of G we know that there exists a Cnite subfamily * of ) such that * is a b-RF of G. Now we shall prove that * is a Q! -open cover of G. Suppose that * is not a Q! -open cover of G. Then there exists a ∩ x! G such that x! *. Hence !G (x) and ∀A ∈ *, !A(x). Further we have that G(x) ∈ Q(!) and ∀A ∈ *, A (x) ∈ Q(!). This shows that b6G(x) and ∀A ∈ *, b6A (x). This contradicts that * is a b-RF of G. (⇐) Suppose that ∀ ∈ M (L), ∀! ∈ ∗ (), each Q -open cover of G has a Cnite subfamily which is a Q! -open cover of G. We shall prove that G is fuzzy compact. Let ∈ M (L) and + is an − -RF of G. Then there exists a ! ∈ ∗ () such that +is a !-RF of G. Take an a ∈ ∗ () such that ! ∈ ∗ (a). By Lemma 4.1(1) we can take a b ∈ ∗ ( Q(!)) ∩ Q∗ (a). Take c ∈ ∗ (b) such that c ∈ Q∗ (a). Now we prove that + is a Qb -open cover of G. In fact, suppose that + is not a Qb -open cover of G. Then there exists a ∩ xb G such that xb + . Hence bG (x) and ∀B ∈ +, bB (x), i.e., G(x)b and B(x)b . This implies that G(x) ∈ Q(b) and B(x) ∈ Q(b). By Lemma 4.1(2) we know that Q(b)¿!. Therefore !6G(x) and ∀B ∈ +; !6B(x). This contradicts that + is a !-RF of G. Thus + has a Cnite subfamily * which is a Qc -open cover of G. Now we shall prove that * is an -RF of G. Suppose * is not an -RF of G. Then there exists x 6G such that x 6 * . Hence 6G(x) that and 6 {A (x) | A ∈ *}. Thus we obtain that a6G(x) and a6 {A (x) | A ∈ *} Further cG (x) and c {A(x) | A ∈ *}. This contradicts that * is a Qc -open cover of G. So we complete the proof. From Theorem 4.3 we easily obtain the following two results. X . Then G is fuzzy compact if and only if ∀ ∈ M (L), Corollary 4.4. Let (LX ; ) be an L-ts and G ∈ L ∗ ∀! ∈ ()and for each ) ⊆ such that G ∨( ))¿, there exists a =nite subfamily + ⊆ ) such that G ∨( +)¿!.
Corollary 4.5. (LX ; ) is fuzzy compact if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each open cover ) of has a =nite subfamily B such that B is an open cover of !. Fuzzy compactness can also be characterized by means of O-cluster points of nets.
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Denition 4.6. Let ∈ M (L). A net {S(n) | n ∈ D} in LX is called an − -net if there exists n0 ∈ D such that ∀n¿n0 , V (S(n))6, where V (S(n)) denotes the height of S(n). A net {S(n)} in LX is said to be constant -net if the height of each S(n) is a constant value . Obviously each constant -net must be an − -net. Theorem 4.7. An L-set G is fuzzy compact in (LX ; ) if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each constant !-net quasi-coinciding with G has an O-cluster point x quasi-coinciding with G. Proof. Suppose that G is fuzzy compact. For ∈ M (L) and ! ∈ ∗ (), let {S(n) | n ∈ D} is a constant !-net quasi-coinciding with G. Suppose that S has no any O-cluster point x quasi-coinciding with G. Then for each x G , there exist Ux ∈ No (x ) and nx ∈ D such that ∀n¿nx ; S(n)Ux . Take ) = {Ux | x G }, then ) is a Q -open cover of G. Since G is fuzzy compact, ) has a Cnite subfamily + = {Uxi | i = 1; 2; : : : ; k} such that + is a Q! -open cover of G. Since D is a directed set, there exists n0 ∈ D such that n0 ¿nxi for each i6k. Thus we can obtain that ∀n¿n0 ; S(n) {Uxi | i = 1; 2; : : : ; k}. This contradicts that + is a Q! -open cover of G. Therefore S has at least an O-cluster point x G . Conversely suppose that ∀ ∈ M (L), ∀! ∈ ∗ (), each constant !-net quasi-coinciding with G has an O-cluster point x G . We now prove that G is fuzzy compact. Let ) be a Q -open cover of G. If there exists ! ∈ ∗ () such that each Cnite subfamily + of ) is not a Q! -open cover of G, then for eachCnite subfamily + of ), there exists S(+) ∈ M (LX ) with height ! such that S(+)G and S(+) +. Take S = {S(+) | +is a Cnite subfamily of )}: Then S is a constant !-net quasi-coinciding with G. By ! ∈ ∗ () we can take s ∈ ∗ () such that ! ∈ ∗ (s). Then S has an O-cluster point xs quasi-coinciding with G. Hence for each Cnite subfamily + of ) we have that xs +, in particular, xs B for each B ∈ ). But since ) be a Q -open cover of G, we know that there exists B ∈ ) such that xs 6B, this is a contradiction. So G is fuzzy compact. Theorem 4.8. An L-set G is fuzzy compact in (LX ; ) if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each !− -net quasi-coinciding with G has an O-cluster point x quasi-coinciding with G. Proof. The su?ciency is obvious. We need only prove the necessity. Let G be fuzzy compact, ∈ M (L); ! ∈ ∗ () and {S(n) | n ∈ D} an !− -net quasi-coinciding with G. Then there exists n0 ∈ D such that ∀n¿n0 ; S(n)6!. Put E = {n ∈ D | n¿n0 } and T = {T (n) | n ∈ E; V (T (n)) = !; support point of T (n) is the same as S(n)}: Then T is a constant !-net quasi-coinciding with G . Let x is an O-cluster point of T . It is easy to see that x is also an O-cluster point of S. Remark 4.9. Because KubiMak deCnition of fuzzy compactness depends on each elements a ∈ L, but the deCnition of nets in [5,12] was only deCned in M (L), we do not know whether KubiMak deCnition of fuzzy compactness can be characterized by O-cluster points. We leave it as an open problem.
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Acknowledgements The authors would like to thank S.E. Rodabaugh and referees for their valuable comments and suggestions. References [1] J.J. Chadwick, A generalized form of compactness in fuzzy topological spaces, J. Math. Anal. Appl. 162 (1991) 92–110. [2] G. Gierz, et al., A Compendium of Continuous Lattices, Springer, Berlin, 1980. [3] B. Hutton, Uniformities in fuzzy topological spaces, J. Math. Anal. Appl. 58 (1977) 559–571. [4] T. KubiMak, The topological modiCcation of the L-fuzzy unit interval, in: S.E. Rodabaugh, E.P. Klement, U. HUohle (Eds.), Applications of Category Theory to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1992, pp. 275–305 (Chap. 11). [5] Y.-M. Liu, M.K. Luo, Advances in Fuzzy Systems—Applications and Theory, in: Fuzzy Topology, Vol. 9, World ScientiCc Publishing Co. Pte. Ltd, Singapore, 1997. [6] R. Lowen, A comparison of diRerent compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446–454. [7] P.-M. Pu, Y.-M. Liu, Fuzzy topology I, neighborhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571–599. [8] F.-G. Shi, Theory and applications of L -nest sets and L -nest sets, Fuzzy Systems Math. 4 (1995) 65–72 (in Chinese). [9] F.-G. Shi, A note on fuzzy compactness in L-topological spaces, Fuzzy Sets and Systems 119 (2001) 547–548. [10] G.-J. Wang, A new fuzzy compactness deCned by fuzzy nets, J. Math. Anal. Appl. 94 (1983) 1–23. [11] G.-J. Wang, On the structure of fuzzy lattices, Acta Math. Sinica 4 (1986) 539–543. [12] G.-J. Wang, Theory of L-fuzzy Topological Space, Shaanxi Normal University Publishers, Xian, 1988 (in Chinese). [13] J.J. Xu, On fuzzy compactness in L-fuzzy topological spaces, Chinese Quart. J. Math. 2 (1990) 104–105 (in Chinese). [14] D.S. Zhao, The N-compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128 (1987) 64–70.
O-convergence of fuzzy nets and its applications Fu-Gui Shia;∗ , Chong-You Zhengb;1 a
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China b Department of Mathematics, Capital Normal University, Beijing 100037, People’s Republic of China Received 19 September 2000; received in revised form 25 November 2002; accepted 16 January 2003
Abstract In this paper, an O-convergence theory of fuzzy nets is built in L-topological spaces by means of neighborhoods of fuzzy points. It has many nice properties. It can be used to characterize the closed set, open set, T2 separation axiom and fuzzy compactness. c 2003 Elsevier B.V. All rights reserved. Keywords: L-topology; O-cluster point; O-limit point; O-convergence; T2 separation axiom; Fuzzy compactness
1. Introduction Pu and Liu [7] introduced the concept of Q-neighborhoods of a fuzzy point and built successfully a theory of Moore–Smith convergence of fuzzy nets. Adherence points and accumulation points of fuzzy sets can be characterized by means of fuzzy nets. In [5] it was pointed out that L-convergence classes and L-topologies completely determined each other. In this paper, O-convergence theory of fuzzy nets is presented in terms of neighborhoods of fuzzy points. Although it is not so ideal as the convergence theory in [5], it still has many nice properties. In particular we can characterize closed sets, open sets, T2 separation axiom and fuzzy compactness by means of O-convergence and O-cluster points of fuzzy nets. Hence we overcome the di?culty which the neighborhood method meets. In 1976, the concept of fuzzy compactness was introduced in [0,1]-topological spaces by R. Lowen [6]. Subsequently its characterization was given by G.J. Wang in terms of -net in [10] and by Chadwick in term of Clter in [1]. In 1988, Wang extended Lowen’s fuzzy compactness into L-topology in terms of -nets in [12], where is a nonzero co-prime element in lattice value ∗
Corresponding author. Tel.: +86-1-068940246. E-mail addresses: [email protected], [email protected] (F.-G. Shi). 1 The Project is supported by the National Natural Science Foundation of China.
c 2003 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(03)00019-8
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L of LX . Wang’s deCnition of fuzzy compactness is not equivalent to KubiMak’s deCnition of fuzzy compactness in [4]. In fact, KubiMak’s deCnition of fuzzy compactness implies Wang’s deCnition of fuzzy compactness, but the inverse is not true. Up till now we have not known whether KubiMak’s deCnition of fuzzy compactness can be characterized by means of fuzzy nets. Since the set of all nonzero co-prime elements in a completely distributive lattice is -generating, based on the action of -nets ( is a nonzero co-prime element), we need a completely distributive lattice L in LX . This assumption is not too strong. In fact if L is a continuous lattice with a quasi-complementation, then Lop is also a continuous lattice. Thus our assumption of completely distributivity is only added a distributive law, i.e. a ∧ (b ∨ c)= (a ∧ b) ∨ (a ∧ c). As pointed out by KubiMak [4], for a continuous lattice L and a topological space (X; T ), T = L !L (T ) is not true in general, but by Proposition 3.5 in [4] we know that one su?cient condition of T = L !L (T ) is that L is completely distributive.
2. Preliminaries Throughout this paper (L; ; ; ) is a completely distributive lattice with an order-reversing involution “ ”. X is a nonempty set. The smallest element and the largest element in L are denoted by 0 and 1. LX is the set of all L-fuzzy sets (or L-set for short) on X . The smallest element and the largest element in LX are denoted by 0 and 1. An element a is called co-prime [2] if a6b ∨ c implies a6b or a6c. The set of nonzero co-prime elements in L is denoted by M (L). Note that the set of all nonzero co-prime elements in LX are exactly the set of all L-fuzzy points x ( ∈ M (L)) deCned by x (x) = and x (y) = 0 otherwise. We shall not distinguish a crisp set and its character function. An L-topological space (L-ts) is a pair (LX ; ), where (⊆ LX ) contains 0 and 1 and is closed for any suprema and Cnite inCma. Members of are called open, and their quasi-complements are called closed. A closed L-set P is called a closed remote-neighborhood(or closed R-neighborhood) of x if x P. An open L-set Q is called an open Q-neighborhood of x if Q is a closed Rneighborhood of x . An open L-set U is called an open neighborhood of x ∈ M (LX ) if x 6U . All closed R-neighborhoods of x are denoted by − (x ). All open Q-neighborhoods of x are denoted by Qo (x ). All open neighborhoods of x are denoted by No (x ). x ∈ M (LX ) is said to be quasi-coincident with B ∈ LX if x B . As proved by Hutton [3], if L is a completely distributive lattice and a ∈ L, then there exists B ⊆ L such that: (1) a = B, and (2) If A ⊆ L and a = A, then for each b ∈ B there is a c ∈ A such that b6c. But for a complete lattice L, if ∀a ∈ L, there exists B ⊆ L satisfying (1) and (2), then in general L is not a completely distributive lattice. To turn L into a completely distributive lattice, Wang revised condition (2) as follows. (2 ) If A ⊆ L and a6 A, then for each b ∈ B there is a c ∈ A such that b6c. 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 (1) and (2 ). Such a set B is called a minimal set of a by Wang [11]. Let (a) denote the union of all minimal set of a and ∗ (a) = (a) ∩ M (L). One easily sees that both (a) and ∗ (a) are minimal set of a (see [5,11]).
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Proposition 2.1 (Gierz et al. [2], Liu and Luo [5], Wang [11]). If L is a completely distributive lattice, then M (L) is a join-generating set of L. In a completely distributive lattice L, it is easy to see that b ∈ ∗ (a) ⇔ b a, where is the way below relation [2] in L. Denition 2.2 (Shi [9]). Let A ∈ LX and a ∈ L, we deCne A(a) = {x ∈ X | a ∈ (A(x))}; A(a) = {x ∈ X | A(x) a}: It is easy to prove that A(0) = A(0) ; A(a) ⊂ A[a] ; A(a) = ba A[b] = ba A(b) . A[a] = {x ∈ X | A(x) ¿ a};
Denition 2.3 (Liu and Luo [5]). If D is a directed set, then every mapping S : D→M (LX ) is called a net in LX . A net T : E→M (LX ) is called a subnet of S : D→M (LX ) if there exists a mapping N : E→D such that (i) T = S ◦ N ; (ii) For each n0 ∈ D, there exists m0 ∈ E such that N (m)¿n0 for m¿m0 . Since for a co-prime element , ∗ () is a directed set, we know that S = {xr | r ∈ ∗ ()} is a net in LX . Denition 2.4 (Liu and Luo [5]). An L-ts (LX ; ) is called a T2 space (or HausdorR space) if for any x ; y ∈ M (LX ) with x = y, there exist U ∈ Qo (x ); V ∈ Qo (y ) such that U ∧ V = 0. Denition 2.5 (Wang [12], Zhao [14]). Let (LX ; ) be an L-ts and B ∈ LX . A ⊆ is called an -Rneighborhood family of B, brieSy -RF of B, if for each point x 6B, there exists A ∈ A such that x A. A is called an − -R-neighborhood family of B, brieSy − -RF of B, if there exists ! ∈ ∗ () such that A is a !-RF of B, where ∈ M (L). In 1983, Wang presented a characterization of Lowen fuzzy compactness by means of fuzzy nets. In 1988, he extended it into L-topology as follows. Denition 2.6 (Wang [12]). Let (LX ; ) be an L-ts and D ∈ LX . D is called fuzzy compact if for all ∈ M (L) and for all ! ∈ ∗ (), every constant -net in D has a cluster point x! 6D. The following theorem gives two characterizations of DeCnition 2.6. Theorem 2.7 (Shi [8]). Let (LX ; ) be an L-ts and D ∈ LX . Then the following statements are equivalent. (1) D is fuzzy compact. (2) For all ∈ M (L), each − -RF of D has a =nite subfamily which is an -RF of D. (3) For all ∈ M (L), each − -RF of D has a =nite subfamily which is an − -RF of D. Condition (3) in Theorem 2.7 is exactly the deCnition in [13].
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KubiMak also extended Lowen fuzzy compactness into L-topology, but Kubiak’s deCnition is not equivalent to DeCnition 2.6. Moreover we have not known whether KubiMak’s fuzzy compactness can be characterized by means of fuzzy nets.
3. O-convergence of nets Denition 3.1. A net S with index set D is also denoted by {S(n) | n ∈ D} or {S(n)}n∈D . For A ∈ LX , a net S is said to quasi-coincide with A if ∀n ∈ D; S(n)A . Denition 3.2. Let {S(n) | n ∈ D} be a net in (LX ; ), x ∈ M (LX ). S eventually possesses the property P, if there exists n0 ∈ D such that ∀n¿n0 , S(n) always possesses the property P. S frequently possesses the property P, if for every n ∈ D, there always exists n0 ∈ D such that n0 ¿n and S(n0 ) possesses the property P. x is an O-cluster point of S, if ∀U ∈ No (x ), S is frequently in U . x is an O-limit point of S, if ∀U ∈ No (x ), S is eventually in U , in this case we also say that S O-converges to x , denoted O by S → x . It is easy to prove the following theorem. Theorem 3.3. Let S be a net in (LX ; ), T a subnet of S and x ; x ∈ M (LX ). Then (1) S = {xr | r ∈ ∗ ( )} O-converges to x . O (2) S → x implies that x is an O-cluster point of S. (3) If x 6x and x is an O-cluster point of S, then x is also an O-cluster point of S. O
O
(4) S → x 6x ⇒ S → x . O
O
(5) S → x ⇒ T → x . (6) x is an O-cluster point of T ⇒ x is an O-cluster point of S. O (7) x is an O-cluster point of S if and only if S has a subnet R such that R → x . Proof. It is simple and is omitted. The following theorem characterizes the closure of an L-set. Theorem 3.4. Let x ∈ M (LX ); B ∈ LX . Then x quasi-coincides with B− if and only if there exists O a net S quasi-coinciding with B such that S → x . Proof. Necessity. Suppose that x quasi-coincides with B− . Then ∀U ∈ No (x ); U (B− ) . Further B− U . Hence BU . This implies U B . Take S(U ) ∈ M (LX ) such that S(U )6U; S(U )B . We obtain a net {S(U ) | U ∈ No (x )} O-converging to x and it quasi-coincides with B. O Su?ciency. Let {S(n)} be a net quasi-coinciding with B and S → x . If x 6(B− ) , then ∃n0 ∈ D such that ∀n¿n0 ; S(n)6(B− ) 6B , contradicts that S quasi-coincides with B.
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The following two corollaries are obvious. Corollary 3.5. Let (LX ; ) be an L-ts and A ∈ LX . Then the following conditions are equivalent: (1) A is closed. (2) For each net S quasi-coinciding with A, if x is an O-cluster point of S, then x A . O (3) For each net S quasi-coinciding with A, if S → x , then x A . Corollary 3.6. Let (LX ; ) be an L-ts and A ∈ LX . Then the following conditions are equivalent: (1) A is open. O (2) ∀x 6A, S → x implies S is eventually in A. (3) ∀x 6A, if x is O-cluster point of S, then S is frequently in A. Moreover we can easily prove the following result. Theorem 3.7. Let f : (LX ; )→(LY ; %) be an L-value Zadeh’s type mapping. Then the following conditions are equivalent. (1) f is continuous. (2) For any net S in LX , if x is an O-cluster point of S, then f(x ) is an O-cluster point of f(S). O O (3) For any net S in LX , if S → x , then f(S) → f(x ). The following theorem gives a characterization of a T2 L-ts. Theorem 3.8. An L-ts (LX ; ) is T2 if and only if for any x ; y ∈ M (LX ) with x = y, there exist U ∈ No (x ); V ∈ No (y ) such that U ∧ V = 0. Proof. (⇒) Suppose that (LX ; ) is T2 and x ; y ∈ M (LX ) with x = y. Then for any co-prime element a and any co-prime element b , there exist U (a) ∈ Qo (xa ); V (b) ∈ Qo (yb ) such that U ∧ V = 0. Let U = a U (a) and V = b V (b), then U ∈ No (x ); V ∈ No (y ) and U ∧ V = 0. (⇐) Suppose that for any xa ; yb ∈ M (LX ) with x = y, there exist U ∈ No (xa ); V ∈ No (yb ) such that U ∧ V = 0. To prove that (LX ; ) is T2 , suppose that x ; y ∈ M (LX ) with x = y. Then for any o co-prime element a and anyco-prime element b No (yb ) , there exist U (a) ∈ No (xa ); V (b) ∈ such that U ∧ V = 0. Let U = a U (a) and V = b V (b), then U ∈ Q (x ); V ∈ Qo (y ) and U ∧ V = 0. This shows that (LX ; ) is T2 . A T2 L-ts can be characterized by O-convergence of nets. Theorem 3.9. An L-ts (LX ; ) is T2 if and only if for each net S in LX having O-limit points, support points of O-limit points of S is unique.
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Proof. Suppose that (LX ; ) is T2 and x ; y are two O-limit points of a net S(n). If x = y, then there exists U ∈ No (x ); V ∈ No (y ) such that U ∧ V = 0. Since x ; y are two O-limit points of S(n), we can take n0 such that when n¿n0 , S(n)6U and S(n)6V , further S(n)6U ∧ V . But U ∧ V = 0. This is a contradiction. Therefore it follows that x = y. Conversely suppose that support points of O-limit points of each net in LX having O-limit points is unique. We prove that (LX ; ) is T2 . Suppose that it is not T2 . Then there exists x ; y ∈ M (LX ) with x = y such that ∀U ∈ No (x ); ∀V ∈ No (y ), U ∧ V = 0. ∀U ∈ No (x ); ∀V ∈ No (y ), take S(UV ) ∈ M (LX ) such that S(UV )6U ∧ V and let S = {S(UV ) | U ∈ No (x ); V ∈ No (y )}: ∀(U1 ; V1 ); (U2 ; V2 ) ∈ No (x ) × No (y ) we deCne (U1 ; V1 )6(U2 ; V2 ) ⇔ U1 ¿U2 ; V1 ¿V2 . Then No (x ) × No (y ) is a directed set. It is easy to see that S is a net O-converging to x and y . Thus we obtain that x = y, this contradicts x = y. The proof is obtained.
4. Characterizations of fuzzy compactness KubiMak pointed out that his deCnition of fuzzy compactness cannot be restated in terms of open L-sets by simply applying the order-reversing involution on LX , since this involution need not be order-reversing with respect to the way-below relation. In this section, we shall present some characterizations of Wang’s deCnition of fuzzy compactness by means of open L-sets and O-cluster points. We Crst prove the following Lemma. Lemma 4.1. For each a ∈ L − {0}, let Q(a) = {b ∈ L | ba }; Q∗ (a) = Q(a) ∩ M (L). Then the following conditions are true. ∗ () ⇒ ∗ ( Q(!)) ∩ Q ∗ () = ∅. (1) For each ∈ L, ! ∈ (2) b ∈ ∗ ( Q(a)) ⇒ Q(b)¿a. Proof. (1) For ∈ L, ! ∈ ∗ (), we suppose that ∗ ( Q(!)) ∩ Q∗ () = ∅. Then ∀e ∈ ∗ ( Q(!)); e 6 . Hence Q(!)6 , i.e. 6 {c | c ∈ Q(!)}. Since ! ∈ ∗ (), there exists a c ∈ Q(!) such that !6c . This is a contradiction. (2) Suppose that b ∈ ∗ ( Q(a)). Then ∀ca , b6c. Hence ∀d ∈ Q(b); dc . This implies that a6d. Further we obtain that Q(b)¿a. To characterize fuzzy compactness, the following deCnition is useful. Denition 4.2. Let (LX ; ) be an L-ts, G∈ LX and ∈ M (L). ) ⊆ is called a Q -open cover of G, if for each x G , it follows that x 6 ) = {A | A ∈ )}. It is obvious that ) is a Q -open cover of 1 if and only if ) is an open cover of constant fuzzy set .) is a Q -open cover of G if and only if ) is an open cover of ∧ G ( ) if and only if G ∨( ))¿.
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Theorem 4.3. Let (LX ; ) be an L-ts and G ∈ LX . Then G is fuzzy compact if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each Q -open cover ) of G has a =nite subfamily B such that B is a Q! -open cover of G. Proof. (⇒) Suppose that G is fuzzy compact, ∈ M (L) and ) is a Q -open cover of G.For each ! ∈ ∗ (), take an a ∈ ∗ () such that ! ∈ ∗ (a). By Lemma 4.1(1) we can take b ∈ ∗ ( Q(!)) ∩ Q∗ (a). Take c ∈ ∗ (b) such that c ∈ Q∗ (a). Now we prove that ) is a c-RF of G. Infact, suppose that ) is not a c-RF of G. Then there exists a ∩ xc 6G such that xc 6 ) , i.e., c6 {A (x) | A ∈ )}. Hence a {A(x) | A ∈ )} and aG(x) . This shows that ) is not a Qa -open cover of G. This contradicts that ) is a Q -open cover of G. Thus ) is a b− -RF of G. Hence by fuzzy compactness of G we know that there exists a Cnite subfamily * of ) such that * is a b-RF of G. Now we shall prove that * is a Q! -open cover of G. Suppose that * is not a Q! -open cover of G. Then there exists a ∩ x! G such that x! *. Hence !G (x) and ∀A ∈ *, !A(x). Further we have that G(x) ∈ Q(!) and ∀A ∈ *, A (x) ∈ Q(!). This shows that b6G(x) and ∀A ∈ *, b6A (x). This contradicts that * is a b-RF of G. (⇐) Suppose that ∀ ∈ M (L), ∀! ∈ ∗ (), each Q -open cover of G has a Cnite subfamily which is a Q! -open cover of G. We shall prove that G is fuzzy compact. Let ∈ M (L) and + is an − -RF of G. Then there exists a ! ∈ ∗ () such that +is a !-RF of G. Take an a ∈ ∗ () such that ! ∈ ∗ (a). By Lemma 4.1(1) we can take a b ∈ ∗ ( Q(!)) ∩ Q∗ (a). Take c ∈ ∗ (b) such that c ∈ Q∗ (a). Now we prove that + is a Qb -open cover of G. In fact, suppose that + is not a Qb -open cover of G. Then there exists a ∩ xb G such that xb + . Hence bG (x) and ∀B ∈ +, bB (x), i.e., G(x)b and B(x)b . This implies that G(x) ∈ Q(b) and B(x) ∈ Q(b). By Lemma 4.1(2) we know that Q(b)¿!. Therefore !6G(x) and ∀B ∈ +; !6B(x). This contradicts that + is a !-RF of G. Thus + has a Cnite subfamily * which is a Qc -open cover of G. Now we shall prove that * is an -RF of G. Suppose * is not an -RF of G. Then there exists x 6G such that x 6 * . Hence 6G(x) that and 6 {A (x) | A ∈ *}. Thus we obtain that a6G(x) and a6 {A (x) | A ∈ *} Further cG (x) and c {A(x) | A ∈ *}. This contradicts that * is a Qc -open cover of G. So we complete the proof. From Theorem 4.3 we easily obtain the following two results. X . Then G is fuzzy compact if and only if ∀ ∈ M (L), Corollary 4.4. Let (LX ; ) be an L-ts and G ∈ L ∗ ∀! ∈ ()and for each ) ⊆ such that G ∨( ))¿, there exists a =nite subfamily + ⊆ ) such that G ∨( +)¿!.
Corollary 4.5. (LX ; ) is fuzzy compact if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each open cover ) of has a =nite subfamily B such that B is an open cover of !. Fuzzy compactness can also be characterized by means of O-cluster points of nets.
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Denition 4.6. Let ∈ M (L). A net {S(n) | n ∈ D} in LX is called an − -net if there exists n0 ∈ D such that ∀n¿n0 , V (S(n))6, where V (S(n)) denotes the height of S(n). A net {S(n)} in LX is said to be constant -net if the height of each S(n) is a constant value . Obviously each constant -net must be an − -net. Theorem 4.7. An L-set G is fuzzy compact in (LX ; ) if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each constant !-net quasi-coinciding with G has an O-cluster point x quasi-coinciding with G. Proof. Suppose that G is fuzzy compact. For ∈ M (L) and ! ∈ ∗ (), let {S(n) | n ∈ D} is a constant !-net quasi-coinciding with G. Suppose that S has no any O-cluster point x quasi-coinciding with G. Then for each x G , there exist Ux ∈ No (x ) and nx ∈ D such that ∀n¿nx ; S(n)Ux . Take ) = {Ux | x G }, then ) is a Q -open cover of G. Since G is fuzzy compact, ) has a Cnite subfamily + = {Uxi | i = 1; 2; : : : ; k} such that + is a Q! -open cover of G. Since D is a directed set, there exists n0 ∈ D such that n0 ¿nxi for each i6k. Thus we can obtain that ∀n¿n0 ; S(n) {Uxi | i = 1; 2; : : : ; k}. This contradicts that + is a Q! -open cover of G. Therefore S has at least an O-cluster point x G . Conversely suppose that ∀ ∈ M (L), ∀! ∈ ∗ (), each constant !-net quasi-coinciding with G has an O-cluster point x G . We now prove that G is fuzzy compact. Let ) be a Q -open cover of G. If there exists ! ∈ ∗ () such that each Cnite subfamily + of ) is not a Q! -open cover of G, then for eachCnite subfamily + of ), there exists S(+) ∈ M (LX ) with height ! such that S(+)G and S(+) +. Take S = {S(+) | +is a Cnite subfamily of )}: Then S is a constant !-net quasi-coinciding with G. By ! ∈ ∗ () we can take s ∈ ∗ () such that ! ∈ ∗ (s). Then S has an O-cluster point xs quasi-coinciding with G. Hence for each Cnite subfamily + of ) we have that xs +, in particular, xs B for each B ∈ ). But since ) be a Q -open cover of G, we know that there exists B ∈ ) such that xs 6B, this is a contradiction. So G is fuzzy compact. Theorem 4.8. An L-set G is fuzzy compact in (LX ; ) if and only if ∀ ∈ M (L), ∀! ∈ ∗ (), each !− -net quasi-coinciding with G has an O-cluster point x quasi-coinciding with G. Proof. The su?ciency is obvious. We need only prove the necessity. Let G be fuzzy compact, ∈ M (L); ! ∈ ∗ () and {S(n) | n ∈ D} an !− -net quasi-coinciding with G. Then there exists n0 ∈ D such that ∀n¿n0 ; S(n)6!. Put E = {n ∈ D | n¿n0 } and T = {T (n) | n ∈ E; V (T (n)) = !; support point of T (n) is the same as S(n)}: Then T is a constant !-net quasi-coinciding with G . Let x is an O-cluster point of T . It is easy to see that x is also an O-cluster point of S. Remark 4.9. Because KubiMak deCnition of fuzzy compactness depends on each elements a ∈ L, but the deCnition of nets in [5,12] was only deCned in M (L), we do not know whether KubiMak deCnition of fuzzy compactness can be characterized by O-cluster points. We leave it as an open problem.
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Acknowledgements The authors would like to thank S.E. Rodabaugh and referees for their valuable comments and suggestions. References [1] J.J. Chadwick, A generalized form of compactness in fuzzy topological spaces, J. Math. Anal. Appl. 162 (1991) 92–110. [2] G. Gierz, et al., A Compendium of Continuous Lattices, Springer, Berlin, 1980. [3] B. Hutton, Uniformities in fuzzy topological spaces, J. Math. Anal. Appl. 58 (1977) 559–571. [4] T. KubiMak, The topological modiCcation of the L-fuzzy unit interval, in: S.E. Rodabaugh, E.P. Klement, U. HUohle (Eds.), Applications of Category Theory to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1992, pp. 275–305 (Chap. 11). [5] Y.-M. Liu, M.K. Luo, Advances in Fuzzy Systems—Applications and Theory, in: Fuzzy Topology, Vol. 9, World ScientiCc Publishing Co. Pte. Ltd, Singapore, 1997. [6] R. Lowen, A comparison of diRerent compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446–454. [7] P.-M. Pu, Y.-M. Liu, Fuzzy topology I, neighborhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571–599. [8] F.-G. Shi, Theory and applications of L -nest sets and L -nest sets, Fuzzy Systems Math. 4 (1995) 65–72 (in Chinese). [9] F.-G. Shi, A note on fuzzy compactness in L-topological spaces, Fuzzy Sets and Systems 119 (2001) 547–548. [10] G.-J. Wang, A new fuzzy compactness deCned by fuzzy nets, J. Math. Anal. Appl. 94 (1983) 1–23. [11] G.-J. Wang, On the structure of fuzzy lattices, Acta Math. Sinica 4 (1986) 539–543. [12] G.-J. Wang, Theory of L-fuzzy Topological Space, Shaanxi Normal University Publishers, Xian, 1988 (in Chinese). [13] J.J. Xu, On fuzzy compactness in L-fuzzy topological spaces, Chinese Quart. J. Math. 2 (1990) 104–105 (in Chinese). [14] D.S. Zhao, The N-compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128 (1987) 64–70.
