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Inverse Limits in Category Ltop http://www.paper.edu.cn Fuzzy Sets and Systems 109 (2000) 291–299 Inverse limits in category LTop (II) 1 Sheng-Gang Li Department of Mathematics, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China Received October 1995; received in revised form October 1997 Abstract This paper is a sequel to Li (1999). For some classes F of L-valued Zadeh functions, we will discuss two problems connected with inverse limits in LTop, K1HLTop and K2HLTop: (1) Will the condition that all L-valued Zadeh functions between factor L-fts of two given inverse systems are in F imply that the induced limit LTop-morphism is in F? (2) Will the condition that all bonding L-valued Zadeh functions are in F imply that all projective L-valued Zadeh functions are in F? c 2000 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy topology; Inverse limit; Projective L-valued Zadeh-function; Mapping between inverse systems; Limit LTop-morphism. 1. Introduction and preliminaries same problems in the subcategories K1HLTop and K2HLTop of LTop. Similar to the case of point set topology, the theory Our terminology and notions coincide with those of inverse limit of fuzzy topological spaces is an es- used in [9]. However, we have some other concepts sential part of fuzzy topology. In [9], we studied the to ÿx. structures of inverse limits in LTop, and investigated An L-fuzzy set A ∈ LX is said to be connected a series of properties of such limits, and we also de- [16] in L-fts (LX ;) if there exists no pair (B; C) ∈ X X ÿned the mappings between arbitrary inverse systems L × L −{(0X ; 0X )} satisfying A = B ∨ C and − − and discussed some of their properties for the case of A ∧ B = B ∧ A =0X . A connected L-fuzzy set A LTop. in L-fts (LX ;) is said to be a component of an L- The present paper is a sequel to [9]. Here we will fuzzy set E ∈ LX if A ⊂ E and there is no connected further discuss the inverse limits of L-fuzzy topolog- L-fuzzy set F satisfying A6 F 6E and A 6= F.An X ical spaces, i.e. we will show how the projective L- L-fts (L ;) is said to be connected if 1X is con- valued Zadeh-functions depend on bonding L-valued nected in (LX ;), or equivalently, there is no pair Zadeh-functions in LTop and how the limit LTop- (A; B) ∈ × −{(0X ; 0X )} satisfying A ∨ B =1X morphism depends on the mapping of inverse sys- and A ∧ B =0X . tems inducing it in LTop. We also investigated the Deÿnition 1.1. An LTop-morphism F :(LX ;) → Y 1 This work was supported by the National Natural Science (L ;) is said to be Foundation of China (Grant No. 19701020). (1) closed if F(A) ∈ for every A ∈ ; 0165-0114/00/$ – see front matter c 2000 Elsevier Science B.V. All rights reserved. 转载 PII: S 0165-0114(98)00002-5 中国科技论文在线 http://www.paper.edu.cn 292 S.-G. Li / Fuzzy Sets and Systems 109 (2000) 291–299 (2) open if for every B ∈ LY and every A ∈ satis- Hausdor and compact), and that (LX ;) is fying A¿F−1(B), there exists a C ∈ such that connected and weakly induced implies that −1 C¿B and F (C)6 A; (X; ÃL()) is connected. −1 X (3) (Xu [21]) perfect if it is closed and F (e)isN- (2) (X; T) is connected if and only if (L ;!L(T)) compact in (LX ;) for every e ∈ M(LY ). is connected. X (3) Every component of 1X in (L ;) is closed crisp Lemma 1.2. Let (LX ;) be a Hausdor and F- set. Y compact L-fts,(L ;) be a T1 L-fts, and Lemma 1.5 (Xu [21]). The Cartesian product of X Y F :(L ;) −→ (L ;) a family of perfect LTop-morphisms is perfect. be an LTop-morphism. Then F−1(e) is N-compact in (LX ;) for each e ∈ M(LY ). 2. Some properties of limit mappings Y Y Proof. Let e = y ∈ M(L ). Since (L ;)isT1, there In this section, we use all symbols as used in [9, exists a y ∈ such that ¿ . Since F is contin- Section 4], and determine when, for a given class of uous and (LX ;)isF-compact, F−1(y )= is E LTop-morphisms F, the condition that all Fi belong closed, and thus F-compact (cf. [9, Lemma 1.4(3)]), to F implies that lim{’; Fi} belongs to F. We will −1 ← where E = f (y). From the deÿnition of F-compact consider the following classes of LTop-morphisms: −1 L-fuzzy set, we can see that E = F (e)isF- (1) injective, (2) surjective, (3) bijective, (4) mono- X compact. Since (L ;) is Hausdor , by [9, Corollary tone, (5) con uent, (6) N-compact, (7) F-compact, X E 4.5(4)], (L ; ) is Hausdor , and so is (L ; | E). (8) block N-compact, (9) open and (10) perfect. X E Since E is F-compact in (L ;), (L ; |E)isan E F-compact L -fts. By [9, Lemma 1.4(2)], (L ; |E) Theorem 2.1. If all F are injections (resp., bijec- −1 i is N-compact. It follows that F (e)= E is N- tions, LTop-isomorphisms), then F = lim{’; Fi} is compact in (LX ;). ← an injection (resp., bijection, LTop-isomorphism). X Remark 1.3. For every L-fts (L ;), we have a topo- Proof. Similar to [3, Lemma 2.5.9], we can show logical space (X; ÃL()), where ÃL() is the co-topology Theorem 2.1 for the cases of injection and bijection. on X generated by {{x ∈ X | A(x)¿r}|r ∈ L; A ∈ } Suppose that all Fi are LTop-isomorphisms, then F = as a subbase. Conversely, for every topological lim{’; F } is a bijection. For every i ∈ I and every X ← i space (X; T), we have an L-fts (L ;!L()), where X V’(i) ∈ ’(i), by the deÿnition of F (see [9, Theorem !L()istheL-fuzzy co-topology on L generated 4.3]), we have F ◦ (P |Y )=(P |Y ) ◦ F, and thus by {[r] ∨ V | V ∈ T;r∈ L} as a base. It was shown i ’(i) i that ÃL ◦ !L(T)=T for every co-topology T; and h i −1 !L◦ÃL() ⊃ for every L-fuzzy co-topology (cf. [16, F (P’(i)|Y ) (V’(i)) Theorem 2.11.10]). The functor YL : LTop → Top, h i X = F (P |Y )−1(F−1 ◦ F (V )) deÿned by YL((L ;))=(X; ÃL()) for every L-fts ’(i) i i ’(i) X h i (L ;) and YL(F)=f for every LTop-morphism F, =(P |Y )−1 F (V ) : and the functor GL : Top → LTop; deÿned by i i ’(i) X GL((X; T))=(L ;!L(T)) for every topological Since F is an LTop-isomorphism, F (V ) ∈ , space (X; T) and GL(f)=F for every LTop- i i ’(i) i morphism f, are called Lowen functors (cf. [11]). and thus h i F (P |Y )−1(V ) Lemma 1.4 (Wang [16]). Suppose that 1 ∈ M(L); ’(i) ’(i) X (X; T) is a topological space and (L ;) is an L-fts. =(P |Y )−1[F (V )] ∈ |Y: Then the following statements hold: i i ’(i) X (1) That (L ;) is T1 (resp., Hausdor and F- Since F is bijection, we can show that F is compact) implies that (X; ÃL()) is T1 (resp., intersection-preserving. It follows from [9, Theorem 中国科技论文在线 http://www.paper.edu.cn S.-G. Li / Fuzzy Sets and Systems 109 (2000) 291–299 293 3.1 and Theorem 4.3] that F is an LTop-isomorphism. a T1 L-fts and F is surjective, then F = lim{’; F } i ← i is surjective. (2) If {’; Fi} is limit-exact and Fi is surjective for Corollary 2.2. If J is a subset conÿnal in I; then the each i ∈ I; then F = lim{’; F } is surjective. inverse limit of D| J op is LTop-isomorphic with the ← i Y inverse limit (L ;|Y ) of D. Proof. (1) Take an ∈ M(L). Then, by [9, Corollary 4.5(4)], Q ◦ D(i) ∈|K2HL Top| for every i ∈ I, and Deÿnition 2.3. (1) The diagram of sets and mappings Q◦D(i)isaT1 L-fts for every i ∈ I. By Lemma 1.4(1), h ◦ Q ◦ D(i) is a Hausdor and compact space for 1 YLQ A −−−−−−−−−−−−−−−−−−→ D every i ∈ I, and Y ◦ Q ◦ D(i)isaT space for ev- LQ 1 ery i ∈ I. Since all F are surjective, i.e. all f are i i surjective, by [3, Theorem 3.2.14], lim{’; f } is sur- k1 h2 ← i jective, i.e. lim{’; F }, which is exactly the L-valued y y ← i Zadeh-function induced by lim{’; f }, is surjective. C −−−−−−−−−−−−−−−−−−→ B ← i k2 (2) It suces to show that f is surjective. Suppose is said to be exact if it is commutative and that y = {yi} ∈ Y . Take an i ∈ I, by the assump- −1 −1 i∈I h1 (d) ∩ k1 (c) 6= ∅ whenever h2(d)=k2(c), or ’(i) −1 −1 2 tion that Fi is surjective, there exists an x ∈ X’(i) equivalently, k1h1 (E)=k2 h (E) for every E ⊂ D. such that f (x’(i))=y(i) =(p |Y )(y) ∈ X . Since (2) The mapping {’; Fi} from inverse system D to i i i inverse system D is said to be exact if, for each pair {’; Fi} is limit-exact, there exists a y ∈ Y such that i; j ∈ I with i6j, the following diagram is exact: f(y)=y, i.e.
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