Inverse Limits in Category Ltop
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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. f is surjective.
f j X −−−−−−−−−−−−−−−−−−→ X By Corollary 2.2, [9, Lemma 1.1] and [13, p. 58], ’(j) j we have f f Lemma 2.5. If cf (I)6ℵ0 and {’; F } is exact, then ’(j);’(i) j; i i {’; F } is limit-exact. y y i Z X’(i) −−−−−−−−−−−−−−−−−−→ X i Deÿnition 2.6. Let (L ;)beanL-fts, ∈ M(L) and f E i E ⊂ Z.If(L ; | E) is a Hausdor and N-compact it is said to be limit-exact if, for every i ∈ I, the (resp., Hausdor , N-compact and connected) L -fts, following diagram is exact: then E is called a block N-compact L-fuzzy set (resp., continuum) in (LZ ;). f Y −−−−−−−−−−−−−−−−−−→ Y Deÿnition 2.7. An LTop-morphism G : A → B is said to be −1 p |Y p |Y (1) monotone if, for every molecule b in B, G (b) ’(i) i y y is connected in A; (2) con uent if, for every continuum Q of B and every X’(i) −−−−−−−−−−−−−−−−−−→ X i −1 f component C of G (Q), G(C)=Q; i (3) N-compact (resp., F-compact, block N-compact) Even if L = {0; 1} and I = I = N, Theorem if, for every N-compact (resp., F-compact, 2.1 is not true for surjective LTop-morphisms (cf. block N-compact) L-fuzzy set Q in B, G−1(Q) [3, Exercise 2.5.C]). However, we have the following is N-compact (resp., F-compact, block N- theorem. compact) in A.
Theorem 2.4. (1) If, for every i ∈ I and every An L-fts (LZ ;) is said to be fully stratiÿed [10] if i ∈ I; D(i) ∈|K2HLTop| (see Section 4),D(i) is [ ] ∈ for every ∈ L. 中国科技论文在线 http://www.paper.edu.cn
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Lemma 2.8. If (LZ ;) is a fully stratiÿed and Haus- compact and connected L-fuzzy set in D(’(i)) since Z dor L-fts, A is an F-compact L-fuzzy set in (L ;); Fi is monotone. Consider the inverse system then A ∈ : ˜ op D : I → K1HLQTop; ∗ − ∗ − Proof. Let x ∈ ÿ (A ), where ÿ (A ) is the standard − E minimal family of A (see [18, p. 356]). It suces to ˜ i ˜ where D(i)=(L ; ’(i) |Ei) and D(i6j)isthe show that x6A. L -valued Zadeh-function induced by the mapping ∗ − Since x ∈ ÿ (A ), there exists a molecular net S = S ◦ D(’(i)6’(j))|E , (cf. [9, Corollary 4.5(4) and n i {xt(n): n ∈ D} in A such that S → x (see [18, Theo- E ˜ n Theorem 2.1]). Let (L ;) be the inverse limit of D, rem 4.22]), where t(n) is the height of molecule xt(n). −1 T F (y )= T , then we can show that (L ; |T)is Let E ^ _ LTop-isomorphic with (L ;). By Theorems 4.1 and E = t(n); 4.4, (L ;)isN-compact and connected, and so is T m∈D n¿m (L ; |T). Therefore F = lim{’; F } is monotone. ← i Wthen we can show that ¿. Suppose thatW , i.e. n¿m t(n) for some m ∈ D. Let d = n¿m t(n), Z Even if L = {0; 1} and I = I = N, Theorem 2.1 then x [d]. Since (L ;) is fully stratiÿed, [d]isa does not hold for the family of N-compact (resp., F- remote neighborhood of x. Obviously, S is eventually compact) LTop-morphisms (cf. [9, Theorem 4.8]). in [d], i.e. S → x is not true. This is a contradiction. W However we have the following result. Hence ¿, i.e. n¿m t(n)¿ for all m ∈ D. For every pair (r; m) ∈ ÿ∗() × D, by the deÿnition of ÿ∗() [18], there exists an n(r; m) ∈ D such that Theorem 2.10. Suppose that (LY ; |Y ) is Hausdor ∗ n(r; m)¿m and t(n(r; m))¿r: For the set ÿ () × D, and fully stratiÿed, and Y ∈ : Then that all Fi are we deÿne (r1;m1)6(r2;m2)i r16 r2 and m16 m2. N-compact (resp., F-compact, block N-compact) im- Then ÿ∗()×D is a directed set since ÿ∗() is directed plies that F = lim{’; F } is N-compact (resp., F- ← i (cf. [22, Theorem 1.7]). Hence T = S ◦ , where compact, block N-compact). : ÿ∗() × D → D Proof. We only prove Theorem 2.10 for the case is deÿned by ((r; m)) = n(r; m), is a subnet of S, (cf. of N-compact. Let Ch be an N-compacti L-fuzzy [18, Deÿnition 4.24]). Apparently, T is a -net [22] Q set in (LY ; |Y ), A = F−1(P (C)) Y . Then ∗ i∈I i i in A and T → x for every ∈ ÿ (). Since A is F- ∗ F−1(C)6 A. For every i ∈ I, since P (C)isN- compact, for every ∈ ÿ (), T has a cluster z in A. i From the assumption that (LZ ;) is Hausdor , we can compact (see [22, Theorem 4.11]) and Fi is N- W −1 compact, F (P (C)) is N-compact; since Y ∈ , show that x = z, and thus x = z = ∈ÿ∗() z 6 A. i i This completes the proof of Lemma 2.8. A is N-compact (cf. [22, Theorems 5.3 and 4.9]). From Lemma 2.8 and the continuity of F, it follows Even if L = {0; 1} and I = I = N, Theorem 2.1 is that F−1(C) ∈ |Y . Since F−1(C)6A, F−1(C)is not true for the family of monotone LTop-morphisms, N-compact, i.e. F is N-compact. (cf. [13, Example 7]). However, we have the following theorem. Theorem 2.11. If {’; Fi} is limit-exact and Fi is open for every i ∈ I; then F = lim{’; Fi} is open. Theorem 2.9. Suppose that, for every i ∈ I; D(i) is ← Hausdor and F-compact and, for every i ∈ I; D(i) Proof. Since the family of all open LTop-morphisms is T1 and F is monotone. Then F = lim{’; F } is i ← i is isomorphism-closed (cf. Deÿnition 3.2), by Corol- monotone. lary 2.2, we can assume that I = ’(I). Let B ∈ LY and A ∈ |Y satisfying A¿F−1(B), we have to show Y i Proof. Let y ∈ M(L ), where y = {y }i∈I . For that there exists a C ∈ |Y satisfying C¿B and −1 i −1 every i ∈ I, by Lemma 1.2, E = F (y )isanN- F (C)6A. i i 中国科技论文在线 http://www.paper.edu.cn
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op ∗ By [9, TheoremV 3.1], it does lose generality to as- of D|’(I) , then, by [9, Proposition 3.5], Y ∗ ∈ , −1 sume that A = i∈I (P’(i)|Y ) (W’(i)), where W’(i) ∈ and thus for each i ∈ I. Since {’; F } is limit-exact, ’(i) i G |Y ∗ :(LY ∗ ;∗|Y ∗) → (LX ; |Y ); F−1 (P |Y )(B) =(P |Y ) F−1(B) i i ’(i) the restriction of G to Y ∗, is a closed LTop- 6 (P |Y )(A)6W ; morphism. Since each F is perfect, for each ’(i) ’(i) i Q e ∈ M(LY ), (G |Y ∗)−1(e)=( F−1(P (e)))|Y ∗ since F is open, there exists a C ∈ , such i∈I i i i i i is N-compact. Hence G |Y ∗ is a perfect LTop- that C ¿(P |Y )(B) and F−1(C )6W . Let C = V i i i i ’(i) morphism. By Corollary 2.2, there exists an LTop- −1 i∈I (Pi |Y ) (Ci), then we can see that C ∈ |Y isomorphism and C¿B: Since {’; Fi} is limit-exact, we have Y Y ∗ ∗ ∗ ^ H :(L ;|Y ) → (L ; |Y ): −1 −1 −1 F (C)= F [(Pi |Y ) (Ci)] It follows that F = lim{’; F } =(G |Y ∗) ◦ H is a per- i∈ I i ^ ← −1 fect LTop-morphism. 6 (P’(i)|Y ) ◦ (P’(i)|Y ) i∈ I −1 −1 ◦ F ◦ (Pi |Y ) (Ci) 3. Some properties of projections ^ = (P |Y )−1 ◦ F−1 ◦ (P |Y ) ’ (i) ’(i) i Similar to Section 2, in this section, we investigate i∈ I how the projective L-valued Zadeh-functions P |Y de- −1 i ◦ (Pi |Y ) (Ci) pend on bonding L-valued Zadeh-functions F .We ^ ji 6 (P |Y )−1 ◦ F−1 (C ) will use all symbols as used in [9, Theorem 2.3]. ’(i) ’(i) i First, by Theorem 2.1 and [9, Theorem 4.8], we i∈ I ^ have −1 6 (P’(i)|Y ) (W’(i))=A: i∈ I Theorem 3.1. If all Fji are injections (resp., bijec- tions, LTop-isomorphisms), then all Pi |Y are injec- This completes the proof of Theorem 2.11. tions (resp., bijections, LTop-isomorphisms). Corollary 2.12. If cf (I)6ℵ ; {’; F } is exact and F 0 i i Deÿnition 3.2. Let A be a category, F be a family is open for every i ∈ I; then F = lim{’; F } is open. ← i of A-morphisms. If every A-morphism, which is the composition of an A-isomorphism and some f ∈ F; Corollary 2.12 follows from Lemma 2.5 and The- belongs to F; then we say F is isomorphism-closed. orem 2.11. It should be noticed that Corollary 2.12 is not true for the case of cf (I) ¿ ℵ0, even if L = {0; 1}, Theorem 3.3. Let F be an isomorphism-closed fam- (cf. [13, Example 4]). ily of LTop-morphisms, then the following statement (1) implies the statement (2): Theorem 2.13. If D(i) is Hausdor for each i ∈ I; (1) If the mapping {’; Fi} from an inverse system D and Fi is a perfect LTop-morphism for each i ∈ I: in LTop to an inverse system D in LTop is limit- Then F = lim{’; Fi} is a perfect LTop-morphism. exact, and all F ∈ F, then lim{’; F }∈F. ← i ← i ˜ op Q (2) For every inverse sequence D : N → LTop, X ∗ ∗ X Proof. Let G = i∈I Fi :(L ; ) → (L ; ), where N is the set of all natural numbers, if all X ∗ ∗ where (L ; ) is the product L-fts of the family bonding L-valued Zadeh-functions Fmn belong to F; then all projective L-valued Zadeh-functions {D(’(i))} i∈I . Then, by Lemma 1.5, G is a perfect ∗ ˜ ˜ ˜ ˜ LTop-morphism. Let (LY ;∗|Y ∗) be the inverse limit Pn|Y of the limit D to D(n) belong to F. 中国科技论文在线 http://www.paper.edu.cn
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Proof. Assume (1). For each n ∈ N, by [9, Theorem are compact. However, the inverse limit of D is not 4.8], there exists an inverse system compact, and thus each Pn|Y is not compact. D˜ : J op → LTop; n n Question 3.8 Whether Theorems 2.1 and 3.1 are true ˜ ˜ a mapping { ;Fm} from D to Dn, and an LTop- for con uent LTop-morphism? ˜ ˜ ˜ ˜ isomorphism H : An → D(n), such that Pn|Y = H ◦ lim{ ;Fm}; (m¿n), where Jn = {m ∈ N |m¿n} and ← 4. Inverse limits in K HLTop and K HLTop ˜ ˜ 1 2 An is the limit of Dn. It is easy to verify that { ;Fm} is exact. By Lemma 2.5, { ;Fm} is limit-exact, and by In [9] and previous sections, we have studied the (1), lim{ ;Fm}∈F. Since F is isomorphism-closed, inverse limits and a series of their properties in LTop. ← ˜ ˜ In this section, we will do the same thing in K HLTop Pn|Y = H ◦ lim{ ;Fm}∈F. Hence (1) ⇒ (2). 1 ← and K2HLTop; where K1HLTop (resp., K2HLTop) denotes the full subcategory of LTop whose objects Theorem 3.4. If cf (I)6ℵ and all F are surjective 0 ji are all N-compact (resp., F-compact) Hausdor L-fts. (resp., surjective and open), then all Pi |Y are surjec- tive (resp., surjective and open). Theorem 4.1. Let A ∈{K1HLTop; K2HLTop}; Proof. Theorem 3.4 follows from Lemma 2.5, Theo- D : I op → A rem 2.4(2), Theorem 2.11, [9, Lemma 1.1] and The- X orem 3.3. be an inverse system in A; where D(i)=(L i ;i) and D(i6 j)=Fji : D(j) → D(i) for all i; j ∈ I sat- Remark 3.5. Even if L = {0; 1}, Theorem 3.4 does isfying i6j;(A; Pi) be the A-product of {D(i)}i∈I ; not hold for arbitrary inverse system, (see, [3, Exercise i.e. A =(LX ;) is the product L-fts of the family 2.5.A(b)] and [13, Example 3]). {D(i)}i∈I ; and Pi : A → D(i) are projective L-valued Zadeh-functions; and Y = {y ∈ X |Fji ◦ Pj(y )= Theorem 3.6. If all Fji are block N-compact, then Pi(y ) whenever ∈ M(L);i;j∈ I and i6 j}. Then Y all Pi|Y are block N-compact. ((L ;|Y );Pi|Y ) is the inverse limit of D. In other words, the inverse limit of D in A is existent and it Proof. Let i ∈ I and Ji = {j ∈ I | j¿i}. Take a is the same as in LTop. in D(i), then block N-compact L-fuzzy set Q = Ei i op Q ◦ DE : Ji → LQTop is an inverse system in LQTop, Proof. Since Hausdor property, N-compactness and E i and (L ; |E) is the inverse limit of Q◦DE in LQTop, F-compactness are all multiplicative and hereditary −1 where E =(pi|Y ) (Ei), (cf. [9, Proposition 3.10 with respect to closed crisp set (cf. [22, Theorems 4.9 and Theorem 4.4]). Since all Fji are block N-compact, and 5.3]), Theorem 4.1 follows from [9, Theorem 2.3 i all Q ◦ DE(j) are N-compact and Hausdor . It follows and Proprosition 3.5]. E from Theorem 4.1 that (L ; |E) ∈|K1HLQTop|, −1 op op i.e. E =(Pi|Y ) (Q) is a block N-compact L-fuzzy In the following, D : I → A and D : I → A Y set in (L ;|Y ). Therefore Pi|Y is block N-compact. always denote two inverse systems satisfying D(i)= This completes the proof of Theorem 3.6. Xi X (L ;i); D(i)=(L i ; i), D(i6j)=Fji and D(i6j) = Fj;i for every i; j ∈ I and every i; j ∈ I satisfying Remark 3.7. Even if L = {0; 1} and I = N, Theorem i6j and i6j, where 3.6 hold for neither monotone LTop-morphisms (cf.
[13, Example 9]) nor N-compact LTop-morphisms. A ∈{K1HLTop; K2HLTop}; Consider the inverse system D : N op → Top deÿned Y Y by D(n)=(Xn; Tn); where Xn =[0; 1); Tn = {∅;Xn} (L ;|Y ) (resp., (L ; |Y )) denotes the inverse limit ∪{[1 − 1=2k;1)|k =1; 2; :::; n}; and D(n6m)= of D (resp., D), where (LX ;) (resp., (LX ; )) is the Fm; n = id[0;1), the identity mapping of [0,1) onto it- product L-fts of the family {D(i)}i∈I (resp., the fam- self. Obviously, all D(n) are compact, and all Fm; n ily {D(i)} i∈I ), Y ⊂ X and Y ⊂ X ; {’; Fi} denotes a 中国科技论文在线 http://www.paper.edu.cn
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mapping from inverse system D to inverse system D; Theorem 4.5. If all Fi (resp., all Fji) are surjective, F = lim{’; F } denotes the limit LTop-morphism in- then lim{’; F } is (resp., all Pi|Y are) surjective. ← i ← i duced by {’; Fi}. Remark 4.6. Consider an inverse system Remark 4.2. It can be seen that Theorems 4.7 and D : N op → KHTop 4.8 in [9] and Theorem 3.3 hold for K1HLTop and K HLTop. 2 where D(n)={0}∪{1=k |k¿n} is the subspace of R, the real line with the usual co-topology, and Theorem 4.3. If all D(i) 6= ∅; then Y 6= ∅. D(n6m)=Fm; n is the embedding mapping for all m; n ∈ N satisfying m¿n. Obviously, all F are open Proof. For all i; j ∈ I satisfying i6j; let m; n and Y = {(0; 0;:::)} is a single set, but Pn|Y is not _ open for any n ∈ N. This example and Theorem 3.3 X Aij = {x ∈ M(L ) | Pi(x )=Fji ◦ Pj(x )}: imply that Theorem 4.5 does not hold for the family of open LTop-morphisms, even if L = {0; 1};’= idN
Then Aij = Eij for some Eij ⊂ X: Similar to [9, and I = I = N, where idN is the identity mapping of
Proposition 3.5], we can show that all Aij = Eij N onto itself. However, we have the following result. are closed, and thus F-compact. Take an ∈ M(L),
then, by [9, Corollary 4.5(4)] and Lemma 1.4(1), Theorem 4.7. If all Fji are surjective and open, then Y YL ◦ Q((L ;|Y )) is a compact and Hausdor space, all Pi|Y are surjective and open. Y and allT Eij are compact in YL ◦ Q((L ;|Y )). Since X Y = {Eij |i; j ∈ I; i6j} it suces, by point set Proof. Let i ∈ I; B ∈ L i and A ∈ |Y satisfying −1 topology, to show that Eij 6= ∅ for all i; j ∈ I satisfy- A¿(Pi|Y ) (B), we have to show that there exists a −1 ing i6j. C ∈ i satisfying C¿B and (Pi|Y ) (C)6A. j For each ∈ M(L) and each x ∈ Xj, let By Corollary 2.2 and [9, Theorem 3.1], it does lose j i j supp Fji((x ) )=x , the support of Fji((x ) ), and generality to assume that i j take an x ∈ X such that pi(x)=x and pj(x)=x , ^ −1 then x 6 Aij, i.e. Eij 6= ∅. This completes the proof A = (Pj|Y ) (Wj); of Theorem 4.3. j∈ Ji
Theorem 4.4. If 1 ∈ M(L) and all D(i) are con- where Ji = {j¿i | j ∈ I} and Wj ∈ j for each j ∈ Ji. nected, then (LY ;|Y ) is connected. For each j ∈ Ji, since Pj|Y is surjective (see Theo- −1 −1 −1 rem 4.5) and (Pj|Y ) ◦ Fji (B)=(Pi|Y ) (B)6 A, −1 Proof. By [9, Lemma 1.4(2)] and Lemma 1.4(1), we have Fji (B)6(Pj|Y )(A)6Wj; since Fji is open, there exists a C ∈ , such that C ¿B and ij V i ij Y ◦ D : I op → KHTop F−1(C )6W . Let C = C , then C ∈ ;C¿B L ji ij j j∈Ji ij i and is an inverse system, where KHTop is the category −1 of all compact Hausdor spaces and continuous map- (Pi|Y ) (C) ^ pings, and all YL ◦ D(i) are connected. Similar to [9, −1 = (Pi|Y ) (Cij) Theorem 2.3], we can show that (Y; ÃL()|Y )isthe j∈Ji inverse limit of YL ◦ D. By [3, Theorem 6.1.18] and ^ Y −1 −1 Lemma 1.4(2), (L ;!L ◦ ÃL()|Y ) is connected, and = (Pj|Y ) ◦ (Fji (Cij)) Y by Remark 1.3, (L ;|Y ) is connected. j∈Ji ^ 6 (P |Y )−1(W )=A: From [3, Theorem 3.2.14], [2, Corollary 3.9, p. 218] j j j∈J and the Proof of Theorem 2.4(1), we can see that the i following Theorem 4.5 holds. This completes the proof of Theorem 4.7. 中国科技论文在线 http://www.paper.edu.cn
298 S.-G. Li / Fuzzy Sets and Systems 109 (2000) 291–299
−1 Theorem 4.8. If all Fi (resp., Fji) are closed, then K ⊂ X . Then, we can verify that K 6F ( Q) and lim{’; Fi} is (resp., all Pi|Y are) perfect. F( K )= Q. This completes the proof of Theorem ← 4.10. Proof. Theorem 4.8 follows from Lemma 1.2, Theo- Y rem 2.13, Remark 4.2 and [9, Theorem 4.8]. Theorem 4.11. (1) If (L ; |Y ) is fully stratiÿed and all Fi are N-compact (resp.;F-compact, block N- compact), then lim{’; F } = F is N-compact (resp., Theorem 2.9, Remark 4.2 and [9, Theorem 4.8] ← i imply F-compact, block N-compact). X (2) If all (L i ;i) are fully stratiÿed and all Fji Theorem 4.9. If all Fi (resp., Fji) are monotone, then are N-compact (resp., F-compact), then all Pi|Y are lim{’; F } is (resp., all Pi|Y are) monotone. N-compact (resp., F-compact). ← i Proof. (1) follows from [9, Proposition 3.5] and Theorem 4.10. If all F (resp., Fji) are con uent, i Theorem 2.10, and (2) follows from Remark 4.2, then lim{’; F } is (resp.; all Pi|Y are) con uent. ← i [9, Theorem 4.8] and (1).
Proof. Suppose that all Fi are con uent. Then, by Remark 4.2 and [9, Theorem 4.8], it suces to show References that lim{’; F } = F is con uent. Take a continuum ← i Y Q ∈ L , we will show that there exists a continuum [1] J.J. Charatonik, W.J. Charatonik, On projections and limit −1 mappings of inverse systems of compact spaces, Topology K 6F ( Q) such that F( K )= Q, which suf- ÿces to close the Proof (cf. [9, Theorem 3.8(3)]). Appl. 16 (1983) 1–9. [2] S. Eilenberg, N. Steenrod, Foundations of Algebraic For each i ∈ I, put Q = P ( Q). Then Q is a i i i Topology, Princeton University Press, Princeton, 1952. continuum because both N-compactness and connect- [3] R. Engelking, General topology, Polish Scientiÿc Publishers, Warszawa, 1977. edness are invariants of LTop-morphisms; since Fi is −1 [4] K.R. Gentry, Some properties of induced map, Fund. Math. con uent, there exists a component A of F ( Q ) i i i 66 (1969) 55–59. in D(’(i)) such that F (A )= Q . By the deÿnition i i i [5] H. Herrilich, G.E. Strecker, Category Theory, Heldermann of component, we can see that Ai is also a compo- Verlag, Berlin, 1979. −1 [6] S.-G. Li, A categorical approach to N-compactness and nent of F ( Q )inQ ◦ D(’(i)). By [9, Corollary i i H()-complete regularity, J. Fuzzy Mathematics 3 (2) (1995) 4.5(4) and Lemma 1.4(2)], Q ◦ D(’(i)) is Hausdor 411–418. and N-compact; by Lemma 1.4(3), we can show that [7] S.-G. Li, Minimal Hausdor TMLs, J. Fuzzy Math. 4 (2) A = K is closed in Q ◦ D(’(i)), and thus A is N- (1996) 237–244. i ’(i) i [8] S.-G. Li, Some results on I˜(L) and R˜(L), Fuzzy Sets and compact. Therefore, Ai is continuum. For all i; j ∈ I satisfying i6j, let Systems 82 (1996) 103–110. [9] S.-G. Li, Inverse limits in category LTop (I), Fuzzy Sets and K Systems 108 (1999) 235–241. D˜ (i)=(L ’(i) ; |K ); ’(i) ’(i) [10] Y.M. Liu, M.K. Luo, Induced spaces and fuzzy Stone– CechÄ compactiÿcation, Scientia Sinica (Ser. A) 30 (1987) ˜ D(’(i)6’(j)) = F’(j);’(i)|K’(j): 1034–1044. [11] R. Lowen, Fuzzy topological spaces and fuzzy compactness, Then we have a inverse system D˜ : I op → LTop. Let J. Math. Anal. Appl. 56 (1976) 621–633. (LK ;) be the inverse limit of inverse system [12] R. Nakagava, Categorical topology, in: K. Morita, J. Nagata, (Eds.), Topics in General Topology, Elsevier, Amsterdam, ˜ op 1989, pp. 563–623. Q ◦ D : I → K1HLQTop [13] E. Puzio, Limit mappings and projections of inverse systems, (cf. [9, Lemma 1.4(3) and Corollary 4.5(4)]. Then, Fund. Math. 80 (1973) 57–73. K K [14] G. Preuss, Theory of Topological Structures, Reidel, by Theorem 4.4, (L ;) is connected and (L ;) ∈ Dordrecht, Holland, 1988. |K1HLQTop|, i.e. K is a continuum. It does lose [15] G.J. Wang, Order-homomorphisms on fuzzes, Fuzzy Sets and generality to assume that ’ : I → I is a bijection and Systems 12 (1984) 281–288. 中国科技论文在线 http://www.paper.edu.cn
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