A Note on inverse limits of continuous images of arcs

Publicacions Matem`atiques, Vol 43 (1999), 485–499. A NOTE ON INVERSE LIMITS OF CONTINUOUS IMAGES OF ARCS Ivan Lonˇ car Abstract The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X={Xa,p ab,A}is an inverse system of continuous images of arcs with monotone bonding mappings such that cf(card(A)) =ω1, then X= lim Xis a continuous image of an arc if and only if each proper subsystem {Xa,p ab,B} of Xwith cf(card(B)) = ω1has the limit which is a continuous image of an arc (Theorem 18). 1. Inverse limits of hereditarily locally connected continua An arc (or ordered continuum) is a Hausdorff continuum with exactly two non-separating points. Each separable arc is homeomorphic to the closed interval I=[0,1]. A space Xis said to be an IOK (IOC) if there exists an ordered compact (connected) space Kand a continuous surjection f:K→X. Frequently, we will say that a space Xis a continuous image of an arc if Xis an IOC. The cardinality of a set Awill be denoted by card(A). We assume that card(A) is the initial ordinal number. The cofinality of a cardinal number mwill be denoted by cf(m). Keywords. Inverse system and limit, continuous image of an arc. 1991 Mathematics subject classifications: 54B35, 54C05, 54F50. 486 I. Lonˇ car A continuum Xis said to be hereditarily locally connected if each subcontinuum of Xis locally connected. A continuum Xis said to be finitely Suslinian [17] if there do not exist open sets Uand V, and an infinite collection Kof pairwise disjoint subcontinua of Xsuch that Cl(U)∩Cl(V)=∅and K∩V=∅and K∩U=∅for each Kin K. Each finitely Suslinian continuum is hereditarily locally connected. A continuum Xis rim-finite (rim-countable) if it has a basis Bsuch that card(Bd(U)) <ℵ0(card(Bd(U)) ≤ℵ 0) for each U∈B. Each rimfinite continuum is finitely Suslinian. Each hereditarily locally connected continuum is a continuous image of an arc [11, Theorem 3.4]. In the paper [8, Problem 9.10] the authors asked when the inverse limit of an inverse system X={Xa,p ab,A}of hereditarily locally connected continua with monotone surjective bonding mappings pab is a continuous image of an arc. If X={Xa,p ab,A}is an inverse system of hereditarily locally connected continua, then X= lim Xneed not be a hereditarily locally continuum since each locally connected metric continuum of dimension 1 (= curve) is the limit of an inverse sequence of rim-finite continua with surjective monotone bonding mappings [13, Theorem 2.2]. In the present section we shall define a class of hereditarily locally connected continua such that each inverse limit of such spaces and monotone bonding mappings has a hereditarily locally connected limit. In Appendix we review some definitions and known results needed in this section. We say that an inverse system Y={Ya,q ab,B}is a subsystem of X={Xa,p ab,A}if B⊂A,Ya=Xaand each qab is pab. We start with the following theorem. Theorem 1. Let Xbe the limit of an inverse system X={Xa,p ab,A} of hereditarily locally connected continua Xasuch that the bonding mappings pab :Xb→Xaare monotone surjections. Then Xis a hereditarily locally continuum if and only if each countable inverse subsystem of X has a hereditarily locally connected limit. Proof: By Theorem 29 Xis homeomorphic to the limit of Xσ= {X∆,p ∆Γ,A σ}, where Aσis the family of all nonempty countable directed subsets of A.IfXis hereditarily locally connected then each X∆ is hereditarily locally connected as a monotone image of a hereditarily locally connected continuum X(see Lemma 28). Conversely, if each X∆ is hereditarily locally connected, then Xis hereditarily locally connected (Theorem 27). Inverse limits of continuous images of arcs 487 The following theorem is a generalization of the well-known result of G. T. Whyburn [20, p. 81] which asserts that a metric continuum X is hereditarily locally connected if and only if each cyclic element (see Appendix) Z⊆Xis hereditarily locally connected. Theorem 2. A locally connected continuum Xis hereditarily locally connected if and only if each cyclic element of Xis hereditarily locally connected. Proof: By Theorems 31 and 29 there exists a σ-directed inverse system X={Xa,p ab,A}of metric locally connected spaces such that pab are monotone and Xis homeomorphic to lim X. Let us prove that each Xais hereditarily locally connected. It suffices to prove that each cyclic element Zaof Xais hereditarily locally connected. By Lemma 34 there exists a cyclic element Zof lim Xsuch pa(Z)⊇Za. Since lim Xis homeomorphic to X,Zis hereditarily locally connected. This means that pa(Z) is hereditarily locally connected. It follows that Zais hereditarily locally connected since Za⊆pa(Z). We infer that each Xais hereditarily locally connected since Xais a metric continuum. From Theorem 27 it follows that Xis hereditarily locally connected. A surjective mapping f:X→Yis said to be hereditarily monotone [5, pp. 16–17] if for each subcontinuum Kof Xthe restriction f|K: K→f(K) is monotone. If f:X→Yand g:Y→Zare hereditarily monotone mappings, then gf :X→Zis hereditarily monotone [5, p. 29, (5.3)]. Lemma 3. If Zis a cyclic element of X, then the canonical retraction ρ:X→Zis hereditarily monotone. Proof: Let ρ:X→Z(see Appendix, p. 13) and let Kbe any subcontinuum of X. Let us prove that the restriction ρK:K→ρK(K)⊆Zis monotone. Consider the following cases. a) K⊆Z.NowρKis the identity and is monotone. b) K⊆X\Z. It is clear that Kis a subset of some component J of X\Z. Let {zJ}= Bd(J). Now, ρK(K)={zJ}. This means that ρ−1 K(zJ)=K. Hence ρKis monotone. c) KZ=∅and K(X\Z)=∅. In this case K=KZKJ:Jis a component of X\Z, K J=∅. 488 I. Lonˇ car We infer that ρK(K)=KZ{{zJ}:Jis a component of X\Z, {zJ}=Bd(J)}. If x∈ρK(K) such that x=zJfor all components Jof X\Z, then ρ−1 K(x)={x}. Hence ρ−1 K(x) is connected. If x=zJfor some component J, then ρ−1 K(x)={zJ}KJi:{zJ}= Bd(Ji),i∈I. It suffices to prove that each set Cl(Ji)Kis connected. Suppose that Cl(Ji)Kis not connected. There exists a component Lof Cl(Ji)K such that zJ/∈L. By virtue of the normality of Xit follows that there exists a pair U,Vof disjoint open sets such that L⊆Uand zJ∈V. We may assume that ClU ⊆Jsince Jis open. For each point xof K\(UV) there exists an open set Uxsuch that x∈Uxand UxU=∅. Let Wbe the union of Vand all the sets Ux,x∈K\(UV). It is clear that UW=∅and K⊆UW. This is impossible since Kis connected. Hence, Cl(Ji)Kis connected and ρKis monotone. Theorem 4. Let X={Xa,p ab,A}be an inverse system of continua and hereditarily monotone bonding mappings. Then the projections pa, a∈A, are hereditarily monotone. Moreover, if each Xais hereditarily locally connected, then X= lim Xis hereditarily locally connected. Proof: Let Kbe a subcontinuum of X. For each a∈A,Ka=pa(K) is a subcontinuum of Xa. We have the inverse system K={Ka,p ab | Kb,A}. From the definition of hereditarily monotone mapping it follows that each mapping pab |Kbis a monotone mapping. This means that the projections pa|Kare monotone [2, pp. 462–463]. From [2, Theorem 6.1.28] it follows that Kis locally connected. Thus, Xis hereditarily locally connected. A surjective mapping f:X→Yis said to be cyclically hereditarily monotone if for each cyclic element Zof Xthe restriction f|Zis hereditarily monotone. Theorem 5. Let X={Xa,p ab,A}be an inverse system of hereditarily locally connected continua Xaand cyclically hereditarily monotone bonding mappings pab. Then X= lim Xis hereditarily locally connected. Inverse limits of continuous images of arcs 489 Proof: By Theorem 2 it suffices to prove that each cyclic element Z of Xis hereditarily locally connected. There exists an inverse system (Theorem 35) Z={Za,g ab,A}such that Zais a cyclic element of Xa, gab =ρa◦(fab |Zb) for all a≤b∈A, and Zis homeomorphic to lim Z. This means that gab is hereditarily monotone since ρais hereditarily monotone (Lemma 3). From Theorem 4 it follows that Zis hereditarily locally connected. Thus, Xis hereditarily locally connected. A space Xis in class Hmif Xis a hereditarily locally connected continuum and Xcontains no non-degenerate metric subcontinuum. Each space in class Hmis rim-finite [19, Theorem 1]. Theorem 6. Let X={Xn,p mn,N}be an inverse sequence with monotone surjective bonding mappings. If each Xnis in class Hm, then X= lim Xis in class Hm. Moreover, Xis rim-finite. Proof: Each Xnis a continuous image of an arc [11, Corollary 3.5] since each Xnis hereditarily locally connected. Thus, Xis a continuous image of an arc (Theorem 14). Let us prove that Xcontains no nondegenerate metric subcontinuum. Suppose that Yis a non-degenerate metric subcontinuum of X. Then there exists a n∈Nsuch that pm(Y) is a non-degenerate metric subcontinuum of Xmfor each m≥n. This is impossible since Xmis in class Hm. We infer that Xcontains no non-degenerate metric subcontinua. By virtue of Theorem 21 it follows that Xis hereditarily locally connected. Moreover, from Theorem 1 of [19] it follows that Xis rim-finite. Theorem 7. Let X={Xa,p ab,A}be an inverse system with monotone surjective bonding mappings. If each Xais in class Hm, then X= lim Xis in class Hm. Moreover, Xis rim-finite. Proof: Apply Theorem 1 and Theorem 23. A hereditarily locally connected continuum Xis in class Hzm if each cyclic element Zof Xis in class Hm. Theorem 8. Let X={Xa,p ab,A}be an inverse system with monotone surjective bonding mappings. If each Xais in class Hzm, then X= lim Xis in class Hzm. Proof: Let Zbe a cyclic element of X. By Theorem 35 there exists an inverse system (Zγ,g γγ,Γ) such that Zis homeomorphic to lim inv(Zγ,g γγ,Γ), where Zγis a cyclic element of Xγand gγγis monotone. By Theorem 7 Zis in class Hm. From Theorem 2 it follows that Xis hereditarily locally connected. Hence, Xis in class Hzm. 490 I. Lonˇ car 2. Special classes of continuous images of arcs Theorem 9 [9].Let Xbe a locally connected continuum such that for each pair of distinct points a,bin X, there exists a continuous onto map f:X→[c, d]such that f(a)=cand f(b)=dand [c, d]is a nonmetrizable arc. If Xis rim-metrizable or rim-scattered or monotonically normal, then Xis a continuous image of an arc. A locally connected continuum is said to be a NTT-space if for each pair of distinct points a,bin X, there exists a continuous onto map f: X→[c, d] such that f(a)=cand f(b)=dand [c, d] is a non-metrizable arc. Theorem 10. Let X={Xa,p ab,A}be an inverse system of NTTspaces with monotone surjective bonding mappings. Then X= lim Xis a NTT-space. Proof: It is known that Xis locally connected continuum. Let x,y be a pair of distinct points of X. There exists an a∈Asuch that pb(x)=pb(y) for each b≥a. Since Xbis a NTT-space there exists a non-metrizable arc [c, d] and a surjective mapping f:Xb→[c, d] such that f(x)=cand f(y)=d. Considering the mapping fpb:X→[c, d] we infer that Xis a NTT-space. Theorem 11. Let X={Xa,p ab,A}be an inverse system of NTTspaces and monotone surjective bonding mappings. If X= lim Xis rim-metrizable or rim-scattered or monotonically normal, then Xis a continuous image of an arc. Proof: By Theorem 10 Xis a NTT-space. Apply Theorem 9. Theorem 12. Let X={Xa,p ab,A}be a σ-directed inverse system of spaces Xasuch that for each pair xa,yaof points of Xathe subspace Xa\{xa,y a}is connected, a∈A. If each Xais a continuous image of an arc and each pab is a monotone surjection, then X= lim X is a continuous image of an arc if and only if there exists an a∈Asuch that pb:X→Xbis a homeomorphism for each b≥a(if and only if X is metrizable). Proof: By Theorem 2 of [18] each Xais metrizable. We shall prove that for each pair x,yof points of Xthe subspace Y=X\{x, y} is connected. Suppose that Yis not connected. Then there exists a pair U,Vof disjoint open subsets of Xsuch that Y=U∪V. Moreover, Inverse limits of continuous images of arcs 491 there exists an a∈Asuch that pb(x)=pb(y), b≥a. The sets Ua= {xa:xa∈Xa,p −1 a(xa)⊂U}and Va={xa:xa∈Xa,p −1 a(xa)⊂V}are disjoint and open. Now we have X\{pa(x),p a(y)}=Ua∪Va. This is impossible since X\{pa(x),p a(y)}is connected. Hence, Y=X\{x, y} is connected. It follows that if Xis a continuous image of an arc, then it is metrizable ([18, Theorem 2]). From Theorem 26 it follows that there exists a b∈Asuch that pc:X→Xcis a homeomorphism for every c≥b. Conversely, if such b∈Aexists, then Xis a continuous image of an arc. Theorem 13. Let X={Xa,p ab,A}be an inverse system of spaces Xa such that for each pair xa,yaof points of Xathe subspace Xa\{xa,y a} is connected, a∈A. If each Xais a continuous image of an arc and each pab is a monotone surjection, then X= lim Xis a continuous image of an arc if and only if there exists a countable subsystem Yof Xsuch that lim Yis homeomorphic to X. Proof: Consider the inverse system Xσ={X∆,p ∆Γ,A σ}from Theorem 29 and apply Theorem 12. 3. Inverse systems and subsystems Theorem 14 [8, Theorem 5.1].Let X={Xn,p mn,N}be an inverse sequence with monotone surjective bonding mappings. If each Xnis the continuous image of an arc, then X= lim Xis the continuous image of an arc. Theorem 15 [4, Theorem 2.17].Let X={Xa,p ab,A}be a wellordered inverse system such that cf(A)=ω1. If the mappings pab are monotone surjections and if the spaces Xaare the continuous images of arcs, then X= lim Xis the continuous image of an arc. Remark 16. Theorem 15 is not true if cf(A)=ω1. This is shown by the following example of Nikiel [10]. Let Ldenote the long interval [2, p. 297]. For each ordinal number α,0<α<ω 1, let fα:[0,1] ×L→ [0,1] ×[0,α]Lbe defined by f(s, t)=(s, t)ift≤Lα (s, α)ifα≤Lt. Each Xα=[0,1] ×[0,α]Lis homeomorphic to [0,1] ×[0,1] and it is a continuous image of an arc. Moreover, w(Xa)=ℵ0. Let fαβ =fα| [0,1] ×[0,β]L,β<α. We obtain an inverse system {Xα,f αβ,α < ω 1} whose limit is [0,1] ×Lwhich is not a continuous image of an arc. 492 I. Lonˇ car Theorem 17. Let X={Xa,p ab,A}be an inverse system of compact spaces such that card(A)>ℵ0. There exists a transfinite sequence {Aα: α<card(A)}of directed subsets Aαof Asuch that: 1. card(Aα)<card(A),α<card(A), 2. α<β<card(A)implies Aα⊆Aβ, 3. A={Aα:α<card(A)}, 4. each collection {Xa,p ab,A α}is an inverse system with limit Xα, 5. if α<β<card(A)then there exists a mapping qαβ :Xβ→Xα, 6. lim Xis homeomorphic to lim{Xα,q αβ,α<β <card(A)}, 7. if the mappings pab are monotone, then the mappings qαβ are monotone. Proof: The proof consists of several steps. Step 1 is from [7, pp. 238– 239, Hilfssatz]. For the sake of the completeness we give the proof of Step 1. Step 1: Let νbe any finite subset of A. There exists a δ(ν)∈A such that δ≤δ(ν) for each δ∈ν. For each B⊆Athere exists a set F1(B)=B{δ(ν):ν⊂Band νis finite}. Put Fn+1 =F1(Fn(B)), and F∞(B)={Fn(B):n∈N}. It is clear that F1(B)⊆F2(B)⊆...⊆Fn(B)⊆... The set F∞(B) is directed since each finite subset νof F∞(B) is contained in some Fn(B) and, consequently, δ(ν) is contained in F∞(B). If Bis finite, then card(F∞(B)) = ℵ0. If card(B)≥ℵ 0, then we have card({δ(ν):ν∈B})≤card(B)ℵ0. We infer that card(F1(B)) ≤ card(B)ℵ0. Similarly, card(Fn(B)) ≤card(B)ℵ0. This means that card(F∞(B)) ≤card(B)ℵ0.Thus card(F∞(B)) ≤card(B)ℵ0. Suppose that card(A)>ℵ0. Put Ω = card(A). Hence, A={aα:α< Ω}. Put Bα={aµ:µ<α<Ω}. We have a transfinite sequence {Bα:α<Ω}such that a) card(Bα)<card(A), b) α<β<Ω implies Bα⊆Bβ, c) A={Bα:α<Ω}. Put Aα=F∞(Bα). Inverse limits of continuous images of arcs 493 Step 2: Assertions 1-3 follow from Step 1. Step 3: Assertion 4 follows from the fact that each Aαis directed subset of A. Step 4: Let us prove 5. From assertion 2 it follows that there exists a continuous mapping qαβ :Xβ→Xαsince each point x∈Xβinduces a collection {xa:a∈Aα}which satisfies pab(xb)=xa, i.e., {xa:a∈Aα} is a point of Xα. Step 5: It is obvious that there exists a mapping H: lim X→ lim{Xα,q αβ,α < β < card(A)}since each x=(xa:a∈A)∈lim X induces a collection {xa:a∈Aα}on each Aα. Thus we have the mappings Hα: lim X→Xα, for each α<card(A). The mappings Hαinduce a continuous mapping H: lim X→lim{Xα,q αβ,α < β < card(A)}. It remains to prove that His 1-1 and onto. Let us prove that His 1-1. Let x,y∈lim Xand x=y. There exists an a∈Asuch that xa=ya. From Step 1 it follows that there is an Aαsuch that a∈Aα. Now, xa=yaimplies Hα(x)=Hα(y) (see Step 5). This means that H(x)=H(y). Hence His 1-1. In order to complete the proof it suffices to prove that His onto. Let y=(yα:α<card(A)) be any point of lim{Xα,q αβ,α<β <card(A)}. Then yα∈Xα. Thus, yαis a thread in Xα, i.e., y=(xa,a ∈Aα). We infer that for each a∈Athere exists a point xa∈Xa. It is readily to seen that pab(xb)=xa. Thus, (xa:a∈A) is a thread in lim Xsuch that H(x)=y. Step 6: Let us prove 7. If the mappings pab are monotone, then from Lemma 28 it follows that the mappings qαβ are monotone. Now we shall prove the main theorem of this section which is a generalization of Theorem 15. Theorem 18. Let X={Xa,p ab,A}be an inverse system of continuous images of arcs with monotone bonding mappings. If cf(card(A)) = ω1, then X= lim Xis a continuous image of an arc if and only if each proper subsystem {Xa,p ab,B}of Xwith cf(card(B)) = ω1has the limit which is a continuous image of an arc. Proof: The “only if part”. If Xis a continuous image of an arc, then for each subsystem {Xa,p ab,B}there exists a natural projections fa:X→lim{Xa,p ab,B}. Hence, lim{Xa,p ab,B}is a continuous image of an arc.