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1-DIRECTED INVERSE SYSTEMS OF CONTINUOUS
IMAGES OF ARCS
IVAN LONˇCAR
(Received 8 July 1999)
Abstract.The main purpose of this paper is to prove that ifX= {Xa,pab,A}is a usualℵ1 -directed inverse system of continuous images of arcs with monotone bonding mappings, thenX=limXis a continuous image of an arc (Theorem 2.4). Some applications of this statement are also given.
Keywords and phrases. Approximate inverse systems, continuous image of an arc. 2000 Mathematics Subject Classification. Primary 54B35; Secondary 54D30, 54F15.
1. Introduction. By an approximate inverse system in this paper we will mean an approximate inverse system in the sense of Mardeši´c [11]. See the appendix, Definition 5.1.
Anarc is a continuum with precisely two nonseparating points.
A spaceXis said to be a continuous image of an arc if there exists an arcLand a continuous surjectionf:L→X.
Inverse sequence of continuous images of arcs with monotone bonding mappings was studied in [16], where was proved that the limit of a usual inverse sequence of continuous images of arcs and monotone bonding mappings is the continuous image of an arc.
The well-ordered andσ-directed usual inverse system of continuous images of arcs with monotone bonding mappings were studied in [7].
In this paper, we shall study theℵ1-directed inverse systems of continuous im-ages of arcs. The main theorem of this paper states that if X= {Xa,pab,A} is a usualℵ1-directed inverse system of continuous images of arcs and monotone bond-ing mappbond-ings, thenX=limXis the continuous image of an arc (Theorem 2.4). Using Theorem 2.10 we shall obtain a characterization of continuous image of an arc by its imagesY with w(Y )= ℵ1(Theorem 3.3). Finally, we shall use this characteriza-tion to produce some new theorems concerning the approximate inverse system of continuous images of arcs (Theorems 4.3 and 4.6).
2. ℵ1-directed inverse systems. We start with some lemmas.
Lemma2.1. Letᏹ= {Mµ:µ∈M}be anℵ1-directed (partially ordered by inclusion)
family of compact metric subspacesMµ of a spaceX. ThenN={Mµ: µ∈M}is a compact metrizable subspace ofX.
Proof. Suppose thatw(N)≥ ℵ1. By virtue of [5] (or [17, Theorem 1.1] ifX is a
w(Nλ)≥ ℵ1. For eachx ∈Nλ there exists aMµ(x)∈ᏹsuch that x∈Mµ(x). The familyᏹ1= {Mµ(x):x∈Nλ}has the cardinality≤ ℵ1. By theℵ1-directedness ofᏹ there exists aMν∈ᏹsuch thatMν⊇Mµ(x)for eachx∈Nλ. This means thatNλ⊆Mν. We infer thatw(Nλ)≤ ℵ0 since Mν is a metric subspace ofX. This contradicts the
assumptionw(Nλ)≥ ℵ1. Hence, w(N)≤ ℵ0. There exists a countable dense subset
Z= {zn:N∈N}ofN. For eachznthere is aMµ(n)∈ᏹsuch thatzn∈Mµ(n). It is clear thatW={Mµ(n):n∈N}is dense inN. By virtue of theℵ1-directedness ofᏹ there exists aMν∈ᏹsuch thatMν⊇Mµ(n)for eachMµ(n). We infer thatMµ⊇W
and, consequently, Mµ is dense in N. From the compactness of Mµ it follows that
N=Mµ. Hence,Nis a compact metrizable subspace ofX.
Letτ be an infinite cardinal. We say thatX= {Xa,pab,A}is τ-directed if for each
b⊆Awith card(B)≤card(A)there is ana∈Asuch thata≥bfor eachb∈B. We say thatXisσ-directedifXisℵ0-directed.
Lemma2.2. Let X= {Xa,pab,A}be an approximate τ-directed inverse system of compact spaces with surjective bonding mappings and limitX. LetY be a compact space withw(Y )≤τand letZbe a closed subspace ofX. For each surjective mapping
f:Z→Y there exists ana∈Asuch that for eachb≥athere exists a mappinggb:
pb(Z)→Y such thatf=gb◦(pb|Z). Moreover, iffis a monotone surjection, thengb
is a monotone surjection.
Proof. LetZa=pa(Z)andqa=pa|Z, a∈A. The proof is broken into several steps.
Step1. For each normal coveringᐁofZ there exists ana(ᐁ)∈Asuch that for eachb≥a(ᐁ)there is a normal coveringᐁbofZbsuch thatqb−1(ᐁb)is a refinement
ofᐁ.
For each normal coveringᐁofZthere exists a normal coveringᐂofXsuch that ᐂ|Z= {VZ:V∈ᐂ}is a refinement ofᐁ[1, Theorem 15.14]. By virtue of (B1) (see [13, Theorems 2.8 and 4.2]), [11, Lemma 4] there exists ana(ᐁ)∈Asuch that for each
b≥a(ᐁ)there is a normal coverᐂbofXbwithp−1b (ᐂb)≺ᐂ. Consider the covering ᐁb=ᐂb|Zb. It is clear thatqb−1(ᐁb)is a refinement ofᐁ.
Step2. For each family of normal coveringsᐁofZwith card(ᐁ)≤τ there exists ana∈Asuch that for each b≥athere is a family of normal coverings ᐁb ofZb,
card(ᐁb)≤τsuch thatq−1
b (ᐁb)is a refinement ofᐁ.
This follows from Step 1 and from theτ-directedness ofA.
Step3. For each basisᏮ= {ᐁn:n < τ}of normal coverings ofY there exists an a∈Asuch that for eachb≥athere is a family of normal coveringsᐁnb ofZbsuch
thatq−1
b (ᐁnb)is a refinement ofᐁn,ᐁn∈Ꮾ.
Now, each basis of normal coverings ofYhas the cardinality≤τ(Lemma 5.2). Apply Step 2.
Step4. IfZbis as in Step 3, then for eachxb∈Zbthe setf (q−1
b (xb))is degenerate.
Suppose that there exists a pairu,v of distinct points ofY such thatu,v∈f (q−1
b (xb)). Then there exists a pairx,yof distinct points ofq−1
b (xb)such thatf (x)=u
such thatᐂn≺ {U,V,X\ {u,v}}. We infer that there is a normal coveringᐂnbofZb
such thatq−1
b (ᐂnb)≺f−1(ᐂn). It follows thatqb(x)≠qb(y)sincexandylie in the
disjoint members of the coveringf−1(ᐂn). This is impossible sincex,y∈q−1
b (xb).
Thus,f (q−1
b (xb))is degenerate.
Step5. There exists a mappinggb:Zb→Y. Now we definegbbygb(xb)=f (q−1
b (xb)). It is clear thatgbqb=f.
Let us prove thatgbis continuous. LetU be open inY. Theng−1
b (U)is open since (qb)−1(g−1
b (U))=f−1(U)is open andqbis quotient (as a closed mapping). Step6. If the mappingfis monotone, thengbis monotone.
This follows from the relationgbqb=f.
Lemma2.3. LetX= {Xa,pab,A}be a usualℵ1-directed inverse system of compact
metric spacesXaand surjective bonding mappings. ThenX=limXis a compact metriz-able space.
Proof. Let us prove that iff:X→Y is a continuous surjection, thenYis a metriz-able space. We shall use transfinite induction onw(Y ). Ifw(Y )≤ ℵ1, then there exists an a∈Aand a surjective mapping ga :Xa→Y such that f =gapa (Lemma 2.2). It follows thatY is a metrizable space. Suppose that this is true for each spaceY
with w(Y ) <ℵβ. Let us prove that this is true for the spaces Y with w(Y )= ℵβ.
By virtue of Theorem 5.9 there exists a well-ordered inverse systemY= {Yµ,ρµν,M} such thatY =limYandw(Yµ) <ℵβ. Moreover, by inductive hypothesis it follows that w(Yµ)= ℵ0since there exists a mappingρµf:X→Yµ. By [17, Theorem 2.2] it follows that w(Y )≤ ℵ1. Now, there exists ana∈A and a surjective mappingga :Xa →Y such thatf=gapa. This means thatY is a metrizable space. Finally, we shall prove thatw(X)= ℵ0. By virtue of Theorem 5.9 there exists a well-ordered inverse system Y= {Yµ,ρµν,M}such thatX=limYandw(Yµ) <ℵβ. Moreover,w(Yµ)= ℵ0. By [17, Theorem 2.2] it follows thatw(Y )≤ ℵ1. There exists ana∈Aand a surjective map-pingga:Xa→Y such thatf=gapa. This means thatXis a metrizable space.
LetX be a non-degenerate locally connected continuum. A subsetY ofX is said to be acyclic element ofXifY is connected and maximal with respect to the prop-erty of containing no separating point of itself. A cyclic element of locally connected continuum is again locally connected continuum. We let
LX=Y⊂X:Y is a non-degenerate cyclic element ofX. (2.1)
For other details see [2, 16].
Now we shall prove the main theorem of this paper.
Theorem2.4. LetX= {Xa,pab,A}be a usualℵ1-directed inverse system of
continu-ous images of arcs and monotone bonding mappings. ThenX=limXis the continuous image of an arc.
Proof. By [2, Theorem 1] it suffices to prove that each cyclic elementZ ofXis a continuous image of arc. From [16, Theorem 2.7] it follows that there exists a usual inverse system(Zγ,gγγ,Γ)of cyclic elements ofXγsuch thatZ=lim inv(Zγ,gγγ,Γ).
points ofZ. There exists ana∈Asuch that for eachb≥a gb(x),gb(y),gb(y)are different points of Zb. Moreover there exists a minimal metrizable T-set Tb which containsgb(x),gb(y)andgb(y). For eachb≥athe family{pbc(Tc):c≥b}satisfies the assumptions of Lemma 2.1. This means thatNb={gbc(Tc):c≥b}is a compact metric subspace ofZb. Moreover,gbc(Nc)=Nb. We have a usual inverse systemN= {Nb,gbc|Nc, a≤b≤c}which satisfies the assumptions of Lemma 2.3. It follows that
N=limNis compact and metrizable. Moreover, by [7, Lemma 2.15] it follows thatN
is aT-set inXwhich containsx,y, andz. From [16, Theorem 4.4] it follows thatZis a continuous image of an arc.
Remark2.5. Let us observe that Theorem 2.4 is not true forσ-directed inverse system. This is shown by an example of Nikiel [14, Example 4.3]. The sufficient and necessary condition for the limit ofσ-directed inverse system to be the continuous image of an arc are given in [7, Theorem 2.21].
Lemma2.6. LetBbe an infinite subset of directed setA. There exists a directed subset
F∞(B)ofAsuch thatB⊆F∞(B)andcard(F∞(B))=card(B).
Proof. Letνbe any finite subset ofA. There exists aδ(ν)∈Asuch thatδ≤δ(ν)
for eachδ∈ν. For eachB⊆Athere exists a setF1(B)=B{δ(ν):ν∈B}, whereνis a finite subset ofB. Put
Fn+1=F1Fn(B), F∞(B)=Fn(B):n∈N. (2.2) It is clear that
F1(B)⊆F2(B)⊆ ··· ⊆Fn(B)⊆ ···. (2.3) The setF∞(B)is directed since each finite subset ν ofF∞(B)is contained in some
Fn(B)and, consequently,δ(ν)is contained inF∞(B). From card(B)≥ ℵ0, it follows 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
cardF∞(B)≤card(B)ℵ0. (2.4) We infer that card(F∞(B))=card(B).
LetX= {Xa,pab,A}be a usual inverse system of compact spaces and letτ <card(A)
be an infinite cardinal. Consider the set Aτ of all F∞(B), B⊆ A, card(B)=τ, or-dered by inclusion. It is clear thatAτ is τ-directed. Each elementαof Aτ is some
F∞(B). We defineXαas the limit of{Xa,pab,F∞(B)}. Letα=F∞(B)andβ=F∞(C). If
α⊆β, then there exists the natural projectionqαβ:Xβ=lim{Xa,pab,F∞(C)} →Xα= lim{Xa,pab,F∞(B)}. It is clear thatqαγ=qαβqβγ ifα≤β≤γ. It follows that{Xα,qαβ,
Aω1}is a usual inverse system.
Theorem2.7. LetX= {Xa,pab,A}be a usual inverse system of compact spaces with limitX. For each infinite cardinalτ <card(A)there exists aτ-directed usual inverse system{Xα,qαβ,Aτ}such thatXis homeomorphic tolim{Xα,qαβ,Aτ}.
Corollary2.8. LetX= {Xa,pab,A}be a usual inverse system of compact spaces with limitX. If ℵ1<card(A), then there exists an ℵ1-directed usual inverse system {Xα,qαβ,Aℵ1}such thatXis homeomorphic tolim{Xα,qαβ,Aℵ1}.
Corollary2.9. LetX= {Xa,pab,A}be a usual inverse system of compact spaces with limitX. Ifℵ0<card(A), then there exists aσ-directed usual inverse system{Xα,qαβ,
Aℵ0}such thatXis homeomorphic tolim{Xα,qαβ,Aℵ0}.
At the end of this section, we shall prove the following theorem.
Theorem2.10. LetX= {Xa,pab,A}be a usual inverse system of continuous im-ages of arcs with limit X andcf(card(A))≠ℵ1. If the mappings pab are monotone
surjections, then the following are equivalent: (a)Xis the continuous image of an arc.
(b)Each proper subsystemY= {Xb,pbc,B}ofXwithcard(B)= ℵ1has the limitY
which is the continuous image of an arc.
Proof. (a)⇒(b). Obvious since there exists the natural projection fq: limX→limY.
(b)⇒(a). Inverse system{Xα,qαβ,Aℵ1}isℵ1-directed. Apply Theorem 2.4.
3. Characterizing spaces by images of the weight less than or equal toℵ1. In this section, we shall characterize a compact locally connected space by its images of the weight≤ ℵ1.
A continuumXis said to be hereditarily locally connectedif each subcontinuum of
Xis locally connected.
We start with the following theorem.
Theorem3.1. LetXbe a compact space. The following are equivalent: (a)Xis locally connected.
(b)Iff:X→Y is a surjection andY is a metric space, thenY is locally connected.
Proof. (a)⇒(b). IfXis locally connected, then from [19, Theorem 1.6, page 70] it follows thatY is locally connected.
(b)⇒(a). By Theorem 5.3 there exists aσ-directed usual inverse systemX={Xa,pab, A}of compact metric spacesXaand surjective bonding mappingspabsuch thatXis homeomorphic to limX. By (b) eachXa is locally connected. Finally, by Theorem 5.6 we infer thatXis locally connected.
Theorem3.2. LetXbe a locally connected continuum. The following are equivalent: (a)Xis hereditarily locally connected.
(b)IfZis a subcontinuum ofXand iff:Z→Y is a surjection onto a metric spaceY, thenY is locally connected.
Proof. (a)⇒(b). If X is hereditarily locally connected, then each subcontinuum
Z⊆Xis locally connected. By [19, Theorem 1.6, page 70] it follows thatY is locally connected.
(b)⇒(a). By Theorem 5.4 there exists aσ-directed usual inverse systemX={Xa,pab, A} of compact metric spaces Xa and surjective monotone bonding mappings pab
Namely, for each subcontinuumKofXa we have a subcontinuumZ=p−1
a (K)and a
mappingpa|Z:pa|Z:Z→K. By (b)Kis locally connected. Hence, eachXais hered-itarily locally connected. From Theorem 5.7 it follows that X is hereditarily locally connected.
From Theorem 2.4 it follows the following characterization of continuous images of arcs.
Theorem3.3. LetXbe a locally connected continuumXwithcf(w(X))≠ω1. The
following are equivalent:
(a)Xis a continuous image of an arc.
(b)Iff :X→Y is a continuous surjection and w(Y )= ℵ1, thenY is a continuous
image of an arc.
Proof. (a)⇒(b). Obvious.
(b)⇒(a). By [3, Theorem 2.3.23] the spaceX is embeddable inIw(X). SinceIw(X) is
an inverse limits of finite cubeIn, we infer that there exists a usual inverse system X= {Xa,pab,A}of metric spacesXasuch thatXis homeomorphic to limX. It is clear that there exists no a subsetB which is cofinal inA and card(B)= ℵ1. This means that there exists anℵ1-directed usual inverse system{Xα,qαβ,Aℵ1}such that X is homeomorphic to lim{Xα,qαβ,Aℵ1}andw(Xa)≤ ℵ1. By (b) eachXais the continuous image of an arc. The system{Xα,qαβ,Aℵ1}satisfies the conditions of Theorem 2.10. Hence,Xis the continuous image of an arc.
Theorem3.4. Let X be a locally connected nonmetrizable continuum such that cf(w(Z))≠ω1for each cyclic element Z of X. The following statements are
equiv-alent:
(a)Xis a continuous image of an arc.
(b)IfZ is a non-degenerate cyclic element ofX orZ=X, f :Z→Y is a surjection onto a spaceY withw(Y )=ω1, thenY is a continuous image of an arc.
Proof. (a)⇒(b). LetZbe a cyclic element ofX. By [2, Theorem 1]Zis a continuous image of an arc. It is clear that iff:Z→Yis a surjection, thenYis aYis a continuous image of an arc.
(b)⇒(a). Repeat the proof of the implication (b)⇒(a) of Theorem 3.3 replacingXbyZ.
Corollary3.5. LetX be a locally connected nonmetrizable continuum of regu-lar weightw(Z) > ω1 for each cyclic elementZ ofX. The following statements are
equivalent:
(a)Xis a continuous image of an arc.
(b)IfZ is a non-degenerate cyclic element ofX orZ=X, f :Z→Y is a surjection onto a spaceY withw(Y )=ω1, thenY is a continuous image of an arc.
4. ℵ1-directed approximate inverse systems. In this section, we will apply theo-rems from the last section to obtain some theotheo-rems for approximate limits. At the begin we shall prove the following theorem.
Proof. By Theorem 3.1 it suffices to prove that iff:X→Y is a surjection and
Y is a metric continuum, thenY is locally connected. From Lemma 2.2 (forZ=X) it follows that there exists ana∈Asuch that for eachb≥athere exists a mapping
gb:Xb→Y. We infer thatY is locally connected sinceXais locally connected. Hence,
Xis locally connected.
Theorem4.2. LetX= {Xa,pab,A}be aσ-directed approximate inverse system of hereditarily locally connected continua. ThenX=limXis hereditarily locally connected.
Proof. By Theorem 3.2 it suffices to prove that iff:X→Y is a surjection andY
is a metric continuum, thenY is locally connected. By virtue of Theorem 4.1 it fol-lows thatXis locally connected. From Lemma 2.2 it follows that there exists ana∈A
such that for each b≥athere exists a mapping gb:pb(Z)→Y. Now,pb(Z)is lo-cally connected sinceXb is hereditarily locally connected. We infer that Y is locally connected sincepb(Z)is locally connected. Thus,Xis hereditarily locally connected.
Theorem4.3. LetX= {Xa,pab,A}be an approximate inverse system of continuous images of arcsXawith the limitX. IfXisℵ1-directed andcf(w(X))≠ω1, thenXis a
continuous image of an arc.
Proof. By virtue of Theorem 4.1Xis locally connected. From Theorem 3.3 it fol-lows that it suffices to prove that iff:X→Yis a continuous surjection andw(Y )= ℵ1, thenY is a continuous image of an arc. Now, using Lemma 2.2 we infer that there ex-ists ab∈Aand a surjective mappinggb:Xb→Y. It follows that Y is a continuous image of an arc sinceXbis a continuous image of an arc.
Remark4.4. Let us observe that the bonding mappings in Theorem 4.3 are not assumed to be monotone.
Corollary4.5. LetX= {Xa,pab,A}be an approximate inverse system of contin-uous images of arcsXa with the limitX. IfXisℵ1-directed and ifw(X)is a regular
cardinal> ω1, thenXis a continuous image of an arc.
Theorem4.6. LetX= {Xa,pab,A}be an approximate inverse system of continuous images of arcsXawith the limitX. IfXisℵ1-directed andcf(w(Z))≠ω1for each cyclic
elementZofX, thenXis a continuous image of an arc.
Proof. The proof is broken into several steps.
Step1. LetX= {Xa,pab,A}be an approximate inverse system of continua and let
Ybe a subcontinuum ofX=limX. Ifr (pa(Y ))≤kfor alla∈Aand some fixed integer
k≥0, thenr (Y )≤k.
K andLare all the components ofCD. Second, we prove thatCDhas≤k+1 components. Suppose thatCDhas≥k+2 components K1,...,Kk+2,... .Consider the family{K1,...,Kk+1,Kk+2}. For each pairKi,Kjthere exists anaij∈Asuch that
pb(Ki), pb(Kj)lie in different components ofpb(C)pb(D)for eachb≥aij. From the directedness ofAit follows that there is an indexb∈B, b≥aij, i,j=1,...,k+2. We infer thatpb(C)pb(D)has≥k+2 components.This is impossible and the proof of Step 1 is completed.
Step2. LetX= {Xa,pab,A}be an approximate inverse system of continua and let
Y be a subcontinuum ofX=limX. Ifpa(Y )is hereditarily unicoherent for alla∈A, thenY is hereditarily unicoherent.
Step3. LetX= {Xa,pab,A}be an approximate inverse system of locally connected continuaXa with monotone bonding mappings and letZ be a non-degenerate cyclic element ofX=limX. There exists ana0such thatLb(Zb)≠∅.
Suppose thatLZb= ∅for eachb∈B(i.e., allZbare dendron), whereBis cofinal in A. By Step 2,Zis hereditarily unicoherent. This means thatZis a dendron since it is locally connected. Thus,LZ= ∅. This contradicts the assumption thatZ∈LX.
Step4. LetX= {Xa,pab,A}be an approximate inverse system of locally connected continuaXa with monotone bonding mappings and letZ be a cyclic element ofX= limX. IfYa is a non-degenerate cyclic element ofZa=pa(Z), thenp−1
a (Ya)⊆Zand Ya is a cyclic element ofXa.
By virtue of [8, Corollary 5.6] the projectionspa are monotone. By virtue of [16, Lemma 2.3] there exists a cyclic elementW ofXsuch thatp−1
a (Ya)⊆W. This means
that card(ZW )≥2. We infer thatZ=W. Similarly, ifWa is a cyclic element inXa
containingYa, thenp−1
a (Wa)⊆Z. Thus,Wa⊆Za. Hence,Yais a cyclic element ofXa. Step5. LetX= {Xa,pab,A}be an approximate inverse system of continuous im-ages of arcsXa with monotone bonding mappings and letZ be a cyclic element of
X=limX. Then eachZa=pa(Z)is a continuous image of an arc.
It suffices to prove that each cyclic elementWaofZais a continuous image of an arc. IfWa is degenerate, then it is a continuous image of an arc. IfWais non-degenerate, then by Step 4,Wais a cyclic element ofXa. From [2, Theorem 1] it follows that Wais
a continuous image of an arc. Hence,Zais a continuous image of an arc.
Step6. Finally, let us prove Theorem 4.6. LetZbe any cyclic element ofX,f:Z→Y
a surjection andw(Y )= ℵ1. From Lemma 2.2 it follows that there exists ana∈A and ga :Za→Y. We infer thatY is the continuous image of an arc sinceZa is the continuous image of an arc (Step 5). By Corollary 3.5 we infer thatXis the continuous image of an arc.
We close this section with two questions.
In connection with [16, Theorem 5.1] it is naturally to ask the following question. Question4.7. LetX= {Xn,pnm,N}be an approximate inverse sequence of con-tinuous images of arcs with monotone bonding mappings. Is it true thatX=limXis the continuous image of an arc?
in-verse system of continuous image of arcs and monotone bonding mappings. Does it follow that the approximate inverse limit spaceX=limXis a continuous image of an arc?
5. Appendix. Cov(X)is the set of all normal coverings of a topological spaceX. For other details see [1].
Definition5.1. An approximate inverse system is a collectionX= {Xa,pab,A}, where (A,≤) is a directed preordered set, Xa, a∈ A, is a topological space and
pab :Xb→Xa, a≤b, Xa, a≤b, are mappings such thatpaa=id and the follow-ing condition (A2) is satisfied:
(A2) For eacha∈Aand each normal coverᐁ∈Cov(Xa)there is an indexb≥a
such that(pacpcd,pad)≺ᐁ, whenevera≤b≤c≤d.
Other basic notions, including approximate mapping and the limit of an approximate inverse system are defined as in [11, 13].
An inverse system in the sense [3, page 135] we call ausual inverse system. An approximate inverse systemX= {Xa,pab,A}is said to becommutative[13, Def-inition 1.4] provided it satisfies the commutativity condition:
pabpbc=pac, fora < b < c. (5.1)
Let us observe that ifX= {Xa,pab,A}is a commutative approximate inverse system of Tychonoff spaces, then the limit ofXin the usual sense and the approximate limit coincide [13, Remark 1.15].
A basis of (open) normal coverings of a spaceXis a collectionᏯof normal coverings such that every normal coveringᐁ∈Cov(X)admits a refinementᐂ∈Ꮿ. In this case, we writeᐂ≺ᐁ. We denote by cw(X)(covering weight) the minimal cardinal of a basis of a normal coverings ofX[12, page 181].
Lemma5.2(see [12, Example 2.2]). IfXis a compact Hausdorff space, thencw(X)= w(X).
Theorem5.3. Let X be a compact spaces. There exists aσ-directed usual inverse systemX= {Xa,pab,A}of compact metric spacesXaand surjective bonding mappings
pab such thatXis homeomorphic tolimX.
Proof. Apply [10, pages 152 and 164] and Corollary 2.9.
Theorem5.4. IfXis a locally connected compact space, then there exists a usualσ -directed inverse systemX= {Xa,pab,A}such that eachXais a metric locally connected compact space, each pab is a monotone surjection and X is homeomorphic tolimX. Conversely, the inverse limit of a such system is always a locally connected compact space.
Proof. Apply [10, Theorem 2] and Corollary 2.9.
Theorem5.6(see [6, Theorem 5]). Let X= {Xa,pab,A}be a usualσ-directed in-verse system of locally connected locally compact spacesXaand perfect mappingspab. ThenX=limXis locally connected.
Theorem5.7(see [4, Corollary 3]). LetX= {Xa,pab,A}be aσ-directed inverse sys-tem of hereditarily locally connected continuaXa. ThenX=limXis hereditarily locally connected.
Theorem5.8(see [15, Corollary 2.9]). IfX is a hereditarily locally connected con-tinuum, then there exists aσ-directed inverse systemX= {Xa,pab,A}such that each
Xa is a metrizable hereditarily locally connected continuum, eachpab is a monotone surjection andXis homeomorphic tolimX.
In the next theorem, we assume thatw(X)is an aleph, i.e.,w(X)= ℵτ(X)and that ωτ(X)is the initial ordinal number belonging toℵτ(X)=w(X)[9, page 279].
Theorem5.9(see [9, Theorem 3]). Every nonmetrizable (Hausdorff) compact space
X is homeomorphic with the inverse limit of an inverse system {Xβ,pββ}, where β
ranges through all the ordinalsβ < ωτ(X), whileXβ are (Hausdorff) compact spaces satisfying
(1) dimXβ≤dimX. (2)w(Xβ) < w(X). Moreover,
(3)w(Xβ)≤card(β)=card({α:α < β}), ω0≤β < ωτ(X).
Ifβis a limit ordinal, then (4)
Xβ=limXα, pαα, α < β, (5.2) pαβ:Xβ→Xαbeing the corresponding projections.
Corollary5.10. Every nonmetrizable compact locally connected spaceXis home-omorphic to the inverse limit of a transfinite inverse sequence X= {Xapab,w(X)}, wherepabare monotone surjection,Xaare compact locally connected spaces such that
w(Xa) < w(X)andw(Xa)≤card(a)providedω0≤a < w(X).
Ifc is limit and0< c < w(X), then Xc =lim{(Xa,pab,c)}and the corresponding
mapspab:pab:Xb→Xaare the corresponding projections fora < c.
Proof. Let(Xα,fα
β,w(X))be the inverse system from Theorem 5.9. Applying the
monotone-light factorization [18] to the natural projectionsfα:X→Xα, we get com-pact spacesXa, monotone surjectionma:X→Xa and light surjectionsla:Xa→Xα
such that fα =la◦ma, a < w(X). By [10, Lemma 8] there exist monotone surjec-tions pab :Xa→Xb such that pab◦ma =mb, a≤b < w(X). It follows thatX= {Xa,pab,w(X)} is a transfinite inverse sequence such thatX is homeomorphic to limX. It is obvious that eachXa is locally connected. Moreover, by [10, Theorem 1] it follows thatw(Xa)=w(Xα).
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Ivan Lonˇcar: Faculty of Organization and Informatics, Pavlinska 2, Varaždin, Croatia
