The structure of Valdivia compact lines

The structure of V aldivia compact lines Ond ˇ rej F.K. Kalend a ∗ Charles Univ ersity , F aculty of Mathematics and Physics Department of Mathematical Analysis Sokolo vsk´ a 83, 1 86 7 5 P raha 8, Czech Republic E-mail: k alenda@k arlin.mff.cuni.cz Wies la w Kubi ´ s † Institute of Mathematics of the Academy of Sciences of the Czech Republic ˇ Zitn´ a 25, 1 15 67 P raha 1, Czech Republic and Department of Mathematics, Jan Ko chano wski Universit y ´ Swi¸ etokrzysk a 15, 2 5 -406 Kie lce, Poland Octob er 30, 2018 Abstract W e s tudy linearly or dered spaces which a re V aldivia co mpact in their order top olog y . W e fin d an internal characteriza tio n of these spaces and we prese n t a counter-example disproving a conjecture po sed earlie r by the firs t author. The conjecture asserted that a compact line is V aldivia co mpact if its weigh t do es not exce e d ℵ 1 , every point of un- countable character is isolated fro m one side and every close d first countable s ubspace is metrizable. It turns o ut tha t the last condition is not sufficien t. On the other hand, we show that the conjecture is v alid if the closure of the set of p oints o f unco untable character is s cattered. This improves a n ea rlier r esult o f the first author. Keyw ords: V aldivia compa c t line, incr easing ma p, retractio n. MSC (2 000): primary 54F05 , 54C1 5 ; seconda ry 54 D30, 06 F3 0. 1 In tro duction By a c omp act line w e mean a linearly ordered compact space, i.e. a compact space wh ose top ology is induced by a linear order. W e inv estiga te V aldivia compact lines, i.e., compact ∗ Researc h supp orted by Resea rch pro ject MSM 0021620 839 and partly by Resea rch g rant GA ˇ CR 201/00/ 0018. † Researc h supp orted by MNiSW grant No. N 201 024 32/0904. 1 lines whic h are V aldivia compact spaces. R ecall th at a compact space K is called V aldivia if it is homeomorph ic to some K ′ ⊆ R Γ for a set Γ suc h that { x ∈ K ′ : { γ ∈ Γ : x ( γ ) 6 = ∅} is countable } is den s e in K ′ . V aldivia compact s paces p la y an imp ortan t r ole in the stu d y of the structure of nons ep arable Banac h sp aces. They app eared firs t in [1], the name w as given in [2]. F or a detailed study of th is class w e refer to [7, 9 ]. V aldivia compact lines were addressed in [10, Section 5], [11] and in [9, Section 3]. In [9 ] the follo wing question w as asked. Question 1 . Let K b e a compact line satisfying the follo wing three conditions. (i) K has w eigh t at most ℵ 1 . (ii) Eac h p oin t x ∈ K of uncountable c haracter is isolated fr om one side (i.e, one of the in terv als ( ← , x ] or [ x, → ) is op en in K ). (iii) Eac h closed fi rst coun table sub set of K is metrizable. Is K necessarily V aldivia? It is show ed there that these conditions are n ecessary and that th ey are also sufficient in case K is either scattered or connected. It is conjectured there [9, Conjecture 3.5] that these conditions are sufficient in general. In the presen t pap er we sh ow that the conjecture is false (Example 3.5 b elo w). W e furth er giv e a c h aracteriza tion of zero-dimensional V aldivia compact lines usin g functions d efined on statio nary subsets of ω 1 (Theorem 3.1). In this c haracterization w e u se a strengthening of condition (iii) from the ab o v e question. A c haracterization of general, not necessarily zero-dimensional, V aldivia compact lines is giv en in Section 4. I n S ection 5 w e sho w that the ab o ve question has p ositiv e answer if the p oin ts of u ncoun table c haracter ha ve scatte red closure in K , whic h generalizes the results of [9]. In the last section we study compact lines w h ic h are con tin u ous images of V aldivia compacta. 2 Preliminaries In this section we collect some au x iliary results on V aldivia compacta and n amely on V aldivia compact lines, n eeded in the sequ el. A c omp act line is a compact sp ace K together w ith a linear order wh ic h ind u ces the top ology of K . It is well kno wn that a linearly ord ered set X is compact in its int erv al top ology if and only if it is ord er complete, i.e. every nonempty s ubset of X has b oth the supremum and the infimum. A compact lin e K is zero-dimensional if and only if for ev ery x, y ∈ K with x < y there are x ′ , y ′ suc h that x ≤ x ′ < y ′ ≤ y and the op en interv al ( x ′ , y ′ ) is emp t y . Given a compact line K , w e shall denote b y 0 K and 1 K the minimal and the maximal elemen t of K resp ectiv ely . 2 W e shall use standard notation concerning in terv als in a linearly ord er ed set. F or example: [ a, → ) will denote the closed final in terv al (segmen t) induced by a , i.e. [ a, → ) = { x : a ≤ x } . A su b set G of a linearly ordered set X is c onvex if [ x, y ] ⊆ G wh en ev er x, y ∈ G are suc h that x < y . T he smallest con vex set cont aining A ⊆ X will b e denoted by con v ( A ). A map f : X → Y b et wee n linearly ordered sets is incr e asing if f ( x 0 ) ≤ f ( x 1 ) wh enev er x 0 ≤ x 1 . W e shall often use the simple fact that eve ry increasing surj ection b et ween compact lines is con tinuous. W e treat ordinals as well ordered sets w ith r esp ect to ∈ . In particular, given tw o ord inals α, β , α < β holds iff α ∈ β . Recall that ω 1 denotes the smallest uncountable ordinal, wh ic h is at the same time treated as a linearly ord ered space, end o wed with the order top ology . W e shall denote by ω − 1 1 the set ω 1 with rev ersed orderin g. Note that a set C ⊆ ω 1 is unb ounde d if it has cardinalit y ℵ 1 . Recall that a set S ⊆ ω 1 is stationary if it intersects ev ery closed u n b ound ed subset of ω 1 . The Pr e ssing Down L emma says that giv en a stationary set S ⊆ ω 1 , for ev ery function f : S → ω 1 whic h is r e gr essive , i.e. f ( α ) < α for α ∈ S , there exists a stationary set S ′ ⊆ S on which f is constan t. F or more inform ation concernin g ordinals and set-theoretic notions w e refer to [4] and [14]. An ω 1 -se quenc e in a top ological space X is a fun ction x : ω 1 → X . W e shall often write x α instead of x ( α ). T h e notion of a limit of an ω 1 -sequence x is defined naturally . Namely , p = lim α → ω 1 x α if for ev ery neighb orh o o d U of p there is α < ω 1 suc h that { x ξ : ξ ≥ α } ⊆ U . If X is lin early ordered and the sequence x is increasing then its p ossible limit is the supremum of the set { x α : α < ω 1 } . A sequence is monotone if it is either increasing or decreasing (i.e. increasing with r esp ect to the rev ersed ordering). Let K b e a compact space and A ⊆ K . W e sa y that A is a Σ -subset of K if there is a homeomorphic injection h : K → R Γ suc h that A = h − 1 (Σ(Γ)), where Σ(Γ) = { x ∈ R Γ : { γ ∈ Γ : x ( γ ) 6 = ∅} is countable } . Hence, K is V aldivia if and only if it admits a dense Σ-sub space. F u r ther, if K is a compact line, follo win g [9 ] we denote b y G ( K ) the set of all p oints of K which are either isolated or can b e obtained as the limit of a one-to-one sequence. Th e follo wing lemma wa s p ro ved in [9, L emma 3.1]. Lemma 2.1. L et K b e a c omp act line. Then G ( K ) is dense in K . M or e over, if K is V aldivia, then G ( K ) is the uniqu e dense Σ - subset of K and is forme d by al l G δ p oints of K . W e will also us e the follo win g charact erization of V aldivia compact lines. Lemma 2.2. L et K b e a c omp act line. Then K is V aldivia if and only if ther e is a f amily A of op en F σ -intervals in K satisfying the fol lowing two c onditions: • F amily A sep ar ates p oints of K , i.e . for e ach distinct p oints x, y ∈ K ther e is I ∈ A c ontaining exactly one of them. • Each x ∈ G ( K ) b elongs to c ountably many elements of A . 3 F urther, if A is such a family, we have G ( K ) = { x ∈ K : { I ∈ A : x ∈ I } is c ountable } . If K is mor e over zer o-dimensional, the family A may b e chosen to c onsist of clop en intervals. Pr o of. It follo ws from Lemma 2.3 and [7 , Prop osition 1.9] that b eing V aldivia is equiv alen t to the existence of a family of op en F σ sets satisfying the t w o conditions. W e ha v e also the equalit y giv en ab o ve. Finally , as eac h op en F σ subset of a compact space is Lindel¨ of, it can b e expressed as a counta ble u nion of element s of a giv en b asis. Therefore A ma y b e c hosen to consist of op en F σ -in terv als and in case K is zero-dimensional to consist of clop en in terv als. The n ext result is pro v ed in [9 , Prop osition 3.2] and sh o ws that V aldivia compact lines ha ve rather exceptional str u cture. Lemma 2.3. L et K b e a c omp act line . If K is V aldivia (a c ontinuous image of a V aldivia c omp act sp ac e), then so is e ach close d su bset of K . W e will fur th er need the follo win g result on con tinuous images: Lemma 2.4. L et K b e a V aldivia c omp act line , L a c omp act line and ϕ : K → L an or der- pr eserving c ontinuous surje ction. Supp ose that for e ach y ∈ L either ϕ − 1 ( y ) i s a singleton or ϕ − 1 ( y ) ∩ G ( K ) is dense in ϕ − 1 ( y ) . Then L is V aldivia as wel l. Pr o of. Set E = {h x 1 , x 2 i ∈ K × K : ϕ ( x 1 ) = ϕ ( x 2 ) } . As G ( K ) is a dense Σ-subset of K , b y [6, Lemma 2.9 and Theorem 2.20] it is enough to observe that E ∩ ( G ( K ) × G ( K )) is dense in E . But this easily follo ws from the assumptions. 3 Zero-dimensional V aldivia c ompact lines In this s ection we giv e an internal c haracterizatio n of zero-dimensional V aldivia compact lines. W e first r estrict to the 0-dimensional case as th ere is a d ualit y b et we en compact 0-dimensional lines and linearly ord ered sets. Namely , giv en a 0-dimensional compact line K , let X ( K ) b e the set of all clop en final segments F of K such that 0 K / ∈ F and 1 K ∈ F . Th en X ( K ) is a linearly ordered set (the order b eing defin ed b y inv erse in clusion). Con v ersely , giv en a linearly ordered set X , let K ( X ) b e the set of all fi nal segmen ts end o wed with th e top ology in herited f r om the Can tor cub e { 0 , 1 } X , where eac h final segment is iden tified with its c haracteristic function. Then K ( X ) is a compact 0-dimensional line, the order is given b y inv erse inclusion. Note th at K ( X ( K )) is canonically order-h omeomorph ic to K for eac h zero-dimensional compact line K and X ( K ( X ) ) is canonically order -isomorp h ic to X for eac h linearly ord ered set X . 4 The ab ov e defined op erations naturally extend to con tra v ariant f unctors w hic h witness th e isomorphism b et we en the categ ory of linearly ordered sets with increasing maps and th e catego ry of nonempty compact 0-dimensional lines K with cont in uous increasing maps. The pr omised c h aracterizat ion of zero-dimensional V aldivia compact lines is con tained in the follo wing theorem. Theorem 3.1. L e t X b e a line arly or der e d se t. Then K ( X ) is V aldivia i f and only if the fol lowing thr e e c onditions ar e satisfie d. (1) | X | ≤ ℵ 1 . (2) Every b ounde d monotone ω 1 -se quenc e has a limit in X . (3) F or every stationary set S ⊆ ω 1 and every map f : S → X ther e is a stationary se t T ⊆ S such that f | T is monotone. Let us comment a bit the cond itions in the ab o v e theorem. As the cardinalit y of X is equal to the wei gh t of K ( X ), condition (1) means just that the w eigh t of K ( X ) is at most equal to ℵ 1 . This corresp onds to condition (i) in Question 1. Condition (2) form ulated in m ore detail means that eac h in cr easing ω 1 -sequence whic h is b ound ed from ab o ve h as a suprem um in X and eac h decreasing ω 1 -sequence b ound ed from b elo w has an infim um in X . Sup p osing that (1) holds, the v alidit y of (2) is equiv alent to the v alidit y of condition (ii) f rom Question 1 for the sp ace K ( X ). Indeed, if, sa y , ( x α ) α<ω 1 is an increasing ω 1 -sequence whic h is b ound ed from ab ov e b u t has no sup rem um, th en the final segmen t \ α<ω 1 ( x α , → ) has uncounta ble c haracter in K ( X ) while it is not isolated from either side. Conv ersely , supp ose that k ∈ K ( X ) has uncounta ble c haracter and is not isolated from either s ide. Without loss of generalit y we can supp ose that the c h aracter of k in ( ← , k ] is uncountable. Then there are k α ∈ ( ← , k ), k < ω 1 , isolate d from the left such th at the ω 1 -sequence ( k α ) is increasing and h as limit k . If we set x α = [ k α , → ), w e get an incresing ω 1 -sequence in X whic h is b ounded fr om ab o ve and h a ving n o limit in X . As we will see b elo w, condition (3) is a natural strengthening of condition (iii) from Question 1. W e first prov e the necessit y of a w eake r assumption. Prop osition 3.2. L et X b e a line arly or der e d set. If K ( X ) is V aldivia c omp act then the fol lowing c ondition is satisfie d: (3’) Every unc ountable subset of X c ontains either a c opy of ω 1 or a c opy of ω − 1 1 . Pr o of. Assume Y ⊆ X con tains n either ω 1 nor ω − 1 1 . T hen K ( Y ) is a firs t coun table increasing quotien t of K ( X ), therefore it is Corson compact b y the resu lt of [7]. Nakhm an s on’s theorem [16] implies that K ( Y ) is metrizable, therefore | Y | ≤ ℵ 0 . 5 Note that condition (3’) is w eak er th an (3). In deed, supp ose that (3) holds f or a linearly ordered set X . L et Y ⊆ X b e uncounta ble. Th en there is a one-to-one map f : ω 1 → Y . By (3) there is a stationary set T ⊆ ω 1 suc h that f | T is monotone. If f | T is increasing, then f [ T ] is a copy of ω 1 , otherwise it is a copy of ω − 1 1 . F u r ther, note that condition (3’) implies th e v alidit y of (iii) for the space K ( X ) . Indeed, let L ⊆ K ( X ) b e a closed first countable set. As K ( X ) is zero-dimensional, there is an increasing retraction r : K ( X ) → L . It follo w s that X ( L ) is order-isomorphic to a subset of X . As L is first countable, X ( L ) con tains no copies of ω 1 or ω − 1 1 . By (3’) we get that X ( L ) is countable, so L is metrizable. Belo w (in Example 3.5) we sho w that conditions (1), (2) and (3’) are not sufficient for K ( X ) b eing V aldivia. In p articular, this will d ispro v e the ab o v e conjecture. But b efore pr o ving the example we n eed t wo lemmata. Lemma 3.3. L et X , Y b e line arly or der e d sets such that Y ⊆ X . Denote by f the dual map to the i nc lu si on (henc e f is an incr esing quotient mapp ing of K ( X ) onto K ( Y ) ). The fol lowing c onditions ar e e quivalent: (a) The map f is top olo g i c al ly rig ht-i nv e rtible. (b) Ther e exists an incr e asing map p : conv( Y ) → Y such that p | Y = id Y . (c) Every pr op er gap h A, B i in Y r emains a gap in X , i.e ., whenever A, B ⊆ Y ar e nonempty subsets suc h that A ∪ B = Y , a < b whenever a ∈ A and b ∈ B , A has no maximum and B has no minimum, then ther e i s no x ∈ X with a < x < b for al l a ∈ A and b ∈ B . Pr o of. (a) ⇒ (b) Let g : K ( Y ) → K ( X ) b e a righ t in verse of f , i.e., f ◦ g = id K ( Y ) . It is clear that g must b e in cr easing. W e define p as follo w s. Let y ∈ conv( Y ). Set k = ( y , → ). Then k ∈ K ( X ) and k > 0 K ( X ) . As y ∈ con v ( Y ), there are y 1 , y 2 ∈ Y with y 1 ≤ y ≤ y 2 . Th en 0 K ( Y ) ≤ [ y 1 , → ) ∩ Y < ( y 1 , → ) ∩ Y ≤ ( y , → ) ∩ Y = f ( k ) , hence k > g (0 K ( Y ) ). F urther, give n an y k 1 < k w e hav e f ( k 1 ) ≤ [ y , → ) ∩ Y ≤ [ y 2 → ) ∩ Y < ( y 2 , → ) ∩ Y ≤ 1 K ( Y ) , hence k ≤ g (1 K ( Y ) ). Thus [ k , → ) ∩ g [ K ( Y )] is a clop en fin al segmen t of g [ K ( Y )] not con taining g (0 K ( Y ) ) bu t con taining g (1 K ( Y ) ). So, we can set p ( y ) = f [[ k , → )]. It is n ow clear that p maps con v ( Y ) in to Y and that it is increasing. Finally if y ∈ Y , then [ y , → ) ∩ Y < ( y , → ) ∩ Y and hence f [[ k , → ) ∩ Y ] = [ f ( k ) , → ). The latter clop en interv al corresp onds to y . This completes the argumen t. (b) ⇒ (c) Let p b e su c h a mapping. Let A, B ⊆ Y b e lik e in (c). Sup p ose there is x ∈ X s uc h that a < x < b for eac h a ∈ A and b ∈ B . T hen x ∈ con v ( Y ) and so p ( x ) is defi ned. As Y = A ∪ B necessarily p ( x ) ∈ A or p ( x ) ∈ B . If p ( x ) ∈ A , then a = p ( a ) ≤ p ( x ) for eac h 6 a ∈ A , so p ( x ) is the maximum of A , a con tr ad iction. Sim ilarly , if p ( x ) ∈ B , then p ( x ) is the minim um of B , a contradict ion. (c) ⇒ (a) W e will defin e a right inv erse of f as follo ws. T ak e k ∈ K ( Y ). Th en f − 1 ( k ) is a closed in terv al in K ( X ), say [ α, β ]. If it is a singleton, the d efinition of g ( k ) is clear. Otherwise, if β is isolated from the righ t, set g ( k ) = α . If β is not isolate d fr om the right but α is isolated from th e left, set g ( k ) = β . If this can b e done for eac h k ∈ K ( Y ), it is clear that g is the required right inv erse. It remains to sho w that it is not p ossib le that α < β , α is not isolated from the left and β is n ot isolated from the right . Sup p ose this p ossibilit y tak es place. Let A b e the set of all elements of Y su ch that the resp ectiv e clop en interv al h as m inim um less than k and B b e the set of all elements of Y suc h that the resp ectiv e clop en inte rv al has minimum greater th an k . T hen A and B do satisfy all assump tions giv en in (c). As α < β , and K ( X ) is zero-dimensional, there is a clopen in terv al in K ( X ) with minimum in ( α, β ]. The corresp ond ing elemen t of x pro du ces a con tradiction with (c). Lemma 3.4. L et X b e a line arly or der e d sp ac e. Then K ( X ) is V aldivia if and only if ther e is a family ( X α : α < ω 1 ) satisfying the fol lowing pr op erties: (i) X α is a c ountable subset of X ; (ii) X α ⊆ X β for α < β ; (iii) X λ = S α<λ X α for every limit or dinal λ < ω 1 ; (iv) X = S α<ω 1 X α ; (v) the inclusion X α ⊆ X satisfies c ondition (c) of L e mma 3.3. Pr o of. It follo ws fr om [13, Prop osition 2.6 an d Corollary 4.3] that a compact space K of w eigh t ℵ 1 is V aldivia if and only if th ere is an ω 1 -sequence of retractions ( r α : α < ω 1 ) satisfying • r α [ K ] is m etrizable for eac h α < ω 1 ; • r α ◦ r β = r β ◦ r α = r α for α ≤ β < ω 1 ; • the map α 7→ r α ( x ) is conti nuous (wh en ω 1 is equipp ed with th e ord er top ology) and has limit x for eac h x ∈ K . Moreo ve r, if K is linearly order ed , the retractions may b e c hosen increasing (this follo ws using [11, Prop osition 5.7]). If K = K ( X ) is V aldivia, tak e su c h retractions and set X α = X ( r α [ K ]) canonically em b edded in to X . Th en the family ( X α ) satisfies the required conditions, th e last one follo ws from Lemma 3.3. Con v ersely , let ( X α ) b e a family satisfying the required conditions. Set X ω 1 = X and K α = K ( X α ) for α ≤ ω 1 . If α ≤ β ≤ ω 1 let f β α : K β → K α b e the incresing su rjection dual to 7 the inclusion X α ⊆ X β . Then it is clear that f β α = f γ α ◦ f β γ for α ≤ γ ≤ β ≤ ω 1 , hence w e ha ve an in v erse sequence indexed by ω 1 . Moreo v er, this sequence is contin uous b ecause of condition (iii), all b onding maps are r igh t-inv ertible by L emma 3.3, K α is metrizable for α < ω 1 , therefore the limit K = K ω 1 is V aldivia compact b y [13, Corollary 4.3]. Belo w is the announ ced example. In fact, it is a classical construction d ue to Ku repa [15], generalized by T o dorˇ cevi ´ c in [17, Section 4]. Example 3.5. There is a linearly ordered set Z satisfying conditions (1), (2) and (3’) such that K ( Z ) is n ot V aldivia. Pr o of. Let X = { x ∈ Q ω 1 : | suppt( x ) | < ℵ 0 } b e end o wed with the lexicographic ord er in g, where sup pt( x ) = { α : x ( α ) 6 = 0 } . Th en K ( X ) is a V aldivia compact. Indeed, th e sets X α = { x ∈ X : suppt x ⊂ α } , α < ω 1 , ha ve all prop erties fr om Lemma 3.4. W e shall no w extend X by adding some new elemen ts. Fix a s et S ⊆ ω 1 consisting of limit ordinals. F or eac h δ ∈ S c h o ose a set c δ order isomorph ic to ω and such that su p( c δ ) = δ . No w let Y S = { 1 c δ : δ ∈ S } , where 1 a denotes the c haracteristic function of the set a ⊆ ω 1 . Define X S = X ∪ Y S . Clearly , X S satisfies (1). W e c hec k that X S satisfies (2). W e will use the follo win g easy observ ation: If { a α } α<ω 1 is a monotone ω 1 -sequence in Q ω 1 , then (*) ( ∀ γ < ω 1 )( ∃ α 0 < ω 1 ) { a α ↾ γ } α 0 ≤ α<ω 1 is constan t . Fix a strictly monotone sequence { a α : α < ω 1 } ⊆ X S . Define T = { t ∈ Q <ω 1 : ∃ α < ω 1 ∀ ξ ≥ α ( t ⊆ a ξ ) } . By Q <ω 1 w e mean S α<ω 1 Q α , i.e. fun ctions with rationals v alues whose d omain is a counta ble ordinal. W e consider Q ω 1 ordered by inclusion. Note that T is a chain in Q <ω 1 . Let g = S T . Then either g ∈ T or dom( g ) = ω 1 . The first p ossibility cann ot o ccur, b ecause assuming δ = dom( g ) < ω 1 w e would find (due to (*)) α 0 < ω 1 suc h that a α ↾ δ + 1 is constan t for α ≥ α 0 and then a α 0 ↾ δ + 1 = g ∪ {h δ , a α 0 ( δ ) i} w ould b e an elemen t of T . Th us g ∈ Q ω 1 . It is clear that sup p t( g ) fin ite, b ecause the sequen ce is strictly monotone and hence it con tains at most one of the added elemen ts 1 c δ , δ ∈ S . Thus g ∈ X ⊆ X S . F u rther, w e will show that g is the limit of { a α } α<ω 1 in Q ω 1 . Supp ose that th e sequ en ce { a α } α<ω 1 is increasing. T h en a α ≤ g for all α < ω 1 . Indeed, s u pp ose that there is some α 0 < ω 1 with a α 0 > g . Then for eac h α ≥ α 0 w e hav e a α > g and so there is some γ ( α ) < ω 1 suc h that a α ↾ γ ( α ) = g ↾ γ ( α ) and a α ( γ ( α )) > g ( γ ( α )). As { a α } α<ω 1 is 8 increasing, th e ω 1 -sequence { γ ( α ) } α 0 ≥ α<ω 1 is decreasing. Therefore it is ev entually constant, i.e., there is α 1 ∈ [ α 0 , ω 1 ) and γ < ω 1 suc h that for eac h α ∈ [ α 1 , ω 1 ) w e ha v e γ ( α ) = γ . It follo ws that g ↾ ( γ + 1) / ∈ T , a contradictio n. Finally , it follo ws easily from the definition of g that it is the supr em u m of { a α } α<ω 1 . If { a α } α<ω 1 is decreasing, the pro of is similar. This completes the p ro of of (2). W e n ow sho w that X S satisfies (3’). W e sh all u se the fact that K ( X ) is V aldivia compact. In particular, by Pr op osition 3.2, ev ery uncounta ble su bset of X con tains a cop y of ω 1 or ω − 1 1 . Fix (if p ossible) an u ncoun table set A ⊆ S . Let us denote y δ = 1 c δ for δ ∈ S . W e sh all sho w that { y δ : δ ∈ A } contai ns a monotone subsequence. F or eac h limit ordinal λ < ω 1 fix δ ( λ ) ∈ A suc h th at δ ( λ ) > λ and s et γ ( λ ) = sup suppt( y δ ( λ ) ↾ λ ). By th e Pr essing Do wn Lemma there is a stationary set S and γ 0 < ω 1 suc h that γ ( λ ) = γ 0 for eac h λ ∈ S . Moreo v er, as γ 0 is coun table and sup pt( y δ ( λ ) ↾ λ ) is finite for eac h λ , there are a stationary set S ′ ⊆ S and a finite set F ⊆ γ 0 + 1 such th at su ppt( y δ ( λ ) ↾ λ ) = F for eac h λ ∈ S ′ . F u r ther w e c ho ose λ η ∈ S ′ for η < ω 1 suc h that λ η > δ ( λ θ ) wh en ev er θ < η < ω 1 . It can b e done as S ′ is unb ounded in ω 1 . Finally note that { y δ ( λ η ) } η<ω 1 is decreasing by the defin ition of the lexicographic order. This fi nishes the pro of of (3’). Finally , notice that K ( X S ) is V aldivia compact if and only if S is not stationary . This follo ws from Lemma 3.4. Indeed, s et X δ = { x ∈ X S : ∃ γ < δ : supp t x ⊆ γ } . Then th e family ( X δ : δ < ω 1 ) satisfies all conditions from Lemma 3.4 except for the last one. If th ere is a closed un b ound ed set C ⊆ ω 1 \ S , then the family ( X δ : δ ∈ C ) witnesses that K ( X S ) is V aldivia. Con v ersely , s upp ose that S is stationary and K ( X S ) is V aldivia. L et ( Y δ : δ < ω 1 ) b e the family witnessing it (i.e., satisfying all the conditions from Lemma 3.4). No w, there is a closed unboun ded set C ⊆ ω 1 suc h that X δ = Y δ for eac h δ ∈ C . Cho ose some δ ∈ C ∩ S . Then the sets A = { x ∈ X δ : x < y δ } B = { x ∈ X δ : x > y δ } witness that condition (c) of Lemma 3.3 is violated for Y δ ⊆ X S . Th is is a con tradiction sho wing that K ( X S ) is n ot V aldivia. Before p r o ving Theorem 3.1, we briefly recall the metho d of elemen tary substructures whic h w e use here. In what follo ws, the letter χ w ill denote an uncoun table regular cardinal, b ig enough so that all relev ant ob jects h av e cardinalit y strictly less th an χ . More pr ecisely , denote by H ( χ ) the family of all sets x whose transitive closure tc ( x ) h as card in alit y < χ . R ecall th at tc ( x ) = x ∪ S x ∪ S S x ∪ . . . . No w, sa y in g “ χ is big enough” means th at all ob jects un der consideration (e.g. a giv en top ologica l space, a giv en transformation, etc.) b elong to H ( χ ). 9 The structure h H ( χ ) , ∈i satisfies all the axioms of set theory , except p ossibly the p ow er set axiom. A set M is an elementary substructur e of of h H ( χ ) , ∈ i if M ⊆ H ( χ ) and for ev ery formula ϕ ( x 1 , . . . , x n ), for ev ery a 1 , . . . , a n ∈ M , M | = ϕ ( a 1 , . . . , a n ) if and only if H ( χ ) | = ϕ ( a 1 , . . . , a n ). Here, “ M | = ϕ ” means “ M satisfies ϕ ” in th e usu al sense of mo del theory . The metho d of elemen tary submo dels is based on th e we ll kn o wn L¨ ow enheim-Sk olem Theorem, s aying that ev er y coun table sub set of H ( χ ) can b e enlarged to a countable ele- men tary su bstructure of H ( χ ). As a consequence, give n a countable set S ⊆ H ( χ ), one can easily consrtu ct by indu ction a c h ain { M α } α<ω 1 of coun table element ary su bstructures of h H ( χ ) , ∈i suc h that S ⊆ M 0 and α ⊆ M α for ev ery α < ω 1 . In fact, w e may ev en require that M α ∈ M α +1 and that the c hain b e con tinuous, i.e. M δ = S ξ <δ M ξ for ev ery limit ordinal δ . The last prop ert y follo w s fr om the fact that S ξ <δ M ξ is again elemen tary . Giv en suc h a chain { M α } α<ω 1 and setting δ α = M α ∩ ω 1 , we note that eac h δ α is a countable ordinal and the set C = { δ α : α < ω 1 } is closed and unb ounded in ω 1 . Consequ en tly , if S is a stationary s u bset of ω 1 , there exists α such that M α ∩ ω 1 ∈ S . W e shall use this r emark b elo w. W e refer to [13, 10] for applications of elementa ry su bmo dels in the cont ext of r etractions and V aldivia compacta. More explanations of the m etho d and its use f or finding pr o j ections in Banac h spaces can b e found in [12 ]. L ast but not least, [3] is an imp ortant sur v ey on the use of elemen tary sub structures in general top olog y . Pr o of of The or e m 3.1. Supp ose fi rst that K ( X ) is V aldivia. Th en (1) and (2) are satisfied b y the ab o ve remarks . Let us p ro ve (3). Let ( X α : α < ω 1 ) b e a family with p rop erties from Lemma 3.4. Let p α : conv( X α ) → X α b e an increasing pro jection, i.e. p α ↾ X α = id X α (it exists b y Lemm a 3.3). Let u s extend p α b y setting p α ( x ) = −∞ if x < con v ( X α ) and p α ( x ) = + ∞ if x > con v ( X α ). Assuming −∞ < x < + ∞ for every x ∈ X , th is d efi nes an increasing map from X in to X α ∪ {−∞ , + ∞} . W e shall wr ite y α instead of f ( α ). Fix a suffi cien tly big regular cardinal χ and fix a contin uous c h ain { M α } α<ω 1 of elemen tary substru ctures of h H ( χ ) , ∈i suc h that X ∈ M 0 , f ∈ M 0 , { p α } α<ω 1 ∈ M 0 and α ⊆ M α for α < ω 1 . If β < α < ω 1 , then β ∈ M α , so p β ∈ M α and h ence X β ∈ M α (as X β is the range of p β ). As X β is countable, w e get X β ⊂ M α b y [12, Pr op osition 2]. Therefore X ⊂ S α<ω 1 M α , and so C 1 = { α < ω 1 : X α = X ∩ M α } is a closed un b ound ed set. Let δ α = ω 1 ∩ M α and let C 2 = { α < ω 1 : δ α = α } . Th en C 2 is a closed unb ounded subset of ω 1 , so C 1 ∩ C 2 ∩ S is stationary . Note that eac h δ α is a limit ord in al, therefore p α ( y α ) ∈ X ξ ( α ) ∪ {−∞ , + ∞ } for some ξ ( α ) < α . Using the Pressing Do wn Lemma, we may assume that ξ ( α ) = ξ f or α ∈ S ′ , wh er e S ′ ⊆ C 1 ∩ C 2 ∩ S is s tationary . No w supp ose that for a stationary set S 1 ⊆ S ′ w e hav e th at p α ( y α ) = −∞ . Th en the sequence { y α : α ∈ S 1 } is strictly decreasing. In deed, let α, β ∈ S 1 suc h that α < β . Th en α ∈ M β , so y α ∈ M β . As α ∈ C 1 , we get y α ∈ X β . F urther, p β ( y β ) = −∞ an d so y β < conv( X β ). In particular y β < y α . 10 Similarly , if the set S 2 = { α ∈ S ′ : p α ( y α ) = + ∞} is stationary , we get a strictly increasing sequence y ↾ S 2 . So assu me that the set S ′′ = { α ∈ S ′ : p α ( y α ) ∈ X ξ } is stationary . Using the fact that X ξ is coun table, furth er refining S ′′ w e ma y assume that p α ( y α ) = v ∈ X ξ for all α ∈ S ′′ . No w supp ose α, β ∈ S ′′ are such that α < β and v < y α and v < y β . Then p β ( y β ) = v and p β ( y α ) = y α , b ecause y α ∈ X β . Since p β is order preservin g, necessarily y β < y α . This observ ation sho w s that y ↾ R is strictly decreasing, where R = { α ∈ S ′′ : v < y α } . Similarly , y ↾ L is strictly increasing, where L = { α ∈ S ′′ : y α < v } . One of these sets must b e stationary , un less y has constan t v alue v on a stationary set. Th is completes the p ro of of (3). No w we are going to prov e sufficiency . Let X satisfy conditions (1)–(3). W rite X = S α<ω 1 X α , where ~ x = { X α } α<ω 1 is an increasing c h ain of countable sub sets of X such that X δ = S ξ <δ X ξ whenev er δ is a limit ordinal. Denote by S the set of all ordinals α < ω 1 for w hic h there exist a pr op er gap h A α , B α i in X α and an element y α ∈ X \ X α whic h fills this gap, i.e. a < y α < b whenever a ∈ A α , b ∈ B α . If there exists a closed unboun ded set C ⊆ ω 1 \ S then we are done by Lemma 3.4. So s upp ose S is stationary . Using (3), w e fix a stationary su bset T ⊆ S suc h that ~ y = { y α } α ∈ T is monotone. Note that ~ y cann ot b e constant, b ecause y α / ∈ X α for α ∈ T . Th us, rev ersing the order if necessary , w e may assu m e th at ~ y is str ictly increasing. Fix a big enough r egular cardin al χ and fix a contin uous c h ain { M α } α<ω 1 of elemen tary substru ctures of h H ( χ ) , ∈i suc h that ~ x ∈ M 0 , ~ y ∈ M 0 and α ⊂ M α for α < ω 1 . Similarly as ab o ve w e get that X α ⊂ M β for α < β < ω 1 and hence C 1 = { α < ω 1 : X α = X ∩ M α } is a closed un b ound ed set. Denote again δ α = ω 1 ∩ M α and let C 2 = { α < ω 1 : δ α = α } . Then C 2 is a closed unb ounded subset of ω 1 and so is C 1 ∩ C 2 . Fix δ ∈ C 1 ∩ C 2 ∩ T . C learly , δ is a limit ordinal, so X δ = S ξ <δ X ξ . Recalling that y δ fills th e gap h A δ , B δ i we see that (**) M δ | = ( ∃ b ∈ X )( ∀ α ∈ T ) y α < b. T o show it first note that T ∈ M 0 ⊂ M δ , as T is the domain of ~ y and ~ y ∈ M 0 . F urther, w e kno w that X α ⊂ M δ for eac h α < δ , so X δ ⊂ M δ as w ell. In particular, B δ ⊂ M δ . So, c ho ose 11 some b ∈ B δ . Then b ∈ M . Moreo v er , if α ∈ T ∩ M , then α < δ , so y α < y δ < b . Th is prov es (**). By elemen tarit y , the sequence ~ y is b ounded f rom ab o v e. By (2) th ere exists g ∈ X such that g = sup α ∈ T y α . Find γ ∈ C 1 ∩ C 2 ∩ T suc h that γ ≥ δ and g ∈ X γ = M γ ∩ X . Observe that [ y γ , g ) ∩ X γ = ∅ . Ind eed, if x ∈ X γ ∩ [ y γ , g ) then there w ould exist α ∈ T su c h that x < y α < g ; by elemen tarit y , w e would ha ve x < y β for some β ∈ T ∩ M γ and hence x < y γ , a cont radiction. Recalling that g ∈ X γ , it follo ws that g = min  [ y γ , → ) ∩ X γ  = min B γ . This con tr adicts the fact that y γ fills the gap h A γ , B γ i . 4 The non-zero-dimensional case In this section we giv e a characte rization of not necessarily zero-dimensional V aldivia compact lines. Let K b e a V aldivia compact line. W e in tro duce an equiv alence relatio n ∼ on K b y setting x ∼ y if the int erv al [ x, y ] is connected. It is clear that this is indeed an equiv alence relation and that equiv alence classes are closed interv als. As a closed subset of a V aldivia compact line is again V aldivia by L emm a 2.3 , eac h equiv alence class is a connected V aldivia compact line. By [10, Theorem 5.2] there are only fiv e suc h spaces. Two metrizable ones – the singleton and th e unit interv al [0 , 1] and three others, whic h are denoted by R → + 1, ( R → + 1) − 1 and ← I → in [10]. R → denotes the long lin e, i.e. th e lexicographic pro duct ω 1 · [0 , 1), R → + 1 is its compactificatio n made by add ing the end p oin t. The space ( R → + 1) − 1 is the ord er inv erse of R → + 1. Finally , the space ← I → is the unique connected compact line [ a, b ] s uc h that a < b and for eac h y ∈ ( a, b ) the inte rv al [ y , b ] is ord er homeomorphic to R → + 1 and [ a, y ] is order homeomorphic to ( R → + 1) − 1 . Therefore eac h of the equiv alence classes of ∼ is ord er isomorphic to one of these five s paces. F u r ther, as K has w eight at most equal to ℵ 1 , at most ℵ 1 equiv alence classes con tain more than one p oint. Fin ally , the sp ace K 0 = K \ [ { ( a, b ) : a ∼ b } is V aldivia as we ll (by Lemm a 2.3 ). Moreo ver, it is zero-dimensional, hence the criterion from the pr evious section applies. The space K must also satisfy cond itions (i)–(iii) from Question 1. Hence we hav e pr o ved the necessit y in the follo wing theorem. Theorem 4.1. L et K b e a c omp act line. Define the e quivalenc e ∼ as ab ove and define K 0 by the ab ove formula. The sp ac e K is V aldivia if and only if the fol lowing c onditions ar e satisfie d. • Each e quivalenc e class is a V aldivia c omp act. 12 • The sp ac e K 0 is V aldivia. • Each p oint of unc ountable char acter is isolate d fr om one side. Pr o of. It remains to pro v e the su fficiency . W e will use Lemm a 2.2. Su p p ose th e ab o v e three conditions are satisfied. Let A b e a f amily of clopen int erv als in K 0 suc h that A separates p oint s of K 0 and for eac h x ∈ G ( K 0 ) there are only countably m any elemen ts of A con taining x . F or eac h interv al I ∈ A we define an op en F σ in terv al e I ⊆ K as follo ws. As I is clop en, we ha ve I = [ a, b ] where a is isolated f rom the left and b is isolate d from th e right (in K 0 ). If there is some x < a su ch that x ∼ a , c ho ose some a ′ ∈ ( x, a ). Otherwise set a ′ = a . Similarly , if there is some y > b with y ∼ b , c ho ose b ′ ∈ ( b, y ). Otherwise set b ′ = b . No w set e I =            [ a, b ] , if a ′ = a, b ′ = b, ( a ′ , b ] , if a ′ < a, b ′ = b, [ a, b ′ ) , if a ′ = a, b ′ > b, ( a ′ , b ′ ) , if a ′ < a, b ′ > b. It is clear that e I is an op en F σ in terv al in K . Indeed, if a ′ = a , then a is isolated from the left also in K , and if a ′ < a , then a ′ has countable charact er in K (as it is not isolate d fr om either side). Similarly for b and b ′ . Set e A = { e I : I ∈ A } . Then e A separates p oint s of K 0 and for eac h x ∈ G ( K ) there are only coun tably many elemen ts of e A con taining x . Indeed, if x ∈ G ( K ) ∩ K 0 , then x ∈ G ( K 0 ) and hence it is only in coun tably man y elemen ts of A . As e I ∩ K 0 = I for eac h I ∈ A , x b elongs to only counta bly many elemen ts of e A as w ell. F u r her, s u pp ose that x ∈ G ( K ) \ K 0 . Let [ a, b ] b e the equiv alence class con taining x . Then necessarily a, b ∈ G ( K 0 ). (If , sa y , a has un countable c haracter in K 0 , then it is n ot isolated from either side in K , a con tradiction.) Finally observ e that if x ∈ ˜ I for some I ∈ A , th en necessarily either a ∈ I or b ∈ I . Thus there can b e at most countably many su ch I ’s. No w we extend e A in order to separate p oin ts of K . Fix an y equiv alence class [ a, b ] with a < b . Then a, b ∈ K 0 . Th er e are four p ossibilities: (i) [ a, b ] is order-homeomorph ic to [0 , 1]. Th en let B a,b b e a coun table basis of ( a, b ) consisting of in terv als. (ii) [ a, b ] is order-homeomorphic to ← I → . Th en [ a, b ] is clop en in K . Let B a,b b e a family of op en F σ in terv als in [ a, b ] separating p oin ts of [ a, b ] such that [ a, b ] ∈ B a,b and eac h p oin t of ( a, b ) is contai ned only in coun tably many elemen ts of B a,b . (iii) [ a, b ] is order-homeomorphic to R → + 1. Th en b is isolated fr om the r igh t in K . Let C b e a family of op en F σ in terv als in [ a, b ] separating p oints of [ a, b ] su c h that [ a, b ] ∈ C and eac h p oint of [ a, b ) is con tained only in counta bly many elemen ts of C . Set B a,b = { I ∩ ( a, b ] : I ∈ C } . (iv) [ a, b ] is order-homeomorphic to ( R → + 1) − 1 . Define the family B a,b analogously as in (iii). 13 Finally set U = e A ∪ [ {B a,b : a, b ∈ K 0 , a < b, a ∼ b } . Then U is a family of op en F σ in terv als in K separating p oin ts of K s uc h that eac h p oint of G ( K ) is con tained only in count ably many elemen ts of U . T h is prov es that K is V aldivia. 5 When the con jecture is v alid In this section w e pro v e the follo wing theorem. Theorem 5.1. L e t K b e a c omp act line satisfying the c onditions (i)–(iii) fr om Question 1 such that the closur e of the set of p oints of u nc ountable char acter is sc atter e d. Then K is V aldivia. Pr o of. W e define an equiv alence relation on K by setting a ∼ b if and only if [ a, b ] is V aldivia. W e will prov e, using the conditions (i)–(iii), that it is really an equiv alence relation and th at the quotien t L = K/ ∼ is a connected compact line. F ur ther, if w e denote by q the canonical quotien t mapping of K onto L , w e will sh o w that the image un der q of the set of p oin ts of uncounta ble characte r is dense in L unless L is a singleton. If w e prov e all this, it follo w s that L is a connected compact line whic h is at the same time scattered (as a con tin uous image of a scattered compact space is again scattered). Thus L is a singleton, h ence K is V aldivia. So, it is enough to pr o ve the ab o ve men tioned p r op erties of ∼ . Step 1. ∼ is an equ iv alence relation. First note that by Lemma 2.3 w e hav e x ∼ y when ever x, y ∈ [ a, b ] and a ∼ b . T o pro ve that it is r eally an equiv alence relation it r emains to chec k that a ∼ c wh enev er a < b < c and a ∼ b and b ∼ c . If b is isolated from one side, then [ a, c ] is th e top ologica l sum of t w o V aldivia compact spaces and therefore it is V aldivia as w ell. If b is isolate d from neither side, it has coun table c haracted. As the top ologica l sum [ a, b ] ⊕ [ b, c ] is V aldivia and [ a, c ] is th e qu otien t made by ident ifying the t w o copies of b , we get that [ a, c ] is V aldivia by Lemma 2.4. Step 2. Eac h equiv alence class of ∼ is closed. Let a ∈ K b e arbitrary and let b b e th e su premum of all x ∈ K with a ∼ x . W e will sho w that a ∼ b . Su pp ose b > a . If b is isolated from the left, th en clearly a ∼ b (in this case the sup rem u m in th e definition of b is obvi ously atta ined). So su pp ose that b is not isolated from the left. If there is some x ∈ ( a, b ) s u c h that [ x, b ] is connected, then [ x, b ] is a conn ected compact line satisfying (i)–(iii) and hence it is V aldivia b y [9, Section 3.2], so x ∼ b . As a ∼ x , we get a ∼ b . Next supp ose that b is not isolated fr om th e left and [ x, b ] is connected for no x ∈ ( a, b ). There are t wo p ossibilities: 14 (a) Th ere is a sequence b n in ( a, b ) suc h that b n ր b . Without loss of generalit y w e ma y supp ose that eac h b n is isolated from one s id e. By p assing to a subsequence we ma y s upp ose that all the p oint s b n are isolated fr om th e same side. Sup p ose that eac h b n is isolated from the left. Th en [ a, b ] is homeomorphic to th e one-p oin t compactification of the top olog ical sum of V aldivia compacta [ a, b 1 ), [ b 1 , b 2 ) , [ b 2 , b 3 ) , . . . . Therefore [ a, b ] is V aldivia by [7, Th eorem 3.35] and so a ∼ b . If eac h b n is isolated fr om the righ t we can pr o ceed sim ilarly . (b) b h as un coun table charact er in [ a, b ]. T hen w e can find an in creasing h omeomorphic cop y ( b α : α < ω 1 ) of ω 1 in ( a, b ) with su prem um b . Again, w e can without loss of generalit y supp ose that for eac h isolated ordinal α < ω 1 the p oin t b α is isolated from one side and that all these p oints are isolated fr om the same side. Supp ose th ey are isolated fr om the left. Set X α = [ b α − 1 , b α ) for α < ω 1 isolated (where b − 1 = a . Let X b e the [0 , ω 1 + 1)-sum of these spaces in the sense of [8]. Then X is V aldivia by [8, Pr op osition 3.4]. Moreo v er, X is homeomorphic to the compact line made fr om [ a, b ] by duplicating b α for eac h limit ordinal α < ω 1 . As eac h of these p oin ts has counta ble characte r, by Lemma 2.4 we get that [ a, b ] is V aldivia and so a ∼ b . If b α is isolated from the right f or eac h isolated α < ω 1 , we can p ro ceed similarly . Step 3. L = K/ ∼ is a connected compact line. By Steps 1 and 2 w e get that the quotien t L = K/ ∼ is a Hausdorff compact space, in f act a compact line. Let us pro v e that L is connected. Su pp ose that a, b ∈ L are su c h that a < b and ( a, b ) = ∅ . Let [ x 1 , y 1 ] b e the equiv alence class in K corresp ondin g to a and [ x 2 , y 2 ] b e the equiv alence class corresp onding to b . Then y 1 < x 2 and [ y 1 , x 2 ] = { y 1 , x 2 } and therefore y 1 ∼ x 2 , a cont radiction. Step 4. Denote by q the quotient mapping of K onto L asso ciated to ∼ . T h en the image under q of the set of all p oin ts of un coun table characte r in K is dense in L un less L is a singleton. T o see this, let y ∈ L b e an y p oin t different from the end p oint . Th e inv erse image of y is a closed in terv al [ a, b ] ⊆ K . If b is isolated fr om the r igh t, then there is b + = min( b, → ). Clearly b ∼ b + , hence also a ∼ b + . This is a contradictio n to th e fact that [ a, b ] is an equiv alence class of ∼ . F urther, let c > b b e arbitrary . If [ b, c ] is firs t coun table, then it is metrizable b y (iii) and so b ∼ c . Again w e get a ∼ c wh ich is a con tradiction. Thus [ b, c ] con tains a p oint of uncoutable c haracter. Therefore there is a net d ν of p oin ts of uncountable characte r con verging to b . Finally , q ( d ν ) → y . The pro of is complete. 15 6 Compact lines whic h are con tin uous images of V aldivia c om- pacta In this section w e discuss the class of compact lines whic h are con tinuous images of V aldivia compacta. The main question in this con text is the follo wing one. Question 2 . Let K b e a compact line wh ic h is a con tin u ous image of a V aldivia compact space. Is there a V aldivia compact line L and an order-preserving con tinuous su rjection of L on to K ? In [9] it is conjectured that the answer is p ositiv e (Conjecture 3.6). It follo ws from [9, T heorem 3.7] that the answ er is p ositiv e for scatte red compact lines. Let us n ote th at the w eigh t of a compact line whic h is a conti nuous image of a V aldivia compact cannot exceed ℵ 1 . Th is fact can b e pr o ved b y similar argu m en ts as in [10, Prop. 5.5]. W e do not kno w the answe r to this q u estion. Nonetheless, we pr ovide a c haracterization of order-preserving quotien ts of V aldivia compact lines. Theorem 6.1. L et K b e a c omp act line. Denote by A the set of al l p oints of unc ountable char acter in K which ar e not isolate d fr om any side. Then K is an or der-pr e se rvi ng quotient of a V aldivia c omp act line if and only if the c omp act line made fr om K b y duplic ating al l p oints of A is V aldivia. Pr o of. The if part is ob vious (the order preserving quotien t is m ad e b y collating b ac k th e duplicated p oints). Let us sh o w the only if part. Let L b e a V aldivia compact line and ϕ : L → K an order- preserving con tin uous sur jection. L et ∼ b e the asso ciated equiv alence relation on L , i.e. x ∼ y if and only if ϕ ( x ) = ϕ ( y ). By our assump tions the equ iv alence classes are closed interv als. Set L 1 = L \ [ { ( a, b ) : a ∼ b } . Then L 1 is again a V aldivia compact lin e b y L emma 2.3. Moreo v er, ϕ ( L 1 ) = K and ϕ is at most t wo-to -one. Denote the restriction of ∼ to L 1 again by ∼ . Then eac h eqiv alence class has at most t wo p oin ts. F or eac h equiv alence class { a, b } su c h that a < b and at least one of the p oin ts a, b is isolated in L 1 , c h o ose an isolated p oint y a,b ∈ { a, b } . Denote by L 2 the compact line made from L 1 b y omitting all these p oint s y a,b . Th en L 2 is again a V aldivia compact line (Lemma 2.3) and ϕ ( L 2 ) = K . Denote the restriction of ∼ to L 2 again by ∼ . Then the equiv alence classes hav e at most tw o p oin ts and, moreo v er, if an equiv alence class has tw o p oints, n one of them is isolated in L 2 . Define an equiv alence relation ∼ 2 on L 2 b y the f ollo wing form ula x ∼ 2 y ⇔ x = y or ( x ∼ y an d b oth p oints x, y ha ve countable c h aracter in L 2 ) Then L 3 = L 2 / ∼ 2 is a V aldivia compact line by Lemma 2.4. 16 Finally it is easy to see that L 3 is exactly the compact line m ad e from K b y dup licating all p oint s of un coun table c haracter whic h are not isolated from any side. As a consequence we get that the non-V aldivia compact lines constructed in Example 3.5 are not order-preservin g quotien ts of a V aldivia compact line. Are they con tinuous images of a V aldivia compact space? Ac kno wledgemen ts The second author would like to thank the Departmen t of Mathematica l Analysis of Ch arles Univ ersit y in Prague for su pp orting his visits (F all 2006, F all 2007 ) when most of this work w as b eing done. 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