Fully closed maps and non-metrizable higher-dimensional Anderson-Choquet continua

FULL Y CLOSED MAPS AND NON-METRIZABLE HIGHER-DIMEN SIONAL ANDERSON-CHOQUET CONTINUA JERZY KRZEMPEK Abstract. F edorch uk’s fully closed (con tinuous) maps and resolutions are applied in constructions o f non-metrizable higher-dimensio nal analogues of An- derson, Cho quet, and Co ok’s r igid contin ua. Ce r tain theorems o n dimension- low ering maps a re p roved for inductiv e dimensio ns and fully clos ed maps from spaces that need not b e hereditarily nor mal, and some examples of contin ua hav e non-coinciding dimensions. F ully closed (con tin uous) maps and resolutions app ear in n umerous construc- tions (see S. W atson [48], V.V. F edorc h uk [26] for surv eys), in particular, in con- structions of homogeneous spaces with non-coinciding dimensions. In this pap er w e apply the maps in order to obtain examples of con tin ua with strong, hereditary rigidit y prop erties. A non-degenerate con tin uum X is called — an A nderson-C ho quet c o n tinuum if ev ery no n- degenerate sub contin uum P of X has exactly one em b edding P → X , the iden tit y id P ; — a Co ok c ontinuum if ev ery non-degenerate sub contin uu m P of X has exactly one non-constan t map P → X , the iden tit y id P . Examples w ere constructed—respectiv ely—b y R.D. Anderson, G. Choquet [2] ( a plane hereditarily decomposable con tin uum), and H. Co ok [1 4] (a metric, one- dimensional, hereditarily indecomposable con tin uum). T. Mać k o wiak [37] con- structed a metric, chainable , hereditarily decomp osable Co ok con tin uum. All kno wn e xamples of suc h con tin ua are one-dimensional. A metric Cook con tin uum mus t hav e dimension ≤ 2 (Maćk owiak [38]), and if it is hereditarily indecomposable, then it m ust b e one-dimensional (Kr zemp ek [32]). On the other hand, sev eral authors inv estigated rigidit y prop erties of higher-dimensional con- tin ua (J.J. Charatonik [8], M. Reńsk a [45], E. P ol [40 – 43, 3 4 ], see [32] for more references). In [32 ] the presen t a uthor constructed a metric, n - dimensional (ar- bitrary n > 1 ), hereditarily indecomp osable con tin uum no tw o of whose disjoin t n -dimensional subcontinua are homeomorphic, but he was not able to ensure that the contin uum b e Anderson-Cho quet. 2000 Mathematics Subje ct Classific ation. 54 F15, 54 F4 5. Key wor ds and phr ases. F ully clo s ed map, ring-like map, non-co inciding dimensions, Co ok contin uum, Ander s on-Cho quet, her editarily indecomp osab le, chainable contin uum. The author was partially supp orted by MNiSW grant no . N2 01 03 4 31/ 2717. 1 2 J. KRZEMPEK In this pap er we a c hiev e b etter results f o r t he non-metric case: w e construct examples of non-metrizable higher-dimensional (with resp ect to the dimensions dim, ind, and Ind) Anderson-Cho quet con tin ua and Co ok con tin ua. Our other aim is a study of b eha vior of the Charalambous-Filipp ov-Iv anov inductiv e dimen- sion Ind 0 under fully closed maps. In Sections 1–3 w e gather some facts ab out fully closed maps, ring-lik e maps, and dimensions. Application of Ind 0 simplifies estimating inductiv e dimensions in some we ll-kno wn examples. W e show that, if f is a ful ly close d ma p f r o m a n on- empty c om p act sp ac e X to a first c ountable sp ac e, then Ind 0 X ≤ Ind 0 f X +Ind 0 f . This enables us to pro ve that (1) F edorc h uk’s first coun table compact spaces [20 ] with dim = n < 2 n − 1 ≤ ind ≤ 2 n ha v e also Ind ≤ Ind 0 = 2 n , and (2) V.A. Chaty rk o’s c hainable con tinua and homogeneous con tin ua [11] with dim = 1 and ind = n ha v e also Ind = Ind 0 = n . In Section 3 w e slightly mo dify the F edorc h uk-Emeryk- Chat yrko resolution t heorem, and t hat is our main to ol fo r constructions. Section 4 con ta ins a construction of hereditarily indecomp osable Anderson- Choquet con tinua with dim = n (a r bitra r y n > 1 ) and a construction of a Co ok con tinuum with dim = 2 . In Section 5 w e obtain c hainable (hence, dim = 1 ), hereditarily decomp osable Co ok con tin ua with n ≤ ind ≤ Ind ≤ Ind 0 = n + 1 . All the con tin ua are separable and first countable , and some hav e dim < ind . 1. Preliminaries: continua, maps, and covering dimension By a sp ac e we mean a regular T 1 top ological space, and all considered map s are con tinuous and closed. A c o n tinuum is a non-empty connected compact space. A sub con tin uum A of a space X is said to be terminal if for ev ery contin uum B ⊂ X , either A ∩ B = ∅ , A ⊂ B , or B ⊂ A . A contin uum is said to b e — de c omp o sable if it is the union of t wo prop er subcontin ua; — her e ditarily de c o m p osable (abbrev. HD) if eac h o f its non-degenerate sub con- tin ua is decomposable; — her e ditarily inde c omp osable (abbrev. HI) if none of its sub con tinua is decom- p osable (equiv alen tly: eac h of its sub contin ua is terminal); and — chainable if for ev ery op en co v er, there exists a natural nu m b er n and a close d refinemen t F 1 , F 2 , . . . , F n of the co ver suc h that F i ∩ F j 6 = ∅ iff | i − j | ≤ 1 . A (closed con tinuous ) map f : X → Y is said to b e — irr e ducible if for ev ery closed prop er subset F X , also f F f X ; — monotone [ atomic ] if for ev ery p oin t y ∈ Y , the pre-image f − 1 y is a contin uum [respectiv ely: a terminal con tinuum ]; — ring-like if fo r ev ery p oint x ∈ X and ev ery pair of o p en sets U ∋ x and V ∋ f x , there is an op en set V ′ ∋ f x suc h that V ′ ⊂ V and f − 1 b d V ′ ⊂ U ; — ful ly close d if for ev ery pair of disjoint closed subsets F , G ⊂ X , the in tersection f F ∩ f G is a discrete subspace of Y . 1 1 An extensive survey [26] by F edorch uk is devoted to fully clo sed maps, ring-like maps, and their applications. See [26, Section I I.1] for equiv alent definitions of fully closed ma ps. F or terms FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 3 W e shall frequen tly use the simple fact that, if f is a ful l y clo se d [ ring-li k e ] map fr om a sp ac e X an d X ′ ⊂ X is close d, then also the r estriction f | X ′ is ful ly close d [ r e s p e ctively: ring-like ]. The followi ng prop osition is w ell-kno wn (cf. [26, Prop osition I I.3.10]). 1.1. Prop osition. Supp ose that X an d Y ar e c omp act sp ac es, and f : X → Y is a m ap whose eve ry p oint-inverse is metrizable. If the set C 2 f = { y ∈ Y : card f − 1 y > 1 } is c ountable a n d Y is metrizable, then X is metrizable. The c on v erse is true if mor e over f is a ful ly close d map. Pr o of. If C 2 f is coun table and Y is metrizable, then the diagonal { ( x, x ) ∈ X × X : x ∈ X } is a G δ -subset of X × X . A Shne ˘ ıder theorem (see R. Engelking [18, Exercise 4.2.B]) implies that X is metrizable. Assume that f is fully closed and there is a metric on X . Using sequen tial compactness of X , one easily c hec ks that the set { y ∈ Y : diam f − 1 y ≥ 1 /n } is finite for ev ery n . Th us, C 2 f is coun table.  1.2. Prop osition (A. Emeryk, Z. Horbanowi cz [17, Theorem 1 ]). A map f fr o m a c ontinuum X is atomic i ff A = f − 1 f A for every c on tinuum A ⊂ X such that f A is not a single p oint.  The p oin t in the foregoing prop o sition is that, if the irreducibil it y condition A = f − 1 f A is satisfied for all sub con tin ua A ⊂ X with non-degenerate images, then f is a monotone map. 1.3. It is easily seen that an y ring-lik e map f : X → Y has ev en a stronger prop- ert y: it is c onne cte d irr e ducible, i.e. A = f − 1 f A for ev ery closed subspace A ⊂ X suc h that f A is connected and con tains more than one p oint (see [26, I I.1.15]). In particular, f is irreducible whenev er Y = f X is connected and contains more than one p oin t. 1.4. Prop osition. If f is a ring-l i k e map fr om a c omp act sp ac e X onto a non- de gener ate c on tinuum, then X is a c ontinuum and f is an atomic map . Pr o of. Assume that f : X → Y is a ring-like map, X is compact, and f X is a non- degenerate con tin uum. Supp ose that X = A ∪ B , where A and B are non-empt y , closed, and disjoin t. By the remark 1.3, the complemen t f X \ f A is non-empt y . Consider a n y component M of f X \ f A . By Janiszewski’s b oundary bumping theorem (see K. Kurato wski [35, §47, I I I, Theorem 2]), the closure cl M meets f A , and is not a single p oin t. Since cl M ⊂ f B and f is connected irreducible, w e obtain f − 1 cl M ⊂ B . A con tradiction. Therefore, X is a con tinuum . Finally , f is an atomic map by Prop osition 1.2 and the remark 1.3.  The next useful prop o sition b elongs to fo lklore (see F edorc huk [26, the pro of of Lemma I I I.3.3] and [27, Prop o sition 1.3]). not explicitly defined her ein the reader is directed to the mono graphs by R. Engelking [18, 1 9] and K. Kuratowski [35]. 4 J. KRZEMPEK 1.5. P rop osition. Supp os e that f is a ( close d ) monotone map fr om a sp ac e X , and A, B ⊂ f X ar e disjoi n t close d sets. If L is a p artition 2 in X b e twe e n f − 1 A and f − 1 B , then f L is a p artition in f X b etwe en A and B .  1.6. Prop osition. S upp o se that f : X → Y and g : Y → Z a r e surje ctive ring- like m aps b etwe en c omp act sp ac es X , Y , and Z . If g is a mon otone ma p , then the c omp osition g f is rin g-like. Pr o of. T ake x ∈ X and op en sets U ∋ x , W ∋ z = g f x . W e can assume g f U ⊂ W . Since f is ring-like, there is an op en set V ′ ∋ f x such that cl V ′ ⊂ g − 1 W and f − 1 b d V ′ ⊂ U . If g − 1 z is not a singleton, w e can moreo v er ha ve g − 1 z 6⊂ cl V ′ . Clearly , b d V ′ ⊂ Y \ f ( X \ U ) . There are tw o cases. (1) If g − 1 z is a non-degenerate con tinuum, then there is a p oint y ∈ b d V ′ with g y = z . The set Y \ f ( X \ U ) is an op en neigh b orho o d of y , and there is an op en set W ′ ∋ z suc h that W ′ ⊂ W and g − 1 b d W ′ ⊂ Y \ f ( X \ U ) . Th us, ( g f ) − 1 b d W ′ ⊂ U . (2) If g − 1 z is a single p oin t, then b d V ′ is a partition in Y b etw een f x and g − 1 ( Z \ W ) . By Prop osition 1.5, g b d V ′ is a partition in Z b et w een z and Z \ W . Hence, there is a n op en set T ⊂ cl T ⊂ W such that z ∈ T and b d T ⊂ g b d V ′ . As g is ring-lik e and b d T ⊂ W ∩ g [ Y \ f ( X \ U )] , for each t ∈ b d T there is an op en set W t ⊂ W suc h that g − 1 b d W t ⊂ Y \ f ( X \ U ) . Then, b d T ⊂ W t 1 ∪ . . . ∪ W t n , where t 1 , . . . , t n ∈ b d T . W e put W ′ = T ∪ W t 1 ∪ . . . ∪ W t n ⊂ W , a nd hav e b d W ′ ⊂ b d W t 1 ∪ . . . ∪ b d W t n . Th us, w e obtain g − 1 b d W ′ ⊂ Y \ f ( X \ U ) and ( g f ) − 1 b d W ′ ⊂ U .  1.7. Prop osition (cf. Chat yrk o [11, Prop osition 2]). Supp ose that f : X → Y is a surje ctive ring-li k e map fr om a c omp act sp ac e X , and g : Y → Z is a surje ctive monotone map without de g e ner ate p oint-inverse s . If every p artition in Y c ontains a p oin t-in v erse of g , then every p artition in X c ontains a p oi n t-inverse of the c omp osition g f . Pr o of. Assume that X 6 = ∅ a nd ev ery partition in Y con tains a p oin t-inv erse of g . Since the empt y set is not a partition in Y , Y is a non-degenerate con tin uum. Hence, f is irreducible by 1.3, a nd monoto ne b y Prop osition 1.4. T ak e a pa r titio n L in X . The irreducibilit y of f implie s that L is a partition b et w een some p oin t- in vers es f − 1 a and f − 1 b , where a, b ∈ Y . By Prop osition 1.5, f L is a partitio n in Y b etw een a and b . Then g − 1 z ⊂ f L for some z ∈ Z , and again b y 1.3, f − 1 g − 1 z ⊂ L .  1.8. Prop osition ( implicite Chat yrk o [10]). If f is a ring-like m ap fr om a c om - p act sp a c e X , then dim f X ≤ dim X . Pr o of (cf. Chat yrk o [10], p. 124). W e shall pro v e that for ev ery natural n umber n , the inequalit y n ≤ dim f X implies n ≤ dim X . F or n = 0 , this is ob vious. F or n = 1 , f X con ta ins a non-degenerate con tin uum Y . Then, b y Prop osition 1.4, f − 1 Y is a non- degenerate con tinuum, a nd hence, 1 ≤ dim X . 2 W e s ay that a close d set L ⊂ X is a p artition in X if the co mplemen t X \ L is not connected. Moreov er, L is a p artition in X b etwe en disjoint set s A, B ⊂ X if there a re disjoint op en sets U, V ⊂ X such that X \ L = U ∪ V , A ⊂ U , a nd B ⊂ V . FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 5 Let us recall that a normal space Y has dim Y ≥ n iff there exists an essen tial family ( A 1 , B 1 ) , . . . , ( A n , B n ) in Y , i.e. A i , B i ⊂ Y are disjoint closed subsets for eac h i , and for ev ery part itio ns L i b et w een A i and B i , the in tersection T n i =1 L i is non-empt y (cf. Engelking [19, Theorem 3.2.6]). Let 2 ≤ n ≤ dim f X . Since f X contains a comp onen t of dimension ≥ n , w e can a ssume that f X is a con tin uum. Then, f is a monotone map b y Prop osition 1.4. T ak e an ess en tial family ( A 1 , B 1 ) , . . . , ( A n , B n ) in f X . W e shall sho w that the pre-images ( f − 1 A 1 , f − 1 B 1 ) , . . . , ( f − 1 A n , f − 1 B n ) form an essen tial family in X . If L i are partitio ns in X b et wee n f − 1 A i and f − 1 B i , then f L i are partitio ns in f X b et w een A i and B i (Prop osition 1.5). By Lemma 5.2 in [47], t he in tersection T n i =2 f L i con ta ins a con tinuu m P whic h meets bo th A 1 and B 1 . Since f ( L i ∩ f − 1 P ) = P for i = 2 , . . . , n , the remark 1.3 implies that f − 1 P = L i ∩ f − 1 P and f − 1 P ⊂ T n i =2 L i . As f is monotone, f − 1 P is a con tinuum, f − 1 P meets L 1 , and hence, T n i =1 L i is non-empt y . Therefore, n ≤ dim X .  The fib erwise co ve ring dimension of a map f : X → Y is defined as dim f = sup { dim f − 1 y : y ∈ Y } . Other fib erwise dimension functions ind , Ind , etc for maps are defined similarly . 1.9. Theorem (F edorc huk , see [26, Theorem I I I.2.4]). If f is a ful l y close d map fr om a normal sp ac e X , then dim X ≤ max { dim f X , dim f } .  The followi ng is a conseque nce o f Theorem 1.9 and Prop osition 1.8. 1.10. Corollary . If f is a ring-like ful ly c l o se d map fr om a c omp act sp ac e X , then dim X = max { dim f X, dim f } .  2. Maps t ha t reduce inductive dim ensions Since the theory of Ind is unsatisfactory outside the class of hereditarily nor- mal spaces, we shall use a not her inductiv e dimension function Ind 0 , whic h w as in tro duced by M.G. Charalambous [5, 6] and A.V. Iv anov [31]. 2.1. Definition. F or normal spaces X , Ind 0 X ∈ { − 1 , 0 , 1 , 2 , . . . , ∞} is defined so that (a) Ind 0 X = − 1 iff X = ∅ ; (b) Ind 0 X ≤ n ≥ 0 iff for ev ery pair of disjoin t closed sets A, B ⊂ X , b et w een A and B there is a G δ partition L suc h that Ind 0 L ≤ n − 1 ; (c) Ind 0 X = n iff Ind 0 X ≤ n and it is not true that Ind 0 X ≤ n − 1 ; (d) Ind 0 X = ∞ if for ev ery n ∈ N , it is not true that Ind 0 X ≤ n . It is clear that Ind X ≤ Ind 0 X for ev ery normal space X , and Ind X = Ind 0 X if X is p erfectly normal. 2.2. Countable sum t heorem for Ind 0 (Charalam b ous [6], Iv ano v [31]). Sup- p ose that X = S ∞ i =1 F i is a normal sp ac e, and F i ar e close d G δ -subsets of X . I f Ind 0 F i ≤ n for every i , then Ind 0 X ≤ n .  6 J. KRZEMPEK The assumption that F i are G δ -sets is necess ary in Theorem 2.2 ev en if X is a hereditarily normal compact space, see [31]. Besides [6, 31], see Charalam b ous and Chatyrk o [7] for more (also bibliographical) information on Ind 0 . The follo wing theorem on dimension-lo w ering fully closed maps seems to b e imp ortan t b ecause of its applications. 2.3. Theorem. I f f is a ful ly close d m a p fr om a non-empty normal sp ac e X to a sp ac e whose every discr ete close d subsp ac e is a G δ -set, then Ind 0 X ≤ Ind 0 f X + Ind 0 f . W e shall mo dify the pro of of Theorem I I I.2.8 in [26]. A t first, w e need some standard preparation (see [26, pp. 4213–4 2 16] for details). Let f : X → Y b e a map, and M ⊂ Y b e an arbitrary set. Consider the decomp osition M = { f − 1 y : y ∈ Y \ M } ∪ {{ x } : x ∈ f − 1 M } of X . Let Y M = X/ M b e the quotien t space, f M : X → Y M the natural quotien t pro jection, and π M : Y M → Y the only map suc h that f = π M f M . If f is ful ly close d, then M is upp er s emic ontinuous, Y M is a r e gular sp a c e, and f M , π M ar e ful ly close d m aps. A pro of of this lemma (cf. [19, Lemma 1.2.9]) is routine. 2.4. Lemma. Supp ose that M , A, B ⊂ X ar e close d s ubsets of a normal sp ac e X , A ∩ B = ∅ , and L is a p artition in M b e twe e n M ∩ A and M ∩ B . I f X \ L is a normal sp ac e, then ther e ar e disjoin t op en sets U, V ⊂ X s uch that A ⊂ U , B ⊂ V , M \ L = ( U ∪ V ) ∩ M , and cl U ∩ cl V ⊂ L .  Pr o of of The or em 2.3. W e start with some general construction fo r arbitrary X , f , Y = f X , and disjoint closed sets A, B ⊂ X . W e can assume that p = Ind 0 Y < ∞ and q = Ind 0 f < ∞ . Clearly p, q ≥ 0 . Since f is fully closed, M = f A ∩ f B is a discrete closed subspace of Y . Consider Y M , f M : X → Y M , and π M : Y M → Y . The restriction f M | f − 1 M is a homeomorphism onto N = ( π M ) − 1 M , and we shall construct a G δ partition in Y M b et w een the disjoin t sets f M A and f M B . The pre- image f − 1 M is homeomorphic to the discrete sum of p oin t-inv erses f − 1 y , y ∈ M , and hence, Ind 0 N ≤ q . There is a G δ partition L in N b etw een N ∩ f M A a nd N ∩ f M B , where Ind 0 L ≤ q − 1 . As f M is a closed map, Y M is a normal space. Since M ⊂ Y is a G δ -set, Y M \ N and Y M \ L are F σ -sets in Y M , and hence, they are also normal spaces (see [1 8 , Exercise 2.1.E]). By Lemma 2.4, there are disjoin t op en sets U, V ⊂ Y M suc h that f M A ⊂ U , f M B ⊂ V , N \ L = ( U ∪ V ) ∩ N , and cl U ∩ cl V ⊂ L . As Y \ M is an op en F σ subset of Y , it is a coun ta ble union of closed G δ subsets F i of Y . Since π M | Y M \ N is a homeomorphism on to Y \ M , w e obtain Ind 0 ( Y M \ N ) ≤ p b y Theorem 2.2. Th us, there are disjoin t op en sets U ′ , V ′ ⊂ Y M \ N with cl U \ N ⊂ U ′ , cl V \ N ⊂ V ′ , and L ′ = Y M \ ( N ∪ U ′ ∪ V ′ ) is a G δ -set with Ind 0 L ′ ≤ p − 1 . Observ e that L ∪ L ′ = Y M \ ( U ∪ U ′ ∪ V ∪ V ′ ) is a G δ partition in Y M b et w een f M A and f M B . It follo ws t ha t ( f M ) − 1 ( L ∪ L ′ ) is a G δ p artition in X b etwe en A and B . FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 7 W e pro ceed b y induction on p . If p = 0 , then L ′ = ∅ a nd Ind 0 ( f M ) − 1 ( L ∪ L ′ ) = Ind 0 L ≤ q − 1 . As w e to o k arbitrary sets A and B , w e ha v e Ind 0 X ≤ q = p + q . Assume the theorem is true for fully closed maps whose images ha v e Ind 0 < p > 0 . Then, L ′ is the coun table union o f closed G δ -sets L i = L ′ ∩ ( π M ) − 1 F i ⊂ Y M with Ind 0 L i ≤ p − 1 . The restrictions f M | ( f M ) − 1 L i : ( f M ) − 1 L i → L i are fully closed, and b y the induction hypothesis w e obta in Ind 0 ( f M ) − 1 L i ≤ p + q − 1 . Since L and ( f M ) − 1 L are homeomorphic, w e hav e Ind 0 ( f M ) − 1 ( L ∪ L ′ ) ≤ p + q − 1 b y Theorem 2 .2. W e ha v e sho wn that Ind 0 X ≤ p + q b ecause ( f M ) − 1 ( L ∪ L ′ ) is a G δ partition betw een disjoin t closed sets A and B , whic h w ere tak en arbitrarily .  2.5. Corollary . Supp ose that f is a ful ly close d map fr o m a no n -empty normal sp ac e X on to a p erfe ctly normal sp ac e . I f every p oint-inverse of f is p erfe ctly normal, then Ind X ≤ Ind f X + Ind f .  The foregoing coro llary ma y b e considered as an Ind -analogue of F edorc huk ’s Theorem 4 in [20], whic h w as stated for ind and sp ecial, resolution fully closed maps f . In a recen t pap er [28, pp. 117– 1 20] F edorc h uk prov es the inequalit y for Ind and resolution maps f , where f X are (metric, compact) tw o-manifolds. Theorem 2.3 and Corollary 2.5 enable estimating inductiv e dimens ions in some w ell-know n constructions (see [26] for a surv ey). In particular, F edorc h uk’s con- tin ua B ([2 0 ])—let us write B n instead—w ere the first examples of s e p ar able and first c ountable compact spaces with non- coinciding dimensions dim a nd ind . F e- dorc huk pro v ed that dim B n = n and 2 n − 1 ≤ ind B n ≤ 2 n . Since eac h B n has a fully closed map onto the n -dimensional sphere, and ev ery p oint-in v erse of the map is homeomorphic to the n - dimensional torus, we obtain 2.6. Corollary . F e dor chuk’s c ontinua B n have also Ind B n ≤ Ind 0 B n ≤ 2 n .  (In fact, w e shall see that Ind 0 B n = 2 n by Theorem 2 .12.) Chat yrko [11] constructed separable first coun table con tin ua I n and ( S 1 ) n , and pro v ed that I n are c hainable, ( S 1 ) n are homogeneous, dim I n = dim ( S 1 ) n = 1 , and ind I n = ind( S 1 ) n = n . 2.7. Corollary . Ch atyrko’s c ontinua I n and ( S 1 ) n have also Ind I n = Ind( S 1 ) n = Ind 0 I n = Ind 0 ( S 1 ) n = n. Pr o of. There is a sequence . . . π n +1 n − → I n π n n − 1 − → . . . π 3 2 − → I 2 π 2 1 − → I 1 = [0 , 1] of fully closed on to maps π n +1 n , see [11]. F or eac h n and ev ery t ∈ I n , the pre-image ( π n +1 n ) − 1 t is homeomorphic to [0 , 1] . Using induction and Theorem 2.3, we obtain Ind 0 I n ≤ n . In the case of ( S 1 ) n there exists an ana lo g ous seque nce of maps, whose p oint- in vers es are homeomorphic to a circumference.  In Corolla ry 2.5 one can replace those p erfectly normal spaces by another class of spaces in whic h Ind = Ind 0 . This could b e the class of hereditarily perfectly κ - normal spaces (F edorc h uk [24]); surely , ev ery discre te closed subset of f X should b e G δ in f X . 8 J. KRZEMPEK The assumption that f is fully closed is necessary in Corollary 2.5. Under a set- theoretical assumption consisten t with ZFC , F edorch uk [23] constructed a p erfect map f F : X F → Y F , where X F and Y F are p erfectly normal, lo cally compact, and coun tably compact spaces, dim X F = Ind X F = 1 , Ind Y F = 0 , and Ind f − 1 F y = 0 for ev ery y ∈ Y F . On the o ther hand, it is not sufficien t to assume only that f is fully closed. Chat yrk o [12] has constructed a certain fully closed map f Ch : X Ch → A c from a compact space X Ch with Ind X Ch = Ind 0 X Ch = 2 on to the compact space A c with the o nly accum ulation p oint y 0 , card A c = c . All p oin t-in v erses f − 1 Ch y , where y 0 6 = y ∈ A c , are single p oin ts, and 1 = Ind f − 1 Ch y 0 < Ind 0 f − 1 Ch y 0 = 2 . F or some other maps f : X → Y , ev en Ind X − Ind f X − Ind f > 1 . F or ev ery pair o f natural n umbers m > n ≥ 1 , the presen t author [33] has constructed a compact space X m,n suc h that Ind X m,n = m and ev ery comp onen t of X m,n is homeomorphic to the n - dimensional cub e [0 , 1 ] n . In consequence, if D stands for the decompo sition of X m,n in to its comp onen ts, a nd f K : X m,n → X m,n / D is the natural quotien t map (it is not fully closed), then Ind X m,n − Ind X m,n / D − Ind f K = m − n . 2.8. Lemma. Supp ose that f is a ful ly clos e d map fr om a normal sp ac e X , L ⊂ X is a close d G δ -set, and A, B ⊂ f X ar e dis joint close d sets. Then, (a) f L ∩ f ( X \ L ) is the c ountable union of discr ete close d subsp ac es of f X (cf. [25, Definition 3 and Lemma 2]) . If mor e ove r every discr ete clo s e d subsp ac e of f X is a G δ -subset, then (b) f L is a G δ -set in f X ; an d (c) w henever L is a p artition b etwe en f − 1 A and f − 1 B , ther e is a c ountable family of discr ete close d sets Γ i ⊂ f X \ ( A ∪ B ) such that the union f L ∪ S i Γ i is a G δ p artition in f X b etwe en A and B . Pr o of. (a) There is a sequence of closed sets F i ⊂ X suc h that X \ L = S i F i . The in tersections f F i ∩ f L are discrete and closed, and f ( X \ L ) ∩ f L = S i ( f F i ∩ f L ) . (b) If f F i ∩ f L are G δ in f X , then f F i \ f L are F σ . Since f X \ f L = S i ( f F i \ f L ) , f L is G δ . The assertion (c) app ears to b e implicitly show n in the pro of of [26, Theorem I I I.2.6] if one can use (b). The new p oin t is to apply Lemma 2.8 ( b) and pro v e that, if P = L in [26, p. 4247] is G δ , then f P , U i ∪ f − 1 f P , f ( U i ∪ f − 1 f P ) , and K = f ( U 1 ∪ f − 1 f P ) ∩ f ( U 2 ∪ f − 1 f P ) = f P ∪ S j,k Γ j k are G δ -sets.  Applying induction, Lemma 2.8(c), Theorem 2.2 , and Prop osition 1.4, we o b- tain the follo wing tw o theorems. (The first one is an Ind 0 -analogue of Theorem I I I.2.6 on Ind in [26].) 2.9. Theorem. If f is a ful ly clos e d map fr om a normal sp ac e X to a sp ac e whose every discr ete close d subsp ac e is a G δ -set, then Ind 0 f X ≤ Ind 0 X + 1 .  2.10. Theorem. If f is a ring- l i k e ful ly close d map f r o m a c o mp act sp ac e X to a first c ountable sp ac e, then Ind 0 f X ≤ Ind 0 X .  FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 9 2.11. Lemma. If f is a ring-like ful ly close d map fr om a c o mp act sp ac e X o n to a non-de gener a te c ontinuum, then every G δ p artition in X c ontains a p oint-inv e rse of f . Pr o of. T ake a G δ partition L in X . There a r e tw o cases. (1) If f L is uncoun t a ble, then b y Lemma 2.8(a), f L ∩ f ( X \ L ) is coun ta ble, f L \ f ( X \ L ) ∋ y fo r some y ∈ f X , a nd f − 1 y ⊂ L . (2) Supp o se that f L is countable . L con tains a thin partition F , i.e. suc h that there a re disjoin t non-empt y op en sets U 1 , U 2 ⊂ X with X \ F = U 1 ∪ U 2 and F = b d U 1 = b d U 2 . By the remark 1.3 and Prop osition 1.4, f is a monotone irreducible map, and X is a con tin uum. By the same argumen t as in F edorc hu k [25, the pro of of Lemma 4, p. 16 7], w e infer that f − 1 x ⊂ F for ev ery p oint x ∈ f X isolated in f F .  2.12. Theorem. If f is a ring-like ful ly close d map fr om a non-em pty c omp ac t sp ac e X onto a firs t c ountable sp ac e Y , and Ind 0 f − 1 y = Ind 0 f f o r every y ∈ Y , then Ind 0 X = Ind 0 Y + Ind 0 f . Pr o of. Theorem 2 .3 yields the inequalit y „ ≤ ”. Fix m = Ind 0 f . The inequalit y „ ≥ ” will b e prov ed b y induction on n = Ind 0 X < ∞ . Clearly , n ≥ m . Let n = m , and supp ose that Ind 0 Y > 0 . Then, Y has a non- degenerate comp onen t Y ′ . By Lemma 2.11, ev ery G δ partition L in X ′ = f − 1 Y ′ con ta ins a p oint-in v erse of f and has Ind 0 L ≥ m . Hence, Ind 0 X ′ > n . A contradiction. Th us, Ind 0 Y = 0 and the inequalit y „ ≥ ” is true. Assume n > m . In o rder to sho w that Ind 0 Y ≤ n − m , ta ke a pair of disjoin t closed sets A, B ⊂ Y . There is a G δ partition L in X b etw een f − 1 A and f − 1 B , Ind 0 L ≤ n − 1 . It is a consequen ce of Lemma 2.8(c) that there is a G δ partition K ⊃ f L in Y b et w een A and B , where K \ f L is coun table. Lemma 2.8(a) implies that also K ∩ f ( X \ L ) = { y i : i = 1 , 2 , . . . } . W e ha v e f − 1 K = L ∪ S i f − 1 y i , and Ind 0 f − 1 K ≤ n − 1 b y Theorem 2.2. By the induction hy p othesis, Ind 0 K ≤ n − m − 1 . W e hav e show n that Ind 0 Y ≤ n − m .  The follo wing statemen t may b e considered as a generalization of Theorem 3 in F edorc h uk [20]. 2.13. Theorem. Supp ose that f : X → Y is a surje c tive ring-like map b etwe en c omp act sp ac es X and Y , and dim Y ≥ 1 . If ind f − 1 y ≥ m for every y ∈ Y , then ind X ≥ dim Y + m − 1 . Pr o of. If dim Y = ∞ , then ind X ≥ dim X = ∞ by Prop osition 1.8 (a nd [19, Theorem 3.1.2 9 ]). W e can assume that dim Y < ∞ . It suffices to prov e b y induction that for ev ery natural n umber k ≥ 1 , the inequalit y dim Y ≥ k implies ind X ≥ k + m − 1 . F or k = 1 the implication is obvious . W e shall use the following classic al notion. A compact space M with dim M = n is called an n -dimension a l Cantor m a nifold pro vided that ev ery partition L in M has dim L ≥ n − 1 . P .S. Alexandroff [1] pro ved that ev ery compact space Z with 1 ≤ n = dim Z < ∞ contains an n -dimensional Can tor manifold (see also [19, Theorem 1.9.9 and Exercise 3.2.F]). 10 J. KRZEMPEK Let n = dim Y ≥ k ≥ 2 . W e can assume that Y is an n -dimensional Cantor manifold. Then, X is a con tin uum and f is an irreducible monotone map b y 1.3 and Prop osition 1 .4. If L is a partition in X , then f L is a partition in Y (Prop osition 1.5), and dim f L ≥ n − 1 ≥ k − 1 . No w, f L con tains a comp onen t P with dim P ≥ n − 1 . Since P = f ( L ∩ f − 1 P ) , t he remark 1.3 implies f − 1 P ⊂ L . By the ob vious induction hy p othesis, ind L ≥ ind f − 1 P ≥ k + m − 2 . Therefore, ind X ≥ k + m − 1 .  3. Main tools f or c onstr uctions F or our constructions w e need ingredien ts of t w o types. The followin g resolu- tion theorem (the first t yp e) is a mo dification of well-kn ow n results. 3.1. Theorem (cf. Chat yrk o [10], Emeryk [15], F edorc hu k [20]). Supp ose that X is a first c o untable c ontinuum, and for e v ery x ∈ X , Y x is a metrizable c on- tinuum. Then, ther e exists a first c ountable c ontinuum Z = Z ( X , Y x ) with a map π : Z → X such that (a) f o r every x ∈ X , the pr e - i mage π − 1 x is home omorphic to Y x ; and (b) π is ring-li ke and ful ly clo se d. Mor e over, the c on junction of (a) and (b) implies that (c) dim Z = max { dim X , dim π } ; (d) if X is p erfe c tly normal, then Ind Z ≤ Ind X + Ind π ; (e) i f X is sep ar able, then so is Z ; (f ) if al l Y x ar e non-d e gener ate c ontinua, and P ⊂ Z is a metrizable c ontinuum, then the im a ge π P is a single p oi n t; and (g) i f X and al l Y x ar e he r e ditarily inde c omp osab l e [ her e ditarily de c omp osable ] , then so i s Z . Sketch of pr o of. Since Chat yrk o’s pap er [10] has not b een translated in to English, w e sk etc h his construction for the con v enience of the reader 3 . W e can assume that eac h Y x is a subspace of the Hilb ert cub e [0 , 1] ∞ . Using the lo cal connectedness of [0 , 1] ∞ , one constructs a map g x : (0 , 1] → [0 , 1] ∞ suc h that for eac h natura l n umber n , Y x ⊂ cl g x (0 , 1 /n ] ⊂ B( Y x , 1 /n ) , where B stands for a ball. One tak es a map f x : X → [0 , 1] with f − 1 0 = { x } , and writes h x : X \ { x } → [0 , 1 ] ∞ for the comp osition g x ( f x | X \ { x } ) . Z is the set S {{ x } × Y x : x ∈ X } ⊂ X × [0 , 1] ∞ , and π : Z → X , π ( x, y ) = x . The top ology on Z is generated b y a ba se of neigh b o r ho o ds at an y p o in t ( x, y ) ∈ Z ; the base consists of all sets W ( V , U ) = [ { x } × ( V ∩ Y x )] ∪ π − 1 ( U ∩ h − 1 x V ) , where U ⊂ X and V ⊂ [0 , 1] ∞ are neigh b orho o ds of x ∈ X and y ∈ Y x respec- tiv ely . The ab ov e is a generalization of F edorch uk’s construction [20] (cf. also 3 In a forthcoming pap e r , joint with M.G. Charala m b ous, we des crib e a genera lization of Chatyrk o’s construction in more detail. FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 11 F edorc h uk [25, the pro of of Lemma 1] and [26, Section I I I.1]). One che c ks that Z is a first coun table con tin uum, and π satisfies (a) and (b). Corollary 1.10 yields the equality (c), a nd Corollary 2.5 yields (d). The state- men t (e) is an easy consequ ence of the fact that π is an irreducible map (see the remark 1.3). The statemen t (f ) follo ws from 1.3 and Prop osition 1.1, and (g) is a simple prop ert y of atomic maps.  3.2. Remarks. (1 ) If a sub con tinuum P of Z = Z ( X , Y x ) in Theore m 3.1 is non-metrizable, then P = π − 1 π P by 1.3. Hence, if al l Y x ar e non-de gen e r ate c ontinua, then for every non-d e gener ate c o ntinuum P ⊂ Z and a n y p oint z ∈ P , ther e is a non-de ge n er ate m etrizable c ontinuum Q such that z ∈ Q ⊂ P . (2) O ne can combin e the pro ofs by Chaty rk o [10] and F edorc h uk [2 5 , Lemma 1] in order to obtain a map π that satisfies also the assertion (2) o f Lemma 1 in [25]. This enables one to construct the con tinuum Z = Z ( X , Y x ) under the con tin uum h yp ot hesis (see [25, pp. 166-167]) so that ( † ) if CH is true, and the c on tinuum X in The or em 3.1 is p erfe ctly normal and her e ditarily sep ar abl e , then Z is p erfe ctly normal and her e ditarily sep ar a ble. W e shall also need Co ok con tinua (the second t yp e of ingredien ts), whose sub- con tinua will b e tak en as the Y x ’s of Theorem 3.1. 3.3. Example (Co ok [14]; see also A. Pultr, V. T rnk o v á [44, App endix A] for a detailed construction). Ther e e xists a metrizable, one-dimension a l, her e ditarily inde c omp o s a ble Co ok c on tinuum M 1 that do es not c ontain non- de gener ate c on- tinuous images of plane c ontinua. Pr o of. J.W. Rogers, Jr. [46] observ ed that Co ok’s con tin uum [14] do es not con tain non-degenerate con tin uous images of plane con tinua.  3.4. Example (Mać ko wiak [37]). Ther e exists a metrizab l e , chainable, her e di- tarily de c om p osable Co ok c o n tinuum.  4. Anderson-Choquet (and similar) continua with dim > 1 The followi ng con tin ua are neither Anderson-Cho quet nor Co ok, but they are HI a na lo g ues of F edorch uk’s spaces with non-coinciding dimensions ([20], cf. also our Corollary 2.6). 4.1. Theorem. F or every natur al numb er n ≥ 1 , ther e exists a non - m etrizable, sep ar able, first c ountable, her e d i tarily i n de c omp osab l e c ontinuum Z such that dim Z = n and 2 n − 1 ≤ ind Z ≤ Ind Z ≤ Ind 0 Z = 2 n . Pr o of. By a theorem of R .H. Bing [3], there exists a metric HI con tin uum X with dim X = n . Let Y x = X for x ∈ X , apply Theorem 3.1, and put Z = Z ( X , Y x ) . The prop erties of Z are consequences of the statemen ts 2.12, 2.13, and 3.1(c– g).  Let us notice that the first examples of non-metrizable HI contin ua w ere con- structed by Emeryk [16]. 12 J. KRZEMPEK 4.2. Theorem. F or every natur al numb er n ≥ 1 , ther e exists a non - m etrizable, sep ar able, first c ountable, her e ditarily ind e c omp osabl e A nderson- C ho quet c o n tin- uum Z with n = dim Z ≤ ind Z ≤ Ind Z ≤ Ind 0 Z = n + 1 . Pr o of. Let X b e a metric n - dimensional HI contin uum (Bing [3]). T ak e a family { C x : x ∈ X } of pairwise disjoin t non-degenerate sub con tinua of Co ok’s con tin- uum M 1 (Example 3.3), apply Theorem 3.1, and put Z = Z ( X , C x ) . Most o f the desired prop erties of Z follow from Theorems 2.12 a nd 3 .1(c-g). There remains to pro v e that Z is an Anderson-Cho quet con tinuum. Let π : Z → X b e the map of Theorem 3.1 , and c ho ose any non-degenerate con tin uum P ⊂ Z and an y embedding ϕ : P → Z . T ak e an a r bitra r y p o int z ∈ P . By Remark 3.2(1), P con ta ins a non-degenerate metrizable con tin uum Q ∋ z . The statemen t (f ) of Theorem 3.1 guaran tees that Q ⊂ π − 1 x and ϕQ ⊂ π − 1 x ′ for some x, x ′ ∈ X . Since π − 1 x and π − 1 x ′ are homeomorphic to C x and C x ′ , respectiv ely , w e obtain x = x ′ , ϕ | Q = id Q , and ϕz = z . W e hav e sho wn that ϕ = id P .  W e shall adapt the fo regoing construction and pro of in order to obtain hered- itarily rigid finite-group actions. W e start with some terminology . Let X b e a space, and G a finite group. W e write H ( X ) for the group of all homeomorphisms X → X . Ev ery homomorphism ξ : G → H ( X ) is called a G -ac tion on X ; the v alue of ξ at g ∈ G will b e denoted by g ξ ∈ H ( X ) . This G -action is said to b e fixe d-p oint-fr e e if for each g ∈ G \ { e } , the homeomorphism g ξ : X → X do es not ha ve a fixed p oin t. Let ζ : G → H ( Y ) b e a G -action o n a space Y . A map f : X → Y is said to b e e quivariant if g ζ f = f g ξ for eac h g ∈ G . 4.3. Theorem. Supp o se that X is a first c ountable c ontinuum, G a finite gr oup, and ξ is a fixe d- p oint-fr e e G -action on X . Then, ther e exists a first c ountable c ontinuum Z with a fi x e d-p oint-fr e e isomorp h ic G -action ζ : G → H ( Z ) and an e quivariant map π : Z → X such that (a) π is a ring-like ful ly close d map, and al l p o int-inverses of π a r e metrizable one-dimensio nal c ontinua; (b) for e v ery non-de gen er ate c ontinuum P ⊂ Z and every emb e ddi n g ϕ : P → Z , ther e is a home omo rphism g ζ ∈ H ( Z ) such that g ζ | P = ϕ . An imp orta n t p oin t in this theorem is that dim Z = dim X by Corollary 1 .1 0, and Ind 0 Z = Ind 0 X + 1 by Theorem 2 .12. In the pro of it will b e seen that we can ensure that all p oin t-inv erses of π are HI (if w e use sub contin ua of Example 3.3 in our construction), or alternativel y , that they are HD ( if w e use Example 3.4). W e may apply this theorem to some standard examples of group actions. It is w ell-know n that for ev ery finite group G and ev ery n ≥ 2 , there exists a (compact metric) n -manifold without b oundary with a fixed-p oin t-free G -action (see J. de Gro ot, R.J. Wille [30, p. 444]). In case n = 1 , there exists a connected finite gr a ph with a fixed-p o in t- free G -action ( G acts o n its Cayle y graph). T w o more simple examples: Using Anderson and Cho quet’s or ig inal HD con tin uum [2], a Cayle y graph of t he group G , and the metho d fr o m [30], one easily constructs a metric FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 13 one-dimensional HD con tinuum Z with a fixed-p oint-free G -action ζ that is an isomorphism G → H ( Z ) and satisfies the assertion (b) of Theorem 4.3. Using our Anderson-Choquet con tin uum (of Theorem 4.2) instead (and the same Cayle y graph metho d), one constructs a non-metrizable, separable, and first coun table con tinuum Z with dim Z = n and a similar G -action ζ on Z . Let us notice that there are n umerous pap ers on g r o up represen tations in top ol- ogy (see de Gro ot [29] for example) and in the more general con text of category theory (see the bibliography in Pultr and T rnk o v á [44]). Pr o of of The or em 4.3. (I) Consider the family D of all orbits { g ξ x : g ∈ G } , x ∈ X , and the quotien t space X ′ = X/ D . The quotien t pro jection q ξ : X → X ′ is a co vering map. T ak e a family { C x ′ : x ′ ∈ X ′ } of pa irwise disjoin t non-degenerate sub contin ua of Cook’s con tinuum M 1 (Example 3.3). Use Theorem 3.1, and take Z ′ = Z ( X ′ , C x ′ ) and π ′ : Z ′ → X ′ , a map that satisfies the statemen ts (a–g) of Theorem 3.1. Consider the set Z = S x ′ ∈ X ′ ( q − 1 ξ x ′ × π ′− 1 x ′ ) ⊂ X × Z ′ . W e define π ( x, t ) = x , q ζ ( x, t ) = t , and g ζ ( x, t ) = ( g ξ x, t ) fo r x ∈ X , t ∈ π ′− 1 q ξ x , and g ∈ G . The followi ng diagram comm utes, X X ′ Z Z ′ X Z ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✿ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✿ ✑ ✑ ✑ ✑ ✰ ✑ ✑ ✑ ✑ ✰ ✲ ✲ ✻ ✻ ✻ q ξ q ζ π π π ′ q ξ q ζ g ξ g ζ and ζ , g 7→ g ζ , is a monomorphism from G to the group of p erm utations o f Z . Finally , we equip Z with the smallest top ology suc h that π and q ζ are contin - uous. Observ e that, if U ⊂ X is a n op en set suc h that q ξ | U : U → q ξ U is a homeomorphism, then q ζ | π − 1 U : π − 1 U → π ′− 1 q ξ U is also a homeomorphism. In- deed, q ζ | π − 1 U is one-to-o ne, all o p en subsets of π − 1 U ha v e the for m q − 1 ζ V ∩ π − 1 U , where V ⊂ Z ′ are op en, and q ζ ( q − 1 ζ V ∩ π − 1 U ) = V ∩ q ζ π − 1 U = V ∩ π ′− 1 q ξ U . It follo ws that q ζ is a closed cov ering map, and hence , Z is a compact space by [18, Theorem 3.7.2]. Similarly , g ζ are homeomorphisms of Z . In view of the ab ov e diagram, π is an equiv ariant map. (I I) Ev ery p oint-in v erse π − 1 x is homeomorphic to q ζ π − 1 x = π ′− 1 q ξ x and C q ξ x . As π is a monotone map, the connectedness of X implies the connectedness of Z . Using the fact that π ′ is ring-lik e, one easily c hec ks t ha t also π is ring-lik e. In order to prov e that π is fully closed tak e disjoin t closed sets A, B ⊂ Z . By virtue of the compactness of X , it is sufficien t to sho w that, if cl U ⊂ X an d q ξ | cl U is one-to-o n e, then cl U c ontain s a finite numb er of p oin ts in π A ∩ π B . Indeed, consider the set F = π − 1 cl U and the restriction π | F . The sets g ζ F , g ∈ G , are pairwise disjoint, and q ζ | F is a one-to-one function. Th us, q ζ ( F ∩ A ) and q ζ ( F ∩ B ) are disjoin t closed subsets o f Z ′ , and the inte rsection π ′ q ζ ( F ∩ A ) ∩ π ′ q ζ ( F ∩ B ) = q ξ π ( F ∩ A ) ∩ q ξ π ( F ∩ B ) = q ξ (cl U ∩ π A ∩ π B ) 14 J. KRZEMPEK is finite since π ′ is a fully closed map. Therefore, cl U ∩ π A ∩ π B is finite, and π is fully closed. Moreo v er, π satisfies statemen t s ana lo g ous to Theorem 3.1(c–g) and Remark 3.2(1). (I I I) Let P ⊂ Z b e a non-degenerate contin uum, a nd ϕ : P → Z a n em b edding. W e claim that for every z ∈ P ther e is a g ∈ G such that ϕz = g ζ z . Indeed, by Remark 3.2(1), P con tains a non- degenerate metrizable contin uum Q ∋ z . By a statemen t analogous to Theorem 3.1(f ), there are x, x ′ ∈ X suc h that Q ⊂ π − 1 x and ϕQ ⊂ π − 1 x ′ . The restrictions q ζ | Q and q ζ ϕ | Q are embeddings into π ′− 1 q ξ x and π ′− 1 q ξ x ′ , resp ectiv ely . Since these p oint-in v erses are homeomorphic to C q ξ x and C q ξ x ′ , resp ectiv ely , we hav e q ξ x = q ξ x ′ , q ζ | Q = q ζ ϕ | Q , and q ζ z = q ζ ϕz . Hence, there is a g ∈ G such that ϕz = g ζ z . The forego ing claim implies that P = S g ∈ G F g , where F g = { z ∈ P : ϕz = g ζ z } . F g are closed, pairwise disjoint, and P is connected. In consequence, only one F g is non-empt y . Th us, there is a g ∈ G such that P = F g and ϕ = g ζ | P . When w e apply the statemen t (b) to P = Z , w e infer that ζ is an isomorphis m on to H ( Z ) .  A non-degenerate contin uum X will b e called a we ak Co ok c ontinuum 4 if for ev ery sub con tin uum P of X , ev ery map f : P → X with P ∩ f P = ∅ is constan t. 4.4. Prop osition (Maćk o wiak [36, Prop osition 29] 5 ). S upp ose that X is a we ak Co ok c ontinuum. If P i s a sub c ontinuum of X and f : P → X is a non-c onstant map, then f P ⊂ P and f is a monotone r etr action.  4.5. Theorem. Ther e exists a non -metrizable, sep ar able, first c ountable Co ok c ontinuum Z with 2 = dim Z ≤ ind Z ≤ Ind Z ≤ Ind 0 Z = 3 . Pr o of. T ake the square [0 , 1] 2 and a family { C x : x ∈ [0 , 1] 2 } of pairwise disjoin t non-degenerate sub con tinua of Coo k’s contin uum M 1 . Put Z = Z ([0 , 1] 2 , C x ) , and let π : Z → [0 , 1] 2 b e the map of Theorem 3.1. Most of the desired prop erties o f Z follow from Theorems 2.12 and 3.1(c-g). W e shall prov e that Z is a weak Co ok contin uum. Assume that P is a sub con tin uum of Z , and f : P → Z is a map with P ∩ f P = ∅ . W e claim that for every metrizable c ontinuum Q ⊂ P , the r estriction f | Q is c onstant. Indeed, f Q is a metrizable sub contin uum of Z . Theorem 3.1(f ) implies that Q and f Q are homeomorphic to disjoint sub con tin ua of Co ok’s M 1 . Hence, f Q is a single p oin t. Thus , if P is metrizable, we a r e done. If not, P = π − 1 π P b y 1.3, and the ab ov e claim implies that there is a factorization f = g ( π | P ) , where g : π P → Z is contin uous. Hence, f P is a metrizable con tinuu m con tained in a p oint-in v erse of π . Since M 1 con ta ins only degenerate images o f plane con tin ua, g a nd f a r e constan t maps. No w, c ho ose a con tin uum P ⊂ Z and a non-constan t map f : P → Z . Supp ose a c ontr ario tha t f 6 = id P . T ak e a p o int z ∈ P with f z 6 = z . It is a consequence of 4 There is so me difference in terminolog y : in [36–38] o ur weak Co ok contin ua are just ca lled Co ok contin ua. 5 The pro of of Prop os ition 29 (i) in [36] works for arbitrar y Hausdo rff contin ua. FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 15 Prop osition 4.4 that A = f − 1 f z and B = f P are non-degenerate con tinua with A ∩ B = { f z } . Since A is a retract o f A ∪ B ∋ z and π − 1 π z is a Co ok contin uum, π ( A ∪ B ) is not a single p oin t. Hence, A ∪ B = π − 1 π ( A ∪ B ) = π − 1 π A ∪ π − 1 π B b y 1.3, π A 6⊂ π B , π B 6⊂ π A , A = π − 1 π A , and B = π − 1 π B . Thus , π − 1 π f z ⊂ A ∩ B . A contradiction. Therefore, f = id P , and Z is a Co ok con tin uum.  5. C hainable Cook continua with ind > 1 Let X b e a c hainable con tin uum. Elemen ts a 6 = b of X are called opp osite end p oints if ev ery op en co v er of X has a finite closed refinemen t F 1 , F 2 , . . . , F k suc h that a ∈ F 1 , b ∈ F k , and F i ∩ F j 6 = ∅ iff | i − j | ≤ 1 . Bing [4, Theorem 15] pro ved that every n on-de gener ate metric chainab le c ontinuum c ontains a chain able c on- tinuum with a p air of opp osite end p oints. The third ingredien t for our construction in this section is the following series of c hainable con tin ua, whic h were men tioned in Corollary 2.7 (now , w e need more detail). 5.1. Example (Chat yrk o [11]; see also [9, 13] for n = 2 , 3 ). The r e exists an inverse se quenc e ( I n , π n m ) ∞ n,m =1 , wher e I n ar e s e p ar able, first c ountable, her e ditarily de c omp osa b le chainable c o ntinua, and π n m : I n → I m ar e surje ctive maps such that (a) I 1 is hom e omorphic to [0 , 1] ; (b) e ach π n +1 n is atomic and ful ly close d , and e ach π n 1 is ring-l i k e; (c) f or e ach n > 1 , m ∈ { 1 , n − 1 } , and eve ry t ∈ I m , the p r e - i m age ( π n m ) − 1 t is home omorphi c to I n − m ; (d) for e ach n and e v ery p air 0 ≤ s < t ≤ 1 , the pr e-image ( π n 1 ) − 1 [ s, t ] is a chainable c ontinuum with opp os i te e nd p oints a s,t ∈ ( π n 1 ) − 1 s, b s,t ∈ ( π n 1 ) − 1 t ; (e) f or e ac h n , every p artition in I n c ontains some p oi n t-inverse ( π n 1 ) − 1 t , wher e t ∈ I 1 ; and (f ) ind I n = Ind I n = Ind 0 I n = n for e ach n (see our Corolla ry 2.7) .  5.2. Lemma (Chatyrk o [11, Lemma 1]). Supp ose that X , Y ar e c omp a c t sp ac es, and f : X → Y , g : Y → [0 , 1] ar e s urje c tive maps such that (a) f is ful ly close d, an d b o th g and g f ar e ring-like; (b) for every t ∈ [0 , 1] , the pr e-imag e ( g f ) − 1 t is a chainable c on tinuum wi th a p air of opp osite end p oi n ts; and (c) f or every p air 0 ≤ s < t ≤ 1 , the p r e -image g − 1 [ s, t ] is a cha inable c ontinuum with opp osite end p oin ts a s,t ∈ g − 1 s, b s,t ∈ g − 1 t . Then, X i s a chainable c ontinuum with a p air of opp osite end p o i n ts a ∈ ( g f ) − 1 0 , b ∈ ( g f ) − 1 1 .  5.3. Theorem. F or every natur al n umb er n ≥ 1 , ther e exis ts a non -metrizable, sep ar able, first c o untabl e , chaina b le, her e ditarily de c o mp osable Co ok c on tinuum Z such that n ≤ ind Z ≤ Ind Z ≤ Ind 0 Z = n + 1 , and every p artition L in Z has ind L ≥ n − 1 . 16 J. KRZEMPEK Pr o of. Let M b e Mać k o wiak’s Co o k con tin uum of Example 3.4, and I n Chat yrko’s con tinuum of Example 5.1. W e claim that M c ontains an unc ountable family o f p airwise disjoint non- d e gener ate sub c ontinua M x , x ∈ I n . Indeed, M is HD, and by a theorem of Bing [4, Theorem 8], there is a monotone surjectiv e map f : M → [0 , 1] . If 0 < t < 1 and the p oin t-in v erse f − 1 t we re a single p oint, f − 1 [0 , t ] w ould b e a retra ct of M , and M w ould not b e a Co ok continuum. Hence, f − 1 t is a non-degenerate con tin uum if 0 < t < 1 . As card I n = 2 ℵ 0 , the claim is pro v ed 6 . By [4, Theorem 15], w e can assume that ev ery M x has a pair of o pp osite end p oin ts. Finally , we apply Theorem 3.1, and tak e Z = Z ( I n , M x ) with π : Z → I n . Note that w e a b o ve describ ed a class of examples Z = Z n , whic h ha ve surjec- tiv e, ring-lik e, fully close d maps π : Z n → I n whose p oint-in v erses are homeo- morphic to pairwise disjoin t non-degenerate sub con tin ua o f M , a nd eac h o f the con tinua has a pair of opp osite end p oin ts. Th us, the statemen t 5 .1(c) implies that for eac h t ∈ [0 , 1] , ( π n 1 π ) − 1 t b elongs to the class of examples Z n − 1 . By Prop osition 1.6, the composition π n 1 π is ring-lik e. Using induction on n , Lemma 5.2 and t he assertions (a–d) of Example 5.1, w e infer that Z is a c hainable con tinuum with a pair of opp osite end p oints . Using induction, Prop osition 1.7, and the assertions (c, e) of Example 5.1, w e infer that ev ery partition L in Z has ind L ≥ n − 1 , and hence, ind Z ≥ n . By Theorem 2.12 applied to π , Ind 0 Z = n + 1 . Similarly as in the pro of of Theorem 4.5, w e shall sho w that Z is a w eak Co o k con tinuum. T ake a contin uum P ⊂ Z and a map f : P → Z with P ∩ f P = ∅ . Supp ose that f is not constan t. Our first claim is that π P is n o t a single p oint and f has a fa c torization f = g ( π | P ) , wher e g : π P → Z is c on tinuous. Indeed, if Q ⊂ P is a metrizable con tin uum, then f Q is metrizable, and is containe d in some p oin t-inv erse π − 1 x , where x ∈ I n . Hence, Q and f Q are homeomorphic to disjoin t sub con tin ua of M , and the r estriction f | Q m ust b e constan t. Thus , P is not metrizable, π P is not a single p o in t by Theorem 3.1(f ), and moreov er, f | π − 1 t is constan t for ev ery t ∈ π P . The map g : π P → Z , g t = f π − 1 t , is w ell defined and con tinuous. Our claim ensures that the b elo w set is non-empt y , and w e define k 0 = min { k : f has a factorization f = g k ( π n k π | P ) , where g k : π n k π P → Z } . No w, observ e that, if Q ⊂ π n k 0 π P ⊂ I k 0 is a metrizable c o n tinuum, then g k 0 | Q is c onstant. Indeed, Q is an arc, g k 0 Q ⊂ Z is metrizable, and is contained in some p oin t-in v erse π − 1 x , where x ∈ I n . As M x do es not con tain arcs, g k 0 | Q m ust b e constan t. This show s that π n k 0 π P is no n- metrizable, k 0 > 1 , and π n k 0 − 1 π P is no t a single point. As π k 0 k 0 − 1 is atomic, ev ery p oin t-inv erse ( π k 0 k 0 − 1 ) − 1 t , t ∈ π n k 0 − 1 π P , is a 6 Recall that a co ntin uum X is said to be Suslinian if every family of pair wise disjoint non- degenerate subcontin ua of X is countable. By a similar argument, we infer that no metric Co ok c ontinuum X is S uslinian. If X co n tains an indeco mpo sable contin uum, then it is o b viously not Suslinia n (see for instance [3 2, Lemma 5.5]). If X is HD, then we can a ssume that it is irreducible by [3 5, §48 I, Theor e m 1], a nd a g ain there is a surjective mono tone map f : X → [0 , 1] b y Kuratowski’s theo rems [3 5, pp. 20 0 and 216]. On the other hand, Ma ćk owiak [36, T heo rem 30 ] constructed an example of a metrizable, Suslinian, chainable, weak Co o k contin uum. FULL Y CLOSED MAPS A ND AN DERSON-CHOQ UET CON TINUA 17 terminal con tinuum, and in consequence, ( π k 0 k 0 − 1 ) − 1 t ⊂ π n k 0 π P . By the o bserv ation emphasized ab o v e, the restriction g k 0 | ( π k 0 k 0 − 1 ) − 1 t is constant f o r eve ry t ∈ π n k 0 − 1 π P . Th us, the map g k 0 − 1 : π n k 0 − 1 π P → Z , g k 0 − 1 t = g k 0 ( π k 0 k 0 − 1 ) − 1 t , is w ell defined and con tinuous. W e hav e f = g k 0 − 1 ( π n k 0 − 1 π | P ) , and this con tra dicts the definition of k 0 . Therefore, f must b e a constan t map. In the same w ay as in the pro of of Theorem 4.5, one show s that Z is a Co ok con tinuum.  6. Remarks and open pr oblems Prop osition 1.7 or alternativ ely Theorem 2.13 allow us to iterate the construc- tions in Section 4 in order to obtain contin ua Z with a rbitrarily large difference ind Z − dim Z > 0 . F or example, one can tak e Y x = [0 , 1] 2 for ev ery x ∈ [0 , 1] 2 = Z 1 , apply Theorem 3.1, and ha v e Z 2 = Z ([0 , 1] 2 , Y x ) , π 2 1 : Z 2 → [0 , 1] 2 , dim Z 2 = 2 . Then, one tak es pairwise disjoint non-degenerate sub contin ua C x of Co ok’s M 1 for x ∈ Z 2 , and puts Z = Z ( Z 2 , C x ) with π : Z → Z 2 . It follo ws from 1.6 and 2.13 that ind Z ≥ 3 > 2 = dim Z . Z is a Co ok contin uum by t he same argumen t as in the pro of of Theorem 5.3. It follows fro m Remark 1.8.(2) that, if CH is true, then al l the examples of c on- tinua Z c ons tructe d in Se ctions 4 – 5 c an b e p erfe ctly normal, her e ditarily sep ar a- ble, an d have ind Z = Ind Z = Ind 0 Z . In Section 5, instead of Chat yrk o’s contin ua I n one should use p erfectly normal c hainable con tinua by A.A. Odints o v [39]. W e suggest the follow ing op en problems. In most of our examples of con tinua Z we hav e got the anno ying difference b et w een a lo w er b ound of ind Z and the exact v alue of Ind 0 Z . 6.1. Question. Supp os e that f : X → Y is a surje ctive, ful ly c l o se d rin g-like map fr om a c ontinuum X , Y is the interval [0 , 1 ] or a n other metrizab l e one - d imension- al c ontinuum, and every p oint-inverse of f is a metrizable one- d imensional [ n - dimensional ] c ontinuum. Can i t happ en that ind X = 1 [ r esp e ctively: ind X = n or even Ind X = n ] ? By Theorem 2.1 2, the ab o ve con tin uum X mus t hav e Ind 0 X = 2 [resp ectiv ely: Ind 0 X = n + 1 ]. 6.2. Question. Do ther e exist [ her e ditarily inde c om p osable ] Co ok c ontinua whose dim = n for n ≥ 3 [ r esp e ctivel y: n ≥ 2 ] ? Suc h con tin ua do not exist in the metric case, see [32] and [38]. 6.3. Question. 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Roger s, Jr., Continua that c ontain only de gener ate c ont inuous images of plane c on- tinua, Duke Math. J. 37 (197 0), 479– 483. [47] L.R. Rubin, R.M. Schori, J.J . W als h, New dimension-the ory te chniques for c onst ructing infinite-dimensional examples, General T op olo gy Appl. 10 (1979 ), 93–10 2. [48] S. W atson, The c onstruction of t op olo gic al sp ac es: planks and r esolutions, in: R e c ent Pr o gr ess in Gener al T op olo gy (M. Hušek and J. v an Mill, eds.), 673–75 7, Nor th-Holland, Amsterdam, 1 9 92. Institute of Ma thema tics, Silesian University of Technolo gy, Kaszu bska 23, 44-100 Gliwice, Poland (Instytut Ma tema tyki, Politechnika Śląska) E-mail addr ess : j .krzem pek@po lsl.pl