On hereditarily indecomposable compacta and factorization of maps

ON HEREDIT ARIL Y INDECOMPOSABLE COMP A CT A AND F A CTORIZA TION OF MAPS KLAAS PIETER HAR T AND EL ˙ ZBIET A POL Abstract. W e prov e a general f actorization theorem for m aps with heredi- tarily i ndecomposable fibers and apply i t to repro ve a theorem of Ma ´ ck oviak on the existen ce of universal hereditarily i ndecomposable cont inua. 1. Introduction All spac es are assumed to be normal. By a map we mea n a c o ntin uous function. W e say that a compac tum X is her e ditaril y inde c omp osable if for every t wo inter- secting contin ua in X one is contained in the other. The main result of this note is the following theorem. Theorem 1.1 . L et f : X → Y b e a p erfe ct map with her e ditarily inde c o mp os- able fib ers fr om a sep ar able metrizable sp ac e X onto a zer o-dimensional sep ar able metrizable sp ac e Y . T hen ther e ar e a her e ditarily inde c omp osable metrizable c om- p actific ation X ⋆ of X with dim X ∗ = dim X and a zer o-dimensional metrizable c omp actific ation Y ∗ of Y such that f c an b e ext ende d to a m ap f ∗ : X ∗ → Y ∗ . Let us note that this r esult, combined with a pseudosusp ensio n metho d, yields a theorem o f Ma ´ cko wiak [1010, 10] o n the existence of universal n - dimens ional her ed- itarily indeco mpo sable contin ua. This theorem was obtained b y Ma´ ck owiak by a quite different metho d based on a subtle us e of inv erse limits. W e commen t on this in Corollary 4.1. Rather unexp ectedly , our pro of uses, in a n e s sential w ay , larg e nonmetrizable compactifications and a considerable strenghtening of Marde ˇ s i´ c’s F a ctorizatio n The- orem (see [33; 3, Theor em 3.4.1]). This strengthening is a dual v ersio n of the L¨ ow enheim-Skolem theor em from model theor y; it a pp ea rs as Theorem 3.1 in [22, 2] and it was put to go o d use in [44, 4] and [1212, 12]. In Section 2 we ex plain some general facts concerning this technique and in sectio n 3 we show how our theo- rem follows from these results. Among other consequence s of this technique is the following theorem, proved in s ection 3. Theorem 1.2. F or every c ar dinal τ and n ∈ { 0 , 1 , . . . , ∞} ther e exists a her e ditar- ily inde c omp osable c omp a ctum X ( n, τ ) of weight τ and dimension n that c ontains a c opy of every her e d itarily inde c omp osable c omp ac tu m of weight not mor e than τ and dimension at most n . Date : Thursda y 21-08-2008 at 10:08:22 (cest). 2000 Mathematics Subje ct Classific ation. Primar y 54F15. Secondary: 54F45 03C98. Key wor ds and phr ases. hereditarily indecomposable compacta, ˇ Cec h-Stone compactificat ion, factorization, L¨ ow enheim-Sk olem theorem. The second author was partially suppor ted by MNi SW Grant Nr N 201 034 31/2717. 1 2 K. P . HAR T AND E. POL The following pro pe rty of a space X was formulated by K rasinkiewicz and Minc [66, 6]: Prop ert y (KM). F o r every t wo disjoint c lo sed sets C a nd D in X and disjoint op en s e ts U and V in X with C ⊂ U and D ⊂ V there exis t closed sets X 0 , X 1 and X 2 in X such that X = X 0 ∪ X 1 ∪ X 2 , C ⊂ X 0 , D ⊂ X 2 , X 0 ∩ X 1 ⊂ V , X 1 ∩ X 2 ⊂ U and X 0 ∩ X 2 = ∅ . T o a void having to wr ite down the six conditions each time w e shall call a triple h X 0 , X 1 , X 2 i a fold of X for the quadruple h C, D , U, V i . Theorem 1.3 ([6 6, 6]) . A c omp act sp ac e is her e d itarily inde c omp osable if and only if it has Pr op erty (K M). 2. A f actoriza tion method The factorization metho d alluded to in the Introductio n is based on a mix of Mo del Theory and Set-Theo retic T op olog y . It works best in the realm of compa ct Hausdorff spaces, as will bec ome clear shortly . The fir st ingre die nt is W a llma n’s repr esentation theorem, [1313, 13], for distribu- tive lattices: if L is such a lattice then the set wL of ultrafilters on L car r ies a natural compact T 1 -top ology . This top o logy has the family { ¯ a : a ∈ L } a s a base for the closed sets, where ¯ a = { u ∈ wL : a ∈ u } . If X is co mpact and T 1 and L is the family of closed subsets of X , with union and intersection as its lattice o p erations then x 7→ u x = { a ∈ L : x ∈ a } is a homeomorphism from X onto w L ; this rema ins true if L is a base for the c losed sets of X that is closed under unions and in tersections. See, e.g., [11, 1] for a short int ro ductio n to W allman r epresentations. F or a normal space X one can obtain the ˇ Cech-Stone compactification, β X , as the W a llma n r epresentation of the lattice o f clo sed sets of X . This is the key to the next theorem. Theorem 2. 1. If X has Pr op erty (KM) then so do es its ˇ Ce ch-Stone c omp actific a- tion β X and, in p articula r, β X is her e ditarily inde c omp osable. Pr o of. T o b egin: it s hould be clear that Pr op erty (KM) can b e (re)formulated in terms of closed sets only and that it is a finitary lattice-theore tic pr op erty; o ne can express it a s an implication in volving seven v ariables. Th us if X has Pro p erty (KM) then the c anonical base, B , for the clos ed sets of β X satisfies this implication. This do es not automatically imply tha t β X has Prop e r ty (KM), b ecaus e that means that the full family of closed sets of β X satisfies the lattice-theore tic form ula. How ever, in the present cas e one c a n sta rt with ar bitrary C , D , U and V and use co mpactness and the fact that B is close d under finite unio ns and intersections to find C ′ , D ′ , U ′ and V ′ such that C ⊆ C ′ ⊆ U ′ ⊆ U a nd D ⊆ D ′ ⊆ V ′ ⊆ V , and such that C ′ , D ′ , β X \ U ′ and β X \ V ′ belo ng to B . One can then find a fold h X 0 , X 1 , X 2 i for h C ′ , D ′ , U ′ , V ′ i in B a nd this will also b e a fold for h C, D , U, V i .  The second ingr edient is the us e of notions from Mo del Theory , esp ecially el- ement ar y substr uctures and the L¨ owenh eim-Skolem theorem. In the context of lattices elementarity is per haps b est ex pla ined in terms of so lutions to eq uations. One can interpret Prop erty (KM) as sa ying that cer tain equations should have solu- tions: the quadruple h C, D , U, V i determines six equations a nd a fold h X 0 , X 1 , X 2 i is a solution to this system. ON HEREDIT ARIL Y INDECOMPOSA BLE COMP ACT A 3 One calls M an element ar y subla ttice of L if every lattice-theor etic eq uation with constants from M that has a solution in L also has a solution in M . T o illustrate its use we pr ov e the fo llowing lemma. Lemma 2.2. Assum e X is a her e ditaril y inde c omp osable c omp act sp ac e and let L b e an elementary sublattic e of the lattic e of close d subsets of X . Then w L is also her e ditarily inde c omp osable. Pr o of. By elemen tarity the la ttice L sa tisfies P rop erty (KM): if C , D , X \ U and X \ V b elong to L then ther e is a fold h X 0 , X 1 , X 2 i in the full family of clo sed sets, henc e there is also suc h a fold in L . Next, in w L the same ar gument as in the pro of of Theo r em 2 .1 applies: an arbitrar y quadr uple can b e expanded to a q uadruple from the base.  The L¨ ow enheim-Skolem Theorem provides us with many elementary substruc- tures: given a la ttice L and some subset A o f L one ca n construct an elementary sublattice L A of L that cont ains A and whose cardinality is at most | A | × ℵ 0 . Theorem 2.3 ([22, 2 1212, 1244, 4]) . L et f : X → Y b e a c ontinuous sur je ction fr om a her e ditarily inde c omp osable c omp act s p ac e onto a c omp act sp ac e. Then ther e ar e a c omp act sp ac e Z and c ontinuous maps g : X → Z and h : Z → Y such that Z is her e ditarily inde c omp osable, dim Z = dim X , w ( Z ) = w ( Y ) and f = h ◦ g . Pr o of. Let B be a bas e for the closed sets of Y of car dinality w ( Y ). Via B 7→ f − 1 [ B ] we can identify B with a subla ttice of the lattice D of closed subsets of X . Apply the L¨ ow enheim-Skolem Theo rem to find an element ar y sublattice C of D that contains B a nd has the same (infinite) car dinality as B ; we let Z = w C . The tw o inclusions B ⊆ C ⊆ D induce co ntin uous surjections g : X → Z and h : Z → Y that, as one readily shows, s atisfy f = h ◦ g . By Lemma 2.2 the spa ce Z is hereditarily indecomp osable. The same ar gument shows that dim Z = dim X : one can use , for example, the Theor em o n Partitions, [33; 3, Theor em 1 .7.9], to turn the statement dim X 6 n into an equation Φ n . B y elementarit y C and D satisfy Φ n for exactly the same v alues of n . The expansion trick applies in this case as well so that dim Z 6 n for exactly the same v alues of n fo r which C satisfies Φ n .  W e refer to [55, 5] for basic information on Mo del Theory . R emark 2.4 . The thesis [1 212, 12] contains a systematic study of pro p erties that are preserved by co nt inuous maps tha t are induced by elementary embeddings. 3. P roofs of the main resul ts W e start with the following Theorem 3.1. L et f : E → F b e a p erfe ct mapping fr om a sp ac e E onto a str ongly zer o-dimensiona l p ar ac omp act sp ac e F such that for every y ∈ F t he fib er f − 1 ( y ) is her e ditarily inde c omp osable. Then E has Pr op erty (K M). Pr o of. Let C and D be disjoint clos ed s ubs ets of E and let U and V disjoint op en subsets of E ar ound C and D resp ectively . Let us fix y ∈ F . W e sha ll find a (clop en) neighbourho o d O y of y and a fold of f − 1 [ O y ] for h C ∩ f − 1 [ O y ] , D ∩ f − 1 [ O y ] , U , V i . Since f − 1 ( y ) is compact and hereditarily indecompo sable, by Theo r em 1 .3 it has Pr op erty (KM) and hence there exists a fold h X 0 , X 1 , X 2 i of f − 1 ( y ) for h C ∩ f − 1 ( y ) , D ∩ f − 1 ( y ) , U, V i . 4 K. P . HAR T AND E. POL Apply [33; 3, Theorem 3.1.1] to find a sequence B = h W 0 , W 1 , W 2 , O U , U V i of op en sets such that their clo s ures form a swelling of the s equence A = h C ∪ X 0 , X 1 , D ∪ X 2 , X \ U, X \ V i , which means that each term of A is a subset of the corres p o nding ter m B and whenever I is suc h that T i ∈ I A i = ∅ then T i ∈ I cl B i = ∅ . Spec ific a lly this means that (1) f − 1 ( y ) ⊆ W 0 ∪ W 1 ∪ W 2 ; (2) cl W 0 ∩ cl W 1 ⊆ V ; (3) cl W 1 ∩ cl W 2 ⊆ U ; (4) cl W 0 ∩ cl W 2 = ∅ . As the map f is p erfect a nd the space F is zero -dimensional we can find a clop en neighbourho o d O y of y such that f − 1 [ O y ] ⊆ W 0 ∪ W 1 ∪ W 2 . It follows that h cl W 0 , cl W 1 , cl W 2 i is a fold of f − 1 [ O y ] for h C ∩ f − 1 [ O y ] , D ∩ f − 1 [ O y ] , U , V i . By strong zero-dimensionality and par acompactness w e c a n find a disjoint clop en refinement O of { O y : y ∈ F } ; it is then a ro utine matter to combine the ‘lo cal’ folds in to one ‘globa l’ fo ld of E for h C, D , U, V i .  W e are now r eady to prov e the fir st main result. Pr o of of The or em 1.1. T o b egin we construct a zero-dimensional compactification Y ∗ of Y , a compactification X 1 of X a nd a contin uous extension f 1 : X 1 → Y ∗ . One way of doing this is b y assuming that X is embedded in the Hilbert cub e I ℵ 0 , that Y is em be dded in the Can tor se t { 0 , 1 } ℵ 0 and then to iden tify X with the graph of f , i.e., X is identified with G ( f ) =  x, f ( x )  : x ∈ X  ⊆ I ℵ 0 × { 0 , 1 } ℵ 0 via x 7→  x, f ( x )  . After this identification f is s imply π 2 ↾ G ( f ), wher e π 2 is the pro jection on to the seco nd factor of the pro duct; we ca n then let X 1 = cl G ( f ) (in the product) and Y ∗ = cl Y (in the Ca nt or set), the des ired extension f 1 of f then is π 2 ↾ X 1 . Next let j : β X → X 1 be the natur a l map (the ex tension of the inclusion of X into X 1 ). By Theorem 3.1 X has Prop er t y (KM) so by Theorem 2.1 β X is hereditarily indecomp osa ble. Apply Theor em 2.3 to obtain a factoriza tio n of j consisting of maps g : β X → X ∗ and h : X ∗ → X 1 in which X ∗ is hereditarily indecomp osable, se cond-countable and satisfies dim X ∗ = dim β X = dim X . Then X ∗ is a metrizable compa c tification of X as g ↾ X is a homeomorphism. It remains to set f ∗ = f 1 ◦ h .  Let us note that since f is p er fect and X ∗ is a c o mpactification o f X , the exten- sion f ∗ satisfies ( f ∗ ) − 1 ( y ) = f − 1 ( y ) for y ∈ Y . T o get universal her editarily indecomp osable compac ta we us e the the fac to riza- tion method a gain. Pr o of of The or em 1.2. Let { X s } s ∈ S be the fa mily of all compact hereditarily in- decomp osable subspa ces of the Tyc honoff cub e I τ whose dimension is not larger than n , and let i s : X s → I τ be the inclusio n. Let X = L s ∈ S X s be the free union of the X s ’s a nd let i : X → I τ be defined by i ( x ) = i s ( x ) for x ∈ X s . Let f : β X → I τ be the extension of i ov er β X . Obviously , X has Pr op erty (KM), so by Theorem 2.1 β X is hereditarily indecomp osable. By Theorem 2.3 f can be factored as h ◦ g , where g : X → Z and h : Z → Y and where Z is hereditarily indecomp osable, w ( Z ) 6 τ and dim Z = dim X . W e ca n take X ( n, τ ) = Z .  ON HEREDIT ARIL Y INDECOMPOSA BLE COMP ACT A 5 4. Corollaries and Remarks Let us no te that as a co rollar y to either Theor em 1.1 or Theorem 1.2 one can obtain the following theorem o f Ma ´ cko wiak [1010, 10]. Corollary 4.1. F or every n ∈ { 1 , 2 , . . . ∞} ther e exists a her e ditarily inde c omp os- able metric c ontinuu m Z n of dimension n c ontaining a c opy of every her e ditari ly inde c omp osable metric c ontinuum of dimension at most n . Pr o of using The or em 1.1. Let P b e the subset of the hyper space 2 I ℵ 0 of the Hilbert cube consisting of all heredita rily indecomp osable contin ua of dimension n o r les s. Then P is a G δ -subset of 2 I ℵ 0 (see [88; 8, § 45, IV, Theor em 4 and § 48 , V, Remark 5]). The r efore there is a contin uous surjection ϕ : Y → P , where Y is the space of the irrationals . T hen let X b e the following s ubset of I ℵ 0 × Y :  ( x, t ) : t ∈ Y and x ∈ ϕ ( t )  and let π : I ℵ 0 × Y → Y b e the pro jection. The restrictio n f = π ↾ X : X → Y is a p erfect map (cf. [77; 7, § 18] or [1111; 1 1, E xercise 1.11.26]) with hereditarily indecomp osable fib ers. By Theor em 1 .1 there e xists a hereditar ily indeco mp o s- able n -dimensional compact space X ∗ that cont ains X and hence a copy of every hereditarily indecomp osable contin uum of dimension n . The decomp ositio n of X ∗ int o its comp onents yields a compact zero- dimensional space. The ps e udo-arc P co nt ains a copy of this decomp osition space (as indeed do es a ny unco untable compact metr iz a ble space). Let q : X ∗ → P b e a map such that A = q [ X ] is that decomp osition space and q : X → A is the quotient map. By Theorem 15 of [99, 9] there exist a hereditarily indecomp osa ble contin uum Z n and an a tomic mapping r from Z n onto P such that r ↾ r − 1 ( P \ A ) is a homeo - morphism a nd r − 1 ( A ) is homeomo rphic to X ∗ ( Z n is a so- called pseudosusp ensio n of X ∗ ov er P by q ). Since dim Z n 6 n by the coun table sum theorem and Z n con- tains X ∗ top ologica lly , the space Z n has the required prop erties.  Pr o of using The or em 1.2. Use the second half o f the pre vious pro of but now take the pseudo susp ension of the space X ( n, ℵ 0 ) ov er P b y q , where q : X ( n, ℵ 0 ) → P is a quo tient map such that A = q [ X ( n, ℵ 0 )] is the decompo sition space of X ( n, ℵ 0 ) int o its comp onents.  R emark 4.2 . If one uses Theorem 2.3 instead of Marde ˇ si´ c’s F actorization Theo rem, and standard top o logical r easoning (see [33; 3, pro o fs of Theore ms 5.4.3 and 3.4.2 ]) one gets the following results. Pr op osition 4.3 . F or every her e ditarily inde c omp osable c omp act sp ac e X such that dim X = n and t he weight of X is e qual to τ , ther e ex ists an inverse system S = { X σ , π σ ρ , Σ } , wher e | Σ | 6 τ , of metrizable her e ditarily inde c omp osable c omp act sp ac es of dimension n whose limit is home omorphic to X . If X is a c ontinuu m, then al l X σ ar e c ontinua. Pr op osition 4.4 . Every normal n -dimensional sp ac e X of weight τ that has Pr op- erty (KM) has a her e d itarily inde c omp osable c omp actific a tion ˜ X of dimension n and of weight τ . R emark 4.5 . The re s ults of this pap er remain v alid if in the formulation of Pr op- erty (KM) one replaces closed sets by zero-sets and o p e n sets by cozer o-sets. This 6 K. P . HAR T AND E. POL implies tha t in Theorem 2.1 one can relax the assumption of normality to co mplete regular ity . References [1] ]cite.Aarts1J. M. A ar ts, Wallman-Shanin Comp actific ation , Encyclopedia of general topology (Klaas Pieter Hart, Jun-iti Nagata, and Jerry E. V aughan, eds.), Elsevier Science Publishers B.V., Amsterdam, 2004, pp. 218–220. [2] ]cite.MR17858372P aul Bankston, Some applic ations of the ultr ap o wer the or em to the the ory of co mp acta , Applied Categorical Structures 8 (2000 ), no. 1-2, 45–66. Papers in honour of Bernhard Banasc hewski (Cap e T o wn, 1996). 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MR 1503392 (Klaas Pieter Hart) F acul ty of Electric al Engineerin g, Ma thema tics and Computer Science, TU Delft, Postbus 5031, 2600 G A Delf t, the Nether lands E-mail addr ess : k.p.hart@tudelf t.nl URL : http://fa.its.t udelft.nl/~hart (El ˙ zbieta Pol) Institute of M a thema tics, University of W arsa w, Banacha 2, 02-197 W arsza w a, Poland E-mail addr ess : pol@mimuw.edu.p l