Leleks problem is not a metric problem
We show that Lelek's problem on the chainability of continua with span zero is not a metric problem: from a non-metric counterexample one can construct a metric one.
Authors: Dana Bartosova, Logan Hoehn, Klaas Pieter Hart
LELEK’S PR OBLEM IS NOT A METRIC PR OBLEM DANA BAR TO ˇ SOV ´ A, KLAAS PIETER HAR T, LOGAN C. HOEHN, AND BERD V AN DER STEEG T o Ken Kunen Abstract. W e sho w that Lelek’s problem on the c hainability of contin ua with span zero is not a metric problem: from a non-metric coun terexample one can construct a metric one. Introduction The notion of span of a metric contin uum was int ro duced by Lelek in [10], where he show ed tha t chainable contin ua hav e span zero, and in [11] he asked whether contin ua with span zero a re ch ainable. This has bec ome one of the classic pro blems of Co n tin uum Theory , see [12] for a rece n t survey . The purp ose o f this pa p er is not to solve Lelek’s problem; our goa l is more mo dest: we sho w that a non-metrizable counterexample to the problem may b e conv erted into a metrizable one. This makes the tools o f infinitar y combinatorics av a ila ble to those sear c hing for a counterexample. Our pro of makes use of metho ds from Mo del Theo r y , most notably the L¨ owen- heim-Skolem theorem. Given a non- metric contin uum one can use this theore m to obtain a metric quotient that shares man y prop erties with the origina l space. Indeed, we shall prov e that the quo tien t will b e chainable iff the original space is and likewise for having span zero. The pro of of one of the fo ur implications is muc h mor e inv olved than t hat of the other s as it relies o n Shelah’s Ultrap ow er Isomorphism theorem fro m [13]. This sug gests an ob vious question that w e shall discuss at the end of this pap er. Section 1 contains some pr eliminaries. W e rep eat the definitions of chainabilit y and the v a r ious for ms o f span. W e also describ e the results from Mo del Theory that will b e used in the pro ofs. In section 2 we prove our main results and in sec tio n 3 we discuss some questions re lated to the pr oofs. 1. Preliminaries 1.1. Chainability and span. Let X b e a contin uum, i.e., a connected compact Hausdorff spa ce. W e say X is chainable if every finite op en cover has a refinement that is a chain, which means t hat it can b e en umer ated a s h V i : i < n i suc h that V i ∩ V j 6 = ∅ iff | i − j | ≤ 1. W e shall dea l with four kinds of span: span, semispa n, surjective span, and sur- jective semispan. E ac h is defined, for a metric contin uum ( X , d ), as the supremum of all ǫ ≥ 0 for which there is a subc o n tin uum Z of X × X with the pro perty that d ( x, y ) ≥ ǫ for all ( x, y ) ∈ Z and Date : T uesday 22-12-2009 at 10:57:04 (cet). 2000 Mathematics Subje ct Classific ation. Pri mary: 54F15. Secondary: 03C20 03C68. Key wor ds and phr ases. con tin uum, chainable, span zero, reflection, L¨ ow enheim-Skolem theo- rem, ul trapow ers. 1 2 D. BAR TO ˇ SOV ´ A, K. P . HAR T, L. C. HOEHN, AND B. V AN DER STEEG • π 1 [ Z ] = π 2 [ Z ], in the ca se of span; • π 1 [ Z ] ⊆ π 2 [ Z ], in the ca se of semispan; • π 1 [ Z ] = π 2 [ Z ] = X , in the cas e of sur jectiv e span; o r • π 2 [ Z ] = X , in the cas e of sur jectiv e semispa n. Thu s an y one of the spans is equal to zero if every sub con tin uum of X × X with the corres p onding prop ert y from the list must intersect the diag onal ∆ X of X . This then yields four definitions o f having span zero fo r general contin ua. There are relations b et ween these four k inds of span zero , co rresp o nding to the inclusion relations betw een the defining collections of subcontin ua o f X × X ; see [7] for a diagra m a nd also for a pro of that chainability implies that all spans a re zero. The dia g ram in [7] also ment ions (surjective) sy mmetric span, but, as re p orted in [3], the dyadic solenoid, which is no t chainable, has symmetric span zero , so that symmetric span zer o do es no t characterize chainability . The r eader will b e a ble to chec k that having (surjective) symmetric span zer o is also cov ered by our re flection results. 1.2. W allman representa tion. In the construction of the metric quotient we employ the W allma n representation of dis tributiv e lattices. W e start with a compact Ha usdorff spa c e X a nd co nsider its la ttice of closed sets 2 X . An y sublattice, L , of 2 X gives rise to a co n tin uous image of X : the space w L of ultrafilters on L . If a ∈ L then ¯ a denotes { u ∈ wL : a ∈ u } ; the family { ¯ a : a ∈ L } is used as a base for the closed sets in w L . In g eneral this yields a T 1 -space; the space wL is Hausdor ff iff L is normal , which mea ns that disjoint elemen ts of L can b e separa ted by disjoin t op en sets that are co mplemen ts of member s o f L . In gener al, a lattice embedding h : L → K yields a contin uous onto map w h : wK → wL , where wh ( u ) is the unique ultrafilter on L tha t contains { a : h ( a ) ∈ u } (this family is a prime filter), so that in o ur case we obtain a c on tin uous o n to map q L : X → wL . It sho uld b e clear that X is the W a llman s pa ce of 2 X . How ev er, o ne space may corres p ond to many lattices. Indeed, if C is a bas e for the closed sets of X that is closed under finite unions and intersections then X = w C . The a rticle [1] gives a go od introductio n to W a llman repres en tations. 1.3. Eleme ntarit y. T o co nstruct the metric quo tien t men tioned in the introduc- tion we need a sp ecial sublattice o f 2 X , an elementary sublattice. In gener al a substructure A of s o me structure B (a group, a field, a la ttice) is said to b e an elementary subs tr ucture if every s en tence in the lang uage for the structure, with para meters from A , that is true in B is also true in A . A sentence is a fo r m ula without free v ar iables and such a formula is tr ue in a structure if it holds with all its qua ntifiers b ound by that s tructure. As a quic k example consider the subfield Q o f R : it is not an elementary subfield bec ause of the following sentence: ( ∃ x )( x 2 = 2) The parameter 2 b elongs to Q ; the sentence ho lds in R but do es not hold in Q . This ex ample illustrates the source of the power o f e le men ta rit y: b ecause a ll exis- ten tial statement s true in the larger structure must b e true in the substructure this substructure is very rich. In fact, a n elemen tary subfield o f R m ust contain all real algebraic num b ers and it is a non-triv ia l result tha t these num ber s do in fact form an elementary subfield o f R . LELEK’S PROB LEM IS NOT A METRIC PROBLEM 3 By a straightforw ard clos ing-off a r gumen t one sho ws that ev ery subset of a struc- ture can b e expa nded to an elementary substructure — this is the L¨ owenheim- Skolem theor e m [8, Co r ollary 3.1.4]. In full it states that a subset, C , of a struc- ture B can b e expa nded to an elementary substructure A whose cardinality is at most ℵ 0 · | C | · |L| , where L is the languag e used to descr ib e the structures. In the case of lattices the languag e is countable: one ne e ds ∧ , ∨ and = as well a s logi- cal sy m bols a nd (co un tably ma n y) v ariables. Thu s every lattice has a countable elementary sublattice. As w e discuss in Section 3 the ex pr essiv e power of the lang uage of lattices is not strong enough fo r our purpo ses; there fore w e consider str uctures for the languag e of Set Theory . Any reasonably large set will do but usually one takes a ‘suitably larg e’ regular cardinal num b er θ and c o nsiders the set H ( θ ) of sets that are hereditarily of cardinality less than θ , which means that they and their e lemen ts a nd their elements’ elements a nd . . . all have cardina lit y less than θ . The adv a n tage o f these sets is that they satisfy all of the ax ioms of Set Theory , ex c e pt p ossibly the p ow er set axiom. What will b e particularly useful to us is that if M is a n elementary substr ucture of H ( θ ) then ω is b oth an e le men t and a subse t of M ; this is b ecause ω and each finite ordinal are uniquely defined in H ( θ ) by a form ula with just one free v a riable; therefore they automa tically b elong to M . As a consequence o f this every finite subset of M is an element of M and this will give us the extr a flex ibility that we need. W e refer to [9 , Chapters IV and V] for information o n the sets H ( θ ) a nd e le - men tarity in the context of Se t Theory . Note that the langua ge of Set Theor y has even fewer no n-logical symbols than that of lattice theory : ∈ a nd =. The lattice op erations, ∩ and ∪ , are derived from these. 1.4. Ultrap o wers and ultracop o w ers. W e shall be using ultrap ow er s of la ttices so we nee d to fix some no tation. Let L b e a lattice; given an ultra filter u on a cardinal n umber κ we define the ultrap o wer Q u L of L by u to b e the quo tien t of L κ by the equiv a le nce r elation ∼ u defined by h x α : α < κ i ∼ u h y α : α < κ i iff { α : x α = y α } ∈ u . W e turn Q u L in to a la ttice by defining the o peratio ns po in t wise. There is an obvious em bedding △ : L → Q u L , the dia gonal em bedding, defined by sending an ele ment a to the (class of the) s equence h a : α < κ i . Dual to this is the notion of ultracop o wer of a compact Hausdorff s pace X by an ultrafilter u . One c a n define it in tw o equiv alen t wa ys. The first is as the W allman representation of the ultrap ow er Q u 2 X of the lattice 2 X by u . The seco nd is via the ˇ Cech-Stone compactification. Consider the pro duct κ × X , where κ ca rries the discrete top ology , and the tw o pro jectio ns π X : κ × X → X and π κ : κ × X → κ . These have extensio ns , β π X : β ( κ × X ) → X and β π κ : β ( κ × X ) → β κ resp ectively . The preimage β π ← κ ( u ) is homeo morphic to the W allman repr e sen tation of Q u 2 X . This follows from the facts that (1) β ( κ × X ) is the W a llma n r epresent ation of 2 κ × X , whic h in turn is is omorphic to (2 X ) κ ; and (2) if F and G are c lo sed subsets of κ × X then the intersections cl β F ∩ β π ← κ ( u ) and cl β G ∩ β π ← κ ( u ) are equal iff the set o f α s for w hich F ∩ ( { α } × X ) = G ∩ ( { α } × X ) be lo ngs to u . The top ological viewp oin t enables us to se e easily that one may use any ba se, C , for the closed sets that is closed under finite unions and finite intersections to construct the ultraco pow er. Indeed, if F and G ar e closed and disjo in t in κ × X then a compactness argument a pplied to { α } × X for each α will yie ld sequences 4 D. BAR TO ˇ SOV ´ A, K. P . HAR T, L. C. HOEHN, AND B. V AN DER STEEG h B α : α < κ i a nd h C α : α < κ i in C such that B α ∩ C α = ∅ for all α , and F ⊆ S α { α } × B α and G ⊆ S α { α } × C α . This then can b e us ed to show that the dual to the inclusion map C κ → (2 X ) κ is injectiv e, so that β ( κ × X ) = w ( C κ ), and, similarly , that the dual to the inclusio n map Q u C → Q u 2 X is injectiv e, which gives us that β π ← κ ( u ) is the W allman representation of Q u C . W e denote the ultra copow er of X by u as ` u X . Also, if h F α : α < κ i is a sequence of closed subs ets of X then w e let F u be the intersection of cl β ( S α { α }× F α ) with ` u X ; in case F α = F for all α the set F u corres p onds to the image of F under the diagona l em b e dding into Q u 2 X . The restriction of β π X to ` u X is induced by the diag onal embedding △ , we shall denote it by ▽ . 2. Reflections W e fix a contin uum X , a suitably lar ge ca rdinal θ and a co un table element ary substructure M of H ( θ ) with X ∈ M ; as θ was ta ken lar ge enoug h the entities X × X , 2 X and 2 X × X belo ng to M as well, by elementarit y . W e let L = M ∩ 2 X and K = M ∩ 2 X × X . The family B L = { w L \ F : F ∈ L } is a base for the o pen sets of L . As M is countable, so a re L a nd K . There fore w L and w K are compact metriz- able s paces. W e shall have proved our main result o nc e w e establish that w L is chainable iff X is and tha t w L has spa n zer o iff X do es. 2.1. Chainability. W e first show that X is chainable if and only if w L is. The forward implication is ea siest to establish. Prop osition 2. 1 ([14, Sectio n 7.2 ]) . If X is chainable t hen so is w L . Pr o of. Let U be a finite ope n cover of w L . By co mpactness we c a n find a finite subfamily B of B L that refines U . Beca use every finite subset of M b elongs to M we hav e B ∈ M . Now the for mula that expresses ‘ C is a chain refinement of B ’ — with C as its only fre e v a riable — is satisfied by a member of H ( θ ) and hence by a n element of M . The latter consists of member s of B L and is a finite c hain re finement of B , and hence o f U . The co nverse implication is slig h tly harder to e stablish; in the pro of we em- ploy the no tion of a a precis e refinement. A pr e cise refinement of a cover U is a refinement, { V U : U ∈ U } , indexed by U such that V U ⊆ U for a ll U . Prop osition 2. 2 ([14, Sectio n 7.3 ]) . If X is not chainable then neither is w L . Pr o of. There is an op en cov er of X that do es not hav e an op en chain refinement. This statement can b e expressed b y a form ula, with parameters in M , that is quite complicated: ex pr essing that a cov er do es not hav e a chain r e finemen t inv olves a quantification ov er a ll finite sequences o f elements of 2 X . By elementarit y this formula ho lds in M , so we can take a n open cover, U , o f X that b elongs to M and that satisfies the formula with al l quantifiers re stricte d to M , which means that U has no chain r efinemen ts that co nsist of members o f B L . As U is a subset of B L it also for ms an op en cov er of w L . W e must s ho w that U do es not hav e a n y op en chain refinement at all. Let V b e any finite op en refinement o f U . By normality w e can find a closed cover { F V : V ∈ V } of wL such that F V ⊆ V for all V . By compactness we can find finite subfamilies B V of B L such that F V ⊆ S B V ⊆ V for all V . Then W = { S B V : V ∈ V } is a r efinemen t of U that consists o f members o f B L , hence it is no t a chain refinement. As W is a precise refinement of V the latter is not a chain refinement of U either. LELEK’S PROB LEM IS NOT A METRIC PROBLEM 5 2.2. Pro ducts. T o establish that (non-)zero s pan is r eflected we ne e d to explore the relations hip b et ween w L × w L a nd wK . It is clear , by elementarit y , tha t K con tains the families { A × X : A ∈ L } and { X × A : A ∈ L } . W e use L ′ to denote the subla ttice of K generated by these families. W e trust that the r eader will r ecognize the formula implicit in the following pro of. Lemma 2.3. If F and G ar e elements of K with empty int erse ction then ther e ar e F ′ and G ′ in L ′ such that F ⊆ F ′ , G ⊆ G ′ and F ′ ∩ G ′ = ∅ . Pr o of. By compactness there are finite families U a nd V of basic o p en sets s uch that F ⊆ S U , G ⊆ S V and cl S U ∩ cl S V = ∅ . By elementarit y , and b ecause F, G ∈ M there are in M tw o sequences h A i , B i i , i < n and h C j , D j i , j < m of pairs of clo sed sets such that F ⊆ S i
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