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Annals of Pure and Applied Logic
www.elsevier.com/locate/apal
Banach-Stone-like results for combinatorial Banach spaces
✩C. Brecha,∗, C. Piñab,c
a DepartamentodeMatemática,InstitutodeMatemáticaeEstatística,UniversidadedeSãoPaulo, Rua doMatão,1010, CEP05508-090, SãoPaulo, SP, Brazil
bDepartamentodeMatemáticas,FacultaddeCiencias,UniversidaddelosAndes,C.P.5101,Mérida, Venezuela
cEscueladeMatemáticas,UniversidadIndustrialdeSantander,C.P.680001,Bucaramanga, Colombia
a r t i cl e i n f o a b s t r a c t
Articlehistory:
Received7December2020 Receivedinrevisedform30March 2021
Accepted25April2021 Availableonline28April2021
MSC:
primary46B45 secondary03E75
Keywords:
Banach-Stonetheorem CombinatorialBanachspaces Compactandspreadingfamilies Homeomorphismsinducedby permutations
IsometriesofBanachspaces Schreierfamilies
Weshowthat underacertaintopologicalassumptionon twocompacthereditary familiesFandGonsomeinfinitecardinalκ,thecorrespondingcombinatorialspaces XF and XG areisometric if andonly if there is a permutationof κ inducing a homeomorphismbetweenFandG.Wealsoprovethattwodifferentregularfamilies F andGonωcannotbepermutedonetotheother.Boththeseresultsstrengthen themainresultof[5].
©2021ElsevierB.V.Allrightsreserved.
The classicalBanach-Stone theorem states that,given two compact Hausdorffspaces K and L, C(K) and C(L) are isometric Banachspaces if and only ifK and L are homeomorphic (see e.g. [13, Theorem 7.8.4]). Ourmain purpose inthis paper is to proveversions of this resultinthe context of combinatorial Banachspaces.
Given afamily F offinite subsets of aninfinite cardinalκ, itcanbe seenas atopological subspaceof 2κ. IfF containsallsingletons ofκandisclosed undersubsets,thecombinatorialBanachspace XF (also called the generalized Schreier space) is the completion of c00(κ), the vector space of finitely supported scalarsequences,withrespectto thenorm:
✩ ThefirstauthorwassupportedbyCNPqgrant(308183/2018-5)andbyFAPESPgrant(2016/25574-8).Thesecondauthorwas supportedbythepostdoctoralprogramofVicerrectoríadeInvestigaciónyExtensióndelaUIS(RC2020000096).
* Correspondingauthor.
E-mailaddresses:brech@ime.usp.br(C. Brech),cpinarangel@gmail.com(C. Piña).
https://doi.org/10.1016/j.apal.2021.102989 0168-0072/©2021ElsevierB.V.Allrightsreserved.
x= sup
α∈s
|x(α)|:s∈ F
.
Notice thatthesequence ofunitvectors (eα) formsa(long)Schauderbasisof XF, whichis consideredto be thecanonicaloneandis1-unconditional.Moreover,ifF iscompact,thecanonicalbasisiseasilyseento be shrinking(see e.g.[4,Theorem6.3]).
Both spaces XF and C(F) can be seenas Banachspaces counterparts of thefamily F and they share some properties: for instance, both contain homeomorphic copies of F in the dual unit sphere equipped with the weak∗ topology. Banach-Stone theorem guarantees thatthe isometric structure of C(F) carries exactlythetopological structureof F.InSection2weproveourfirstmain result,Theorem7,andgivein Corollary 15sufficienttopologicalassumptions ontwofamiliesF andG,so thatXF andXG are isometric ifandonlyifF and Gareπ-homeomorphic(thatis,there isahomeomorphismbetweenF andG whichis inducedbyapermutationofκ,seeSection1).Hence,theisometricstructure ofXF carriesmorethanthe topological structureofF.
Isometries between combinatorialspaces have been recently analyzed in [1,5]. In [1, Theorem 5.1],the authors prove that any isometry of the real combinatorial space of a Schreier family of finite order is determined by a change of signs of the elements of the canonical basis. This has been generalizedin [5]
to real and complex combinatorial spaces of regular families. Regular families are families on the set of theintegersω whicharehereditary(closedundersubsets),compactasasubspaceof2ωandspreading(see Definition10fordetails).OurproofofTheorem7followssimilarlinesastheirs,butweextractedasufficient topologicalconditionwhichmadeitpossibletoenlargetheclassoffamiliesforwhichanyisometrybetween two combinatorialspaces isgiven byasignedpermutationof thecanonicalbases: itincludesnow families on ω whichare notnecessarilyspreading,as well asfamilieson uncountablecardinals κ. Moreover,Arens andKelley[2] gaveasimpleproofoftheBanach-Stonetheoremusingthefactthatisometriestakeextreme pointstoextremepoints.ItwasimprovedtoanoncommutativeversionforC∗-algebrasbyKadison[8].Our proof alsousesthisfact,asdoesthatof[5].
Theorem7statesthatanyisometrybetweenXF andXG isinducedbyapermutationofκandasequence of signs for aclass of families.If we takeF to be thefamily of singletons of ω,we get thatXF = c, the Banach spaceof convergent sequencesof scalars,with thesupremum norm. Taking F to be thefamily of allfinitesubsetsofκmakesXF tobe1(κ).Hence,similarlytowhatismentionedin[5],ourresultscanbe compared to thefactthatisometries ofthespacesc0or p,1≤p<∞, p= 2,areallsignedpermutations of thecanonicalunitbasis(seee.g.[9,Theorem 2.f.14]).
Our second main result, Theorem 14, states that two different regular families on ω are not π- homeomorphic. A permutation of ω which takes F onto G and a sequence of signs induce an isometry between XF and XG. [5, Theorem 10] shows that theconverse is also true for regular families F and G: any isometry between XF and XG is induced bya permutationof ω and asequence of signs. Our Theo- rem 14showsthatunderany ofthefollowing two equivalentassumptions -G being theimageof F under apermutationofωor XF andXG being isometric-F andG arejustthesamefamily.Meaningthatthese morphisms preservenotonlythestructureofthefamily,butthefamilyitself.
Thecrucialdifference betweenthehypotheses ofTheorems7and14istheassumptionthatthefamilies arespreading(seeDefinition10).Acompactfamilycanonlyhappentobespreadingforfamiliesonω.The propertiesofbeing compactorhereditaryarepreservedunderpermutationsofκ,asapermutationinduces a homeomorphism between a family and its image. However,this is not the case for spreading. This led us to suspect thatbeing spreading isa strong assumption for theclassification of theisometries between combinatorialspacesobtainedin[5] andthis isindeedthecase,asguaranteesTheorem14.
AnotherimplicationofafamilybeingspreadingisthattheCantor-Bendixsonindexofthesingletons{n} is amonotone nondecreasingsequence ofordinals. This factisexplored inmoredetail inthelast section, where wealsogivefurtherexamplesandremarks relatedtoboththecountableandthegeneralsetting.
The paper is organized as follows. In Section 1 we introduce definitions, notations and basic facts to be used in thefollowing ones. InSection 2 we present Theorem 7, which characterizes the isometries on combinatorialspaces offamilies inabroaderclass than theclass ofregular families,including familieson uncountable cardinals.In Section3, we show that two different regular families cannot be permutedone to theother:thisis Theorem14and itsproofispurely combinatorial.Wefinishthepaperexploring more aboutthetopologyand givingexamplesandremarkspresented inSection4.
1. Preliminaries
Given aninfinitecardinalκ,we will denoteby [κ]<ω theset of allfinite subsetsofκ. Givenn∈ω and M⊆κ,wedenoteby[M]n thecollectionofallsubsetsofMofsizen,by[M]<nthecollectionofallsubsets ofM ofsizelessthann,and by[M]≤n theunion[M]n∪[M]<n.Ifs∈[κ]n,wewrite s={α1,. . . ,αn}< to indicatethattheenumerationisstrictly increasing.
Weintroducenowsomenotionsaboutfamilies:
Definition1.Afamily F onκisasetF ⊆[κ]<ω and wewillalwaysassumethat[κ]≤1⊆ F.
• WesaythatafamilyF ishereditary ifitisclosedundersubsets:s∈ F whenevert∈ F and s⊆t.
• A familyF on κis atopological spacewhen endowed with the subspacetopology inherited from 2κ, where eachs∈ F isidentified withitscharacteristicfunctionand2κhastheproducttopology.
• Anytopological assumptionaboutF referstothistopology.Forinstance,F iscompactifitisaclosed subspaceof2κ.
GivenafamilyF,letFM AX denotethesetofmaximalelementsofF with respecttotheinclusionand F⊆ denote itsdownwardsclosure,thatis:
FM AX ={s∈ F:t∈ F such thatst} and F⊆={s:∃t∈ F such that s⊆t}.
A permutation of κis simply a bijection π : κ −→ κand if s ⊆ κ, then π[s] ={π(α) : α∈ s}. Any permutation π ofκ induces ahomeomorphism πˆ: 2κ −→ 2κ, defined byπ(x)(α)ˆ =x(π−1(α)).We chose to use hereπ−1 insteadof π inorder to have thefollowing: ifπ[F] denotes theimage of F by ˆπ, thenit coincides with {π[s] : s ∈ F}. We say that two families F and G on κ are π-homeomorphic if there is a permutationπofκsuchthatG=π[F].
ByanisometrybetweenBanachspaceswemeanalinearbijectiveoperatorwhichpreservesthenorm. If X isaBanachspace,X∗ denotesitstopologicaldual,BX istheunitballandBX∗ isthedualunitball.
Throughout thepaperwewill beconcerned aboutconditionsontwofamiliesF and G whichguarantee thatallor someofthefollowing statementsareequivalent andexamplesto showwhentheyarenot:
(i) F=G.
(ii) F andG areπ-homeomorphic.
(iii) XF andXG areisometric.
(iv) XF∗ andXG∗ areisometric.
Noticethatthese statementsaredecreasinginstrength.(i)clearlyimplies(ii)andifπisapermutation of κ witnessing (ii), then, given any sequence (θα) of scalars such that |θα| = 1, there is an isometry T :XF →XG satisfyingT(eα)=θαeπ(α) forallα∈κ, sothat(ii) implies(iii).(iii) implies(iv)easily, as theadjointoperatorofany isometrybetweentwoBanachspacesisanisometrybetweentheirdualspaces.
Moreover, in case T is such thatT(eα) =θαeπ(α) for every α∈ κ, some sequence of scalars (θα) and a permutationπ,thenclearlyT∗(e∗α)=θαe∗π−1(α)forallα∈κ.
Also, we will be mostly working with hereditary families, so that conditions (i) and (ii) are clearly equivalent toconditions(i’)and(ii’)below,andwewillusethem interchangeably:
(i’) FM AX =GM AX.
(ii’) FM AX and GM AX areπ-homeomorphic.
2. Isometriesbetweencombinatorialspaces
Thepurposeofthissectionisto(essentially)improvetheresultsof[5] bygettingthattwofamiliesareπ- homeomorphicfromweakerassumptionsonthem.Ourfirstpurposewastoreplacethespreadingcondition bysomeconditionwhichdoesnottaketheorderintoaccount.Thisledustoamuchmoregeneralversion, for compactand hereditaryfamilies,onany infinitecardinalκ,satisfyingsometopological condition.The existence ofnontrivial compactand hereditaryfamiliesonuncountablecardinals wasalreadyused in[10], and muchmorecomplexfamilieswereexplored in[6]. Werecallsomeof these examplesanddiscuss these briefly inSection4.
Our proof follows the path of the proof of Theorem 10 of [5]. In particular, we will also make use of extreme points in ourproof. Given a Banach space X, recall that xis an extreme point of aconvex set B ⊆X ifthere arenox1 =x2 inB suchthatx=αx1+ (1−α)x2for some0< α <1.LetExt(BX∗) be theset ofextremepointsofBX∗.
We willusethe characterizationoftheextremepoints ofBXF∗ stated in[7], andprovedand usedin[1]
and [5] inmoreparticular cases.Actually, despite beingstated forregular familiesin[5, Proposition5],a carefulanalysisofitsproofshowsthatthecharacterizationholdsforanycompactandhereditaryfamilyF. Forcompleteness purposes,wereproducetheirproofherewiththerequiredadaptations.
LetKbethefieldofscalarsRorC.GivenaBanachspaceX,wesaythatasubsetN⊆BX∗ isnorming if x= sup{|x∗(x)|: x∗ ∈ N}for everyx ∈X, and N is sign invariant if for any sign θ ∈K, we have θN =N.Thefollowing classicallemma wasstated asLemma4of[5]:
Lemma2.LetX beaBanachspaceoverK,andletN ⊆BX∗ beasigninvariantnormingsetforX.Then BX∗ =conv(N)w∗.
To prove thecharacterization of the extreme pointsin ourmore generalsetting, wereplace the weak∗ convergence ofsequencesusedin[5,Proposition5] bythefollowing auxiliarylemma:
Lemma 3.LetF be acompactandhereditaryfamily onκ.Then {
ξ∈s
θξe∗ξ :s∈ F and(θξ)ξ∈s is a sequence of scalars with |θξ|= 1}
is weak∗-closedin XF∗.
Proof. Let
M ={
ξ∈s
θξe∗ξ :s∈ F and (θξ)ξ∈sis a sequence of scalars with|θξ|= 1}.
Since (eξ)ξ∈κ is shrinking, given x∗ ∈ Mw
∗
, we can write x∗ =
ξ∈κθξe∗ξ for some sequence of scalars (θξ)ξ∈κconvergingto0.Foreachξ∈κ,noticethat
|θξ|=|x∗(eξ)| ∈ {|y∗(eξ)|:y∗∈M)}={0,1}.
Hence,s = suppx∗ is finite.Lety ∈ M be such that|x∗(eξ)−y∗(eξ)|< 12 for everyξ ∈s. Inparticular, this impliesthat|y∗(eξ)|= 1 for everyξ∈s, so thats ⊆suppy∗ ∈ F.Since F ishereditary, weget that s∈ F.
Wearenowreadyto reproducetheproof givenin[5,Proposition5] ofthefollowing characterization:
Proposition4.If F isacompactandhereditaryfamily onan infinitecardinalκ,then Ext(BX∗F) =
α∈s
θαe∗α:s∈ FM AX and(θα)α∈s is a sequence of scalars with |θα|= 1
.
Proof. Take N := {
α∈sθαe∗α : s ∈ FM AX,|θα| = 1}and M := {
α∈sθαe∗α : s ∈ F,|θα| = 1}. M is clearlynormingforXF andsigninvariant.ItfollowsfromLemma2thatBXF∗ =conv(M)w
∗
.ByLemma3, M isw∗-closed,sothatbothM andBXF∗ =conv(M)w∗ arecompactinthelocallyconvexspace(XF∗,w∗).
ItfollowsfromMilman’stheorem(see[12],Theorem3.25)thateveryextremepointofBX∗F liesinM.Also, (M\N)∩Ext(BX∗F)=∅,asanyx∈M\N canbewrittenasx=(x+e∗α)+(x2 −e∗α),wheresuppx∪ {α}∈ F. SinceN ⊆Ext(BXF∗) weconcludethatExt(BX∗F)=N.
Weproveanotherauxiliarylemmabeforegettingourimprovedversionof[5,Proposition10]
Lemma 5.Let F be a compact and hereditary family on κ. Given α ∈ κ, if {α} ∈ FM AX, then e∗α ∈ Ext(BXF∗)w
∗
.
Proof. Fixα∈κsuchthat{α}∈ FM AX andletε>0 andx1,. . . ,xn ∈XF. Wehaveto showthatthere isx∗∈Ext(BXF∗) suchthat|e∗α(xi)−x∗(xi)|< εforall1≤i≤n.
Takey1,. . . ,yn∈c00(κ) suchthatyi−xi< ε2forevery1≤i≤nandtakeF =
{suppyi: 1≤i≤n}. SinceF isfiniteand {α}∈ FM AX,thereiss∈ FM AX suchthatα∈sand s∩F ⊆ {α}.
Let x∗ =
ξ∈se∗ξ and notice thatx∗ ∈Ext(BXF∗) by Proposition 4. Also, for every 1≤ i ≤n, since suppyi∩s⊆ {α},wehavethat
|e∗α(yi)−x∗(yi)|=|
ξ∈s\{α}
e∗ξ(yi)|= 0.
Finally,sincee∗α= 1 andx∗= 1,weconcludethat
|e∗α(xi)−x∗(xi)| ≤ |e∗α(xi)−e∗α(yi)|+|e∗α(yi)−x∗(yi)|+|x∗(yi)−x∗(xi)|
≤ e∗α · xi−yi+x∗ · xi−yi< ε, andthisconcludestheproofthate∗α∈Ext(BXF∗)w
∗
.
Proposition6.LetF andGbecompactandhereditaryfamiliesonaninfinite cardinalκwiththeadditional propertythat [κ]1⊆ FM AX.If T :XF∗ →XG∗ isweak∗-weak∗ continuousandpreserves extremepoints,then forevery α∈κthereissα∈ G andasequence(θξα)ξ∈sα with |θαξ|= 1 suchthat T(e∗α)=
ξ∈sαθξαe∗ξ. Proof. Fixα∈κandsince{α}∈ FM AX,thenbyProposition4and Lemma5,
e∗α∈Ext(BX∗ F)w
∗⊆BX∗ F
w∗
=BX∗ F.
Since T is weak∗-weak∗-continuous andtakesextremepointsto extremepoints,wegetthat:
T(e∗α)∈T[Ext(BXF∗)w
∗
]⊆T[Ext(BXF∗)]w
∗ ⊆Ext(BXG∗)w
∗
.
By Lemma3forG we get thatthere issα ∈ G and asequence (θαξ)ξ∈sα with |θαξ|= 1 suchthatT(e∗α)=
ξ∈sαθαξe∗ξ.
Wecannowestablishthemainresultofthissection:
Theorem 7.LetF andG becompactandhereditaryfamilieson κwiththeadditionalproperty that
[κ]1⊆ FM AX∩ GM AX.
If T :XF →XG is anisometry,then thereexistsapermutationπ ofκand asequenceofsigns (θα)α such that T eα=θαeπ(α) forallα∈κ.
Proof. Given anisometry T :XF →XG, theadjointoperator T∗ :XG∗ → XF∗ isweak∗-weak∗ continuous and is also an isometry, so that T∗ takes the extreme points of BXG∗ to the extreme points of BXF∗. By Proposition 6, for eachα∈κ, there issα ∈ F and asequence (θαξ)ξ∈sα such that|θαξ|= 1 and T∗(e∗α)=
ξ∈sαθαξe∗ξ.Theconclusionnowfollows similarlyasTheorem 10of[5].
Letusnowcomparetheresultsofthissectiontothoseof[5].NoticethatbothinthepreviousTheorem7 andProposition6,wehaveaweakerassumptiononthefamiliesFandG:theysupposethemtoberegular familiesandwereplacethespreadingconditionbythestrictlyweakertopologicalconditionthattheFM AX and GM AX containthesingletons.Ontheother hand,we startwithaslightlystrongerassumptiononthe operator: in case of Proposition 6we ask T : XF∗ → XG∗ to be weak∗-weak∗ continuous and to preserve extreme points,whilein[5] they requireonlythelatter.Asaconsequence,westartinTheorem7from an isometry T :XF→XG, whiletheystartfrom anisometryT :XF∗ →XG∗.
Given two compactandhereditary familiesF and G suchthat[κ]1 ⊆ FM AX ∩ GM AX,we donotknow theanswertothefollowing question:
Question 8. Isevery isometry T :XF∗ →XG∗ weak∗-weak∗ continuous?Or, equivalently, isevery isometry T :XF∗ →XG∗ theadjointoperatorof anisometry S:XG →XF?
If the answer to these questions is positive,we could get afull version of Corollary 12 of [5] with the weakerassumptiononthefamilies.Nevertheless,givenaπ-homeomorphismbetweentwohereditaryfamilies F andGinducedbyapermutationπofκ,theuniquelinearcontinuousoperatortakingeαtoeπ(α)isclearly anisometry betweenXF and XG.Hence,wehavethefollowingcorollary fromourpreviousresults:
Corollary 9.LetF andG be compactandhereditary familieson κsuchthat [κ]1⊆ FM AX ∩ GM AX.Then TFAE:
(ii) F andG are π-homeomorphic.
(iii) XF andXG are isometric.
3. Uniquenessofregularfamiliesunder permutations
In theprevioussection weimproved theresultsof [5] bygetting weakerassumptions onfamilieswhich are sufficient to guarantee that the combinatorial spaces are isometric if and only if the families are π-
homeomorphic.Inthissectionwedealwith familiesonthecountableinfinitecardinalωand improvetheir resultsinanotherdirection.Weshowthattwoπ-homeomorphicregularfamiliesareactuallythesame.
Letusconsider thefollowing definitions:
Definition 10.Given s,t∈[ω]<ω, wesay thatt isa spread of sif there isa bijectivemappingσ: s−→t suchthatσ(i)≥iforeveryi∈s;wewillabbreviatethisfactbysayingthatσ[s] is aspreadofs.
Wecall afamilyF spreading if σ[t] ∈ F forevery t∈ F and everyspreadσ[t] of t; and wesay thata familyF isregular ifitiscompact,hereditaryandspreading.
Wefirstaskedourselveswhethertwo homeomorphicregularF and G arealwaysπ-homeomorphic. The followingsimpleexampleanswersthisquestionnegatively.
Example11.LetF = [ω]≤1∪[ω\{1}]2andG= [ω]≤2.Considerthemappingϕ:F −→ Ggivenbyϕ(∅)=∅, ϕ({1})={1,2},ϕ({n})={n−1}ifn>1,ϕ({2,m})={1,m}form>2,andϕ({n,m}<)={n−1,m−1} foreveryn>2.It isclearthatϕisahomeomorphism.Nevertheless,there isno permutationπ:ω −→ω such thatG ={π[s] : s ∈ F}. Indeed, ifsuch a permutation π exists,then π({1}) should be anisolated point in G with size 1, which is impossible. Hence, F and G are homeomorphic compact and hereditary familieswhicharenotπ-homeomorphic. Inparticular, XF and XG arenotisometric.
The main purpose of this section is to show Theorem 14, which states that hereditary and spreading families on ω are invariable under permutations. In particular, different regular families on ω cannot be permutedonetotheother.AsaconsequenceofTheorem10of[5],wegetthatthecombinatorialspacesof twodifferent regularfamiliesarenotisometric.
Westartwiththefollowinglemma,whichgivesalternativeusefuldefinitionsfor“tbeingaspreadofs”.
Lemma12. Given s,t∈[ω]<ω,thefollowingconditions are equivalent:
(i) thereisabijectivemapping σ:s−→t suchthatσ(i)≥i forevery i∈s;
(ii) thereisan increasingbijectivemapping σ:s−→t suchthat σ(i)≥iforevery i∈s;
(iii) there is a bijective mapping σ : s −→ t such that σ(i) ≥ i for every i ∈ s and σ(i) = i for every i∈s∩t.
Proof. (ii)and (iii)clearlyimply(i).
Toprovethat(i)implies(ii), letσ be abijectivemappingσ:s−→t suchthatσ(i)≥i foreveryi∈s and letσ :s−→ t be theonly increasingbijection from s ontot.Suppose bycontradiction that there is somei0∈ssuchthatσ(i0)< i0.Then
|{j∈t:j < i0}| ≥ |{σ(i) :i∈sandi≤i0}|>|{i∈s:i < i0}|
and,bythesurjectivity ofσ, thatthere mustbesomei∈s,i≥i0 suchthatσ(i)< i0 (hence,σ(i)< i),a contradiction.
Finally, to prove that (i)implies (iii), let σ be a bijective mappingσ : s −→ t such that σ(i) ≥i for everyi∈sand letσ:s−→t be suchthatσ(i)=i foreveryi∈s∩tand therestrictionofσ tos\tis theonlyincreasingbijectionfroms\tontot\s. Nowwefollowasimilarargument asabove:assumingby contradictionthatthereissomei0∈ssuchthatσ(i0)< i0,wegetthat
|{j∈t\s:j < i0}| ≥ |{σ(i) :i∈s\tand i≤i0}|>|{i∈s\t:i < i0}|
and,bythesurjectivityofσ,thattheremust besomei∈s\t,i≥i0suchthatσ(i)< i0 (hence,σ(i)< i), acontradiction.
Lemma 13.If F ⊆[ω]<ω isaspreadingfamily, thensoisF⊆.
Proof. LetF ⊆[ω]<ω beaspreadingfamily,s∈ F⊆ andσ[s] aspreadofs.Wemustshowthatσ[s]∈ F⊆. Lett∈ F besuchthats⊆tandlett\s={m1,m2,. . . ,mk}<.Then,considerintegersnk>· · ·> n2>
n1>max(t∪σ[s]) anddefineσ :t−→ω byσ(j)=σ(j) ifj∈s;andσ(mi)=ni for1≤i≤k.Itfollows that σ[t] isaspreadof t and thatσ[s] =σ[s]⊆σ[t]. Bybeing F spreading, weget thatσ[t]∈ F and, therefore, σ[s]∈ F⊆.
Wearenow readytoproveourmain result:
Theorem 14.If F andG arehereditary, spreadingandπ-homeomorphicfamilies1 on ω,thenF=G. Proof. Weprove,byinductiononn, thatforany twohereditary andspreadingfamiliesF andG,ifthere is apermutationπ:ω→ω suchthatG=π[F], thenF ∩[ω]≤n =G ∩[ω]≤n.
Forn= 1,let
I={i∈ω:{i}∈ F}/ andJ ={i∈ω:{i}∈ G}/ .
Noticethat,fromspreading,IandJ areinitialsegmentsofω.Moreover,i∈Iiffπ(i)∈J,sothat|I|=|J|. It followsthatI=J and,therefore, F ∩[ω]1={{i}:i∈/I}={{i}:i∈/J}=G ∩[ω]1.
Assumethatthestatementholdsfornandletusproveitforn+ 1.Givens∈ F ∩[ω]n,let Is={i∈ω:{i} ∪s /∈ F}.
Let usprovesomepropertiesofIs. Itis easyto see,from spreading,thatIsis aninitial segmentofω\s.
Moreover, wehavethefollowing:
Claim. If s∈ F ∩[ω]n andt∈[ω]n isaspreadof s(inparticular, t∈ F),then|It|≤ |Is|. Proof of the claim. Firstnotice that
It\s=It\(s∪t)⊆Is\(s∪t) =Is\t,
wherethetwoequalitiesfollowfromthefactthatIs∩s=It∩t=∅andtheinclusionfollowsfromspreading of F.
Second, let us show that |It∩s|≤ |Is∩t|. Indeed,by Lemma 12, consider σ : s→ t a bijection such that σ(i) ≥ i for every i ∈ s and, additionally, σ(i) =i for every i ∈ s∩t. We show that if j ∈ It∩s, then σ(j) ∈Is∩t. Indeed, fix j ∈It∩s and letσ :{σ(j)}∪s → {j}∪t be defined byσ(i)=σ(i) for i ∈s\ {j,σ(j)},σ(j)=j and σ(σ(j))=σ(j). Notice thatσ isinjective andwitnesses that{j}∪t isa spreadof{σ(j)}∪s.
Since j ∈It, then σ[{σ(j)}∪s] ={j}∪t ∈ F/ . Also,F isspreading, so that {σ(j)}∪s ∈ F/ , that is, σ(j)∈Is.Hence,σ(j)∈Is∩tand weconcludethat|It∩s|≤ |Is∩t|.Summingupwegetthat|It|≤ |Is| and thisconcludestheproof oftheclaim.
Now, foreachk∈ω,let
Fn,k ={s∈ F ∩[ω]n:|Is| ≤k}.
1 InthisresultwedonotactuallyneedtoassumethatF andGcontainallsingletons.Iftheydid,theproofforn= 1 inthe inductionwouldbesimpler,butweoptedtopresentaproofwhichmakessensebothunderthisassumptionorwithoutit.
Itfollows fromtheclaimthatFn,k is aspreadingfamily,foreachk∈ω.
Foreacht∈ G ∩[ω]n andeachk∈ω,define
Jt={i∈ω:{i} ∪t /∈ G}andGn,k ={t∈ G ∩[ω]n:|Jt| ≤k},
andgetsimilar propertiesas forIsand Fn,k.Inparticular,Gn,k isaspreadingfamilyforeveryk∈ω.
Sinceboth Fn,k andGn,k arespreadingandGn,k =π[Fn,k], thenbyLemma13, thedownwardclosures F⊆n,k and G⊆n,k are hereditary and spreading families, and G⊆n,k = π[F⊆n,k]. It follows from the inductive hypothesisthatGn,k =G⊆n,k∩[ω]n=F⊆n,k∩[ω]n=Fn,k,foreveryk∈ω.
Finally,
F ∩[ω]n+1 ={{i} ∪s:i /∈sand∃k∈ω, s∈ Fn,k\ Fn,k−1 andi /∈τk(ω\s)}
={{i} ∪s:i /∈sand∃k∈ω, s∈ Gn,k\ Gn,k−1 andi /∈τk(ω\s)}=G ∩[ω]n+1, whereτk :℘(ω)→[ω]k associatestoany setX itsinitialsegmentτk(X) ofsize k.
The following corollary follows immediately from [5, Corollary 12] and Theorem 14 above,our crucial contributionbeingtheequivalencebetween(i)and(ii).
Corollary15. If F andG are regularfamiliesonω,thenTFAE:
(i) F=G.
(ii) F and G areπ-homeomorphic (iii) XF andXG are isometric.
(iv) XF∗ andXG∗ are isometric.
4. Examplesandfurtherremarks
Thepurposeofthissectionis togivesomeexamplesandproveseveral factswhichputourmain results intoabiggerpicture.Inparticular,atopologicalapproachisuseful.
For mostof theresults inthis sectionwe will use theCantor-Bendixson rank.We recall thatifX is a topological space, then X = {x∈ X :xis not isolated inX}and given an ordinal α, the α−th Cantor- BendixsonderivativeofX isdefinedrecursivelyby:
X(0) =X X(β+1) = (X(β))
X(λ) =
β<λX(β), ifλ >0 is a limit ordinal.
The Cantor-Bendixson rankof X is the minimal ordinal such thatX(α) = X(α+1) and it is denoted by rk(X).
If F ⊆[κ]<ω is acompact family,then its Cantor-Bendixson index is thesmallestordinal αsuch that F(α)=∅anditisthereforeasuccessorordinal.Moreover,rk(F)< κ+.
4.1. Injective Cantor-Bendixsonindex
GivenacompactfamilyF ⊆[κ]<ω,wecandefineafunctionrkF :F →κ+ whichassociatestos∈ F the biggestordinal β for whichs∈ F(β).Notice thattherankof apointcoincides with therankof itsimage
underahomeomorphismbetweentwotopologicalspaces.So,inparticular,therankofasingletoncoincides with therankofitsimageunderaπ-homeomorphismbetweentwo families.
WhenacompactfamilyFonωisspreading,thecorrespondingfunctionrkF : [ω]1→ω1isnondecreasing.
Inparticular, fortheSchreierfamily
S :={s∈[ω]<ω:|s| ≤min(s) + 1} ∪ {∅}
(knowntoberegular)rkS({n})=nforeveryn∈ω,sothatrkS [ω]1isinjective.Thisimposesrestrictions onwhichpermutationsinduceahomeomorphismofagiven family:
Remark16.IfF isacompactfamilyonκsuchthatrkF [κ]1 isinjective,thentheonlypermutationπof κsuchthatπ[F]=F istheidentity map.
It follows fromthisremark andTheorem7thatif,moreover,[κ]1⊆ FM AX,then everyisometry ofXF is givenbyachangeofsignsoftheelementsofitsbasis.Thisgeneralizes[5,Theorem15].
Also,forfamilieswhichhavestrictly increasing rankfunction,wecaneasilygettheconclusionofTheo- rem 14:
Proposition 17.If F andG are π-homeomorphiccompactfamilieson κsuch thatrkF isstrictly increasing andrkG isnondecreasing, thentheonlypermutationof κinducinganhomeomorphism betweenthemisthe identity map. Inparticular, F=G.
Proof. LetF and G beas inthehypothesis, andletπbe apermutationof κsuchthatG=π[F]. Wewill prove thatπis increasingand sinceit isapermutation, ithas to be theidentity map.Assumetowardsa contradiction thatthere areα < β < κsuchthatπ(α)≥π(β).Then
rkF({α}) = rkG({π(α)})≥rkG({π(β)}) = rkF({β}), contradicting thefactthatrkF isstrictlyincreasing.
As an application, we get an example of distinct compact and hereditary families which are π- homeomorphic. ItguaranteesthatTheorem14cannot beimprovedbydropping thespreadingcondition:
Example18.Letπ=IdbeanypermutationofωandF=π[S].FromRemark16wegetthatF =S.Since F isπ-homeomorphictoS, F isalsocompactandhereditaryanditfollows fromCorollary9thatXS and XF areisometric.
4.2. Moreonspreadingfamilies
Inthecontext ofcompactspreadingfamiliesonω, thehypothesesofProposition17arealmost entirely fulfilled, theonlymissing onebeingthattherankofoneof thefamilieshastobe notonlynondecreasing, butstrictlyincreasing. ThenextexampleshowsthatProposition17doesnotnecessarilyholdwhen rkF is notstrictlyincreasing.Italsoshowsthattwofamiliesmaybeπ-homeomorphicwithonlyoneofthembeing spreading.
Example 19.Consider F = [ω]≤2\
{2,3} and G = [ω]≤2\
{1,2} . It is clearthat F iscompact, that rkF({n})= 1 forevery n∈ω, andthat G is spreading.The permutationπ: ω −→ω givenby π(1)= 3, π(2)= 1,π(3)= 2,and π(n)=nforeveryn>3 witnessesthatG=π[F].Nevertheless,F =G.
Itiswellknownthatforeverylimitordinalα < ω1thereisacompactfamilyFonωwithrankα+ 1 such that rkF is strictly increasing; for example, the Schreier families (see [3]) or the closure of an α-uniform front (introduced in [11]). We should also mention the existence of a non hereditary and non spreading compactfamilyF onω withstrictlyincreasing rank:letforexample
F ={s∈ S :{n, n+ 1}is not an initial segment ofs, for everyn∈ω};
then F is not hereditary nor spreading since {3,5,6} ∈ F but {5,6},{4,5,6} ∈ F/ . Thus, although we cannotapplyTheorem 14heretoconcludethatanyspreadingfamilyπ-homeomorphictoF isF itself,we caninsteadgetthatconclusionfrom Proposition17.
Sinceeachregularfamilydeterminesaclassofcompactandhereditaryfamiliesonωwhichhaveisometric combinatorialspaces,itwouldbeniceifeverycompactandhereditaryfamilyonωwouldbeπ-homeomorphic toaregularone,sothatregularfamilieswouldclassifyallisometric-typesofseparablecombinatorialspaces.
However,thisisnotthecase,as showsthefollowingexample.
Example20.ConsiderF= [N]≤2\
{n,n+1}:n∈N whichisclearlycompact,hereditaryandnotspread- ing.WeclaimthatFisnotπ-homeomorphictoanyregularfamily.Indeed,assumetowardsacontradiction thatthereisaregularfamilyGandapermutationofω suchthatG=π[F].Then,{π(n),π(n+ 1)}∈ G/ for everyn< ω.
We will now see thatπ(n) and π(n+ 1) is onethe successor of the other, for every n< ω. Indeed, if there are n,k < ω suchthat π(n)< k < π(n+ 1), let p< ω be such thatπ(p) =k. Since p= n,n+ 1, then {n,p}∈ F and {π(n),k}∈ G. Bybeing G spreading, we havethat{π(n),π(n+ 1)}∈ G, whichis a contradiction.Analogously,weseethatthere arenotn,k < ωsuchthatπ(n+ 1)< k < π(n).
Let m< ω be such thatπ(m)= 1. Then, π(m+ 1)= 2 as it isthe successor of π(m). It follows that π[{m,m+ 1,m+ 2,...}]=ω.Thus,πis theidentity map,contradicting thatF isnotspreading.
4.3. Examplesin theuncountablesetting
Our purpose now is to exploresome examplesof familieson agiven uncountablecardinal κwhichare compact, hereditary and suchthat [κ]1 ⊆ FM AX, as required for both families inTheorem 7. It is clear thatF= [κ]≤n issuchafamily,butweconsider thistobe somewhattrivial,asthenormlooksverymuch like thesupremumnorm andanypermutationofκfixesF.
On theother hand,if we assumeF onsome uncountablecardinalκnot to be contained inany [κ]≤n, then itcannotbe spreading:let(sn)n ⊆ F with|sn|≥nand fixα∈κsuch thatsn ⊆αforeveryn∈ω.
Spreading and hereditary would imply that the intervals [α,α+n] ∈ F. In turn, this would contradict compactness,sincethissequenceconvergesto [α,α+ω),whichisinfiniteanddoesnotbelongtoF. Example 21.Given acardinalκ, letS(κ) bethe familyformed bycopies of theSchreier familyS ineach intervaloftheform [λ,λ+ω),where λ< κisalimitordinal(orzero).Formally,
S(κ) ={s∈[κ]<ω:s⊆[λ, λ+ω) for some limit ordinalλ < κ and|s| ≤min{n∈ω:λ+n∈s}+ 1}. Notice that S(κ) is a compact hereditary family such that {α} ∈ S(κ)M AX for each α ∈ κ, so that it satisfies the hypothesis of Theorem 7. Moreover, S(κ) contains arbitrarily large (finite) elements. The corresponding combinatorialspaceXS(κ)isisometricto an∞-sumofκmanycopiesofXS.
Togetanexamplewhichisabitmoreinvolvedweusetrees.Atreeisapartially orderedset(T,≤) such thatforeveryf ∈T,theset{g∈T :g≤f}iswell-ordered.Hence,wecandefinetheheighthtT(f) off in T astheonlyordinalwhichisorder-isomorphicto{g∈T :g≤f}.Theheightht(T) ofT isthesupremum
of allhtT(f),f ∈T.AchaininT isachainwithrespect to≤,thatis,asubsetC⊆T suchthatforevery f,g∈C,eitherf ≤gor g≤f. AbranchinT isamaximal chaininT.
Finally,wehavethefollowingproposition,whichisimplicitin[6, Lemma2.32]:
Proposition 22.Let κ be an infinite cardinal and let T be a tree of height κ. If there is a compact and hereditary familyF onκsuchthat [κ]1⊆ FM AX,then thereissuchafamily onλ=|T|.
Proof. FirstnoticethatthecardinalκcouldbereplacedbyanyindexsetI inDefinition1.Moreover,itis easy to seethatifπ:κ→I isabijection, thenthe naturallyinducedπ-homeomorphismbetween2κ and 2I takes anycompact hereditaryfamilyF onκsuchthat[κ]1⊆ FM AX ontoacompacthereditaryfamily π[F] onI suchthat[I]1⊆π[F]M AX, andvice-versa.Therefore,theexistenceofafamily,withthese three properties,onsomeindexset Iofcardinalityκisequivalentto theexistenceofsuchafamilyonκitself.
Hence, withoutloss ofgenerality, we mayassume thatF is acompact and hereditaryfamily onκ+ 1 (insteadofκ)suchthat[κ+ 1]1⊆ FM AX.Wedefine
G={C⊆T :C is a chain and{htT(f) :f ∈C} ∈ F}.
It iseasy to see thatG is acompact and hereditary familyon T. Given f ∈T, we fix abranch C which contains f and, given (sn)n ⊆ FM AX convergingto {htT(f)},it iseasy tobuild (tn)n convergingto {f}, where each tn ⊆C and {htT(g):g ∈tn}=sn.Clearly eachtn ∈ GM AX, whichconcludesthe proofthat [T]1⊆ GM AX.Fromthefirstparagraphoftheproof,itfollowsthatthedesiredfamilyonλ=|T|exists.
Example 21 can be seen as a particular case of a family given by Proposition 22, taking T to be κ- many incomparablecopies of ω. Also, if we havea family on κ, we get afamily on 2κ, by taking T the complete binary tree of height κ. This way we get more interesting families for everycardinal belowthe firstinaccessible cardinal.
MotivatedbyRemark16andProposition17,weaskedourselvesifthere isacompacthereditaryfamily F onω1 suchthatrk({α})=αforeveryα < ω1.Thefollowingexampleanswersthis questionpositively:
Example 23.Given α < ω1, consider Fα acompactand hereditary familyonω with rk(Fα)=α+ 1 (the existenceofthese familiesis guaranteed,forexample,in[3]).
Given alimit ordinalλ< ω1,and s∈[ω1]<ω such thats⊆[λ,λ+ω),we will denoteby s−λtheset {γ−λ:γ ∈s}which belongsto [ω]<ω. Inasimilar wayas we havedefinedS(κ) inExample 21,we now define thefamiliesFα(κ),forκatailofω1 andα < ω1,by
Fα(κ) ={s∈[κ]<ω : there isλ < κ a limit ordinal or zero such thats⊆[λ, λ+ω) ands−λ∈ Fα}. Notice thatFα(κ) isacompactandhereditary familyonκwithrk(Fα(κ))=α+ 1.Consider
F:=S ∪
ω≤α<ω1
{α} ∪s:s∈ Fα(ω1\α), andα < γ for everyγ∈s .
It is clear that F is a compact and hereditary family on ω1. Moreover, notice that rkF({α}) = rkFα(ω1\α)(∅)= rk(Fα(ω1\α))−1=α,foreveryω≤α < ω1.
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