WEAKLY INFINITE DIMENSIONAL SUBSETS OF RN LILJANABABINKOSTOVAANDMARIONSCHEEPERS 9 0 0 2 n Abstract. TheContinuumHypothesisimpliesanErdo¨s-Sierpin´skilikedual- a itybetweentheidealoffirstcategorysubsetsofRN,andtheidealofcountable J dimensional subsets of RN. The algebraic sum of a Hurewicz subset - a di- 2 mensiontheoretic analogueofSierpinskisetsandLusinsets -ofRN withany 2 compactly countabledimensionalsubsetofRN hasfirstcategory. ] N 1. Introduction. G The main purpose of this paper is threefold: We point out strong analogies . betweenthe well-knownclassofcountabledimensionalsubsets ofthe Hilbertcube, h and the classes of Lebesgue measure zero subsets of the real line and the class of t a first category subsets of complete separable metric spaces. A classical theorem of m Erdo¨s and Sierpin´ski gives, under the Continuum Hypothesis, some explanation of [ thisanalogy. WeshowalsothatthedimensiontheoreticanalogueofLusinsetsand 2 SierpinskisetshassomeofthedeeperpropertiessharedbyLusinsetsandSierpin´ski v sets. And we give some information about a class of weakly infinite dimensional 1 spaces which is emerging as an important class in dimension theory. In particular 6 weshow,usingtheContinuumHypothesis,thatthemostrestrictiveoftheseclasses 6 contains metric spaces which are not countable dimensional. 3 . RN denotes the Tychonoff product of countably many copies of the real line, R. 1 The symbol [0, 1] denotes the unit interval and the subspace [0,1]N of RN denotes 1 the Hilbert cube. The symbol P denotes the set of irrationalnumbers, viewed as a 8 0 subspace of the real line. : v 2. The countable dimensional subsets of RN i X Hurewicz [9] defined a subset of a topologicalspace to be countable dimensional r a ifitisaunionofcountablymanyzero-dimensionalsubsetsofthespace,andproved that the Hilbert cube, thus also RN is not countable dimensional. Thus, the collection CD of countable dimensional subsets of RN is a σ-ideal. Since each metrizable space of cardinality less than 2ℵ0 is zerodimensional, the cofinality of CD, denoted cof(CD), is 2ℵ0. Smirnov and independently Nagami also showed that every separable metric space is a union of ℵ zero-dimensionalsubsets, and thus the covering 1 number of CD, denoted cov(CD), is ℵ . This is unlike the σ-ideal M of first cate- 1 gorysubsets ofR, andthe σ-idealN ofLebesgue measurezerosubsetsofR, where the cofinality and the covering number are not decided by ZFC. If one assumes the Continuum Hypothesis (CH), these differences vanish and one can give an explanation for the unusually strong analogies between the theory of the countable dimensional subspaces of RN and the theory of the first category subsets of RN: These two collections are dual in the sense of Erdo¨s and Sierpin´ski. Chapter 19 of [17] gives a nice exposition of the following classical theorem: 1 2 LILJANABABINKOSTOVAANDMARIONSCHEEPERS Theorem 1 (Erdös-Sierpin´ski Duality Theorem). Let X be a set of cardinality ℵ . Let I and J be σ-complete ideals on X such that: 1 (1) X =A B where A and B a disjoint sets with A∈I and B ∈J. S (2) cof(I)=ℵ =cof(J). 1 (3) For each I ∈I there is an S ⊂X\I such that |S|=ℵ and S ∈I. 1 (4) For each J ∈J there is an S ⊂X \J such that |S|=ℵ and S ∈J. 1 Then there is a bijective function f :X −→ X which is its own inverse, such that for each set E ⊆X we have E ∈I ⇔f[E]∈J. A classicaltheoremof Tumarkin[28] is animportanttoolinprovingthe duality between countable dimensional sets and first category sets of RN under CH: Theorem 2 (Tumarkin). In a separable metric space each n-dimensional set is contained in an n-dimensional G -set. δ Corollary 3. Assume the Continuum Hypothesis. There is a bijection f :RN −→ RN such that f =f−1 and for each set E ⊂RN, E is countable dimensional if, and only if, f[E] is first category. Proof: InTheorem1takeI tobeCD,theidealofcountabledimensionalsubsets of RN, and J to be M, the ideal of first category subsets of RN. By Hurewicz’s theoremthatRN is notcountabledimensional,CD isaσ-complete idealonRN. By the Baire category theorem M is a σ-complete ideal on RN. SinceRNisseparable,letDbeacountabledenseset. ThenDiszero-dimensional. By Theorem 2 there is a zero-dimensional G set A⊃D. But then B = RN\A is δ a first category set, and so (1) of Theorem 1 is satisfied. Theorem2alsoimpliesthateachcountabledimensionalsubsetofRNiscontained in a countable union of zero-dimensionalG sets, and so CD has a cofinalfamily of δ cardinality 2ℵ0. Since M has a cofinal family of cardinality 2ℵ0, C.H. implies that (2) of Theorem 1 holds. Evidently (3) and (4) also hold. (cid:3) 3. Hurewicz sets. Some strong analogies between the theory of the ideal of countable dimensional subsets of RN and the theories of the ideal N of Lebesgue measure zero subsets of the real line, and the ideal M of first category subsets of uncountable complete separable metric spaces emerge when we consider the following features: Call a subset of RN a Hurewicz set if it is uncountable, but its intersection with every zero-dimensional subset of RN is countable. Hurewicz sets are first category: Take a countable dense set D ⊂ RN. Since D is zerodimensional it is, by Theorem 2, contained in a zerodimensional dense G set E. Since a Hurewicz set meets E in δ only countably many points the Hurewicz set is first category. We shall see below that Hurewicz sets are first category in a strong sense. The notion of a Hurewicz set is analogousto the notions of a Lusin set and of a Sierpinski set: A subset of R is a Lusin set if it is uncountable but its intersection with any first category subset of R is countable. A subset of the real line is a Sierpinski set if it is uncountable but its intersection with each Lebesgue measure zero set is countable. Mahlo[15]andLusin[14]independentlyshowedthattheContinuumHypothesis implies the existence of a Lusin set of cardinality 2ℵ0, and Sierpinski [26] showed WEAKLY INFINITE DIMENSIONAL SUBSETS OF RN 3 that the Continuum Hypothesis implies the existence of a Sierpinski set of cardinality 2ℵ0. It is known that the existence of a Lusin set of cardinality 2ℵ0 does not imply the Continuum Hypothesis, and that the existence of a Sierpin´ski set of cardinality 2ℵ0 does not imply the Continuum Hypothesis. But Rothberger [25] provedthat the simultaneousexistence ofboth a Lusinsetof cardinality2ℵ0 anda Sierpinskisetofcardinality2ℵ0 isequivalentto2ℵ0 =ℵ1. W.Hurewicz[10]showed that the existence of a Hurewicz set is equivalent to the Continuum Hypothesis. SomedeeperpropertiesoftheLusinsetsandthe Sierpin´skisetsaretiedupwith the algebraic properties of the real line viewed as a topological group. Theorem 4 (Galvin-Mycielski-Solovay). If L ⊂ R is a Lusin set, then for each first category set M ⊂R, L+M 6=R. Galvin,MycielskiandSolovay[6]provedasignificantlystrongertheorem: Strong measure zero sets are characterized as sets whose translates by first category sets donotcoverthe realline. Theorem4followsfromthis sinceSierpin´skiprovedthat Lusin sets are strong measure zero. Pawlikowski [19] proved the counterpart for Sierpin´ski sets: Theorem 5 (Pawlikowski). If S ⊂ R is a Sierpin´ski set, then for each Lebesgue measure zero set N ⊂R, S+N 6=R. WeexpectthatasimilarstatementistrueaboutHurewiczsetsinthetopological group (RN,+): Conjecture 1. If H ⊂RN is a Hurewicz set, then for each countable dimensional set C ⊂RN, H +C 6=RN. We have a partial result on this conjecture: Call a subset of RN strongly countable dimensional if it is a union of countably many closed, finite dimensional subsets. This notion was introduced by Nagata [16] and Smirnov [27]. Every strongly countable dimensional set is countable dimensional, but not conversely. Thus, the σ-ideal SCD generated by the strongly countable dimensional subsets of RN is a proper subideal of CD. Indeed, every strongly countable dimensional set is of first category in RN, and thus SCD is also a subideal of the σ-ideal of first category subsets of RN. Call a subset of RN compactly countable dimensional if it is a union of countably many compact finite dimensional sets. The σ-deal KCD generated by the compact finite dimensional sets is a proper subideal of SCD. We shall show Theorem 6. If H is a Hurewicz subset of RN, then for every compactly countable dimensional set C ⊂RN, H +C is a first category subset of RN. Sincetheunionofcountablymanyfirstcategorysetsisfirstcategory,Theorem6 followsdirectlyfromLemma8below. ThisLemmausesanotherimportantclassical result: Theorem 7 (Hurewicz, Tumarkin). In a separable metric space the union of countably many closed, n-dimensional sets, is an n-dimensional set. Lemma 8. If H ⊆RN is a Hurewicz set and C ⊂RN is a compact n-dimensional set, then H +C is a first category subset of RN. Proof: Let D ⊂ RN be a countable dense set. Then D − C is a union of countably many closed n-dimensional sets, so by Theorem 7 it is n-dimensional. 4 LILJANABABINKOSTOVAANDMARIONSCHEEPERS By Theorem 2 choose open sets U ⊃ U ⊃ ··· ⊃ U ⊃ ··· ⊃ D −C such that 1 2 k ∞ U is n-dimensional. Then put Y = {x ∈ RN : x−C ⊆ ∞ U }. Then Y Tk=1 k Tk=1 k is dense in RN. Note that Y is a G -set: Consider a k, and an x ∈ Y. Since C is δ compact there is an open neighborhood V (x) of x such that V (x)−C ⊂U . But k k k then V = ∪ V (x) is an open set containing Y, and V −C ⊂ U . It follows k x∈Y k k k that Y =∩ V . k<∞ k Next, put ∞ X =H ∩(\ Uk). k=1 Since H is a Hurewicz set and ∞ U is n-dimensional, X is a countable set. Tk=1 k Define Z = Y \ (X + C). Since X + C is a union of countably many closed, n-dimensional sets, it is an n-dimensional F -set and thus first-category in RN. σ Consequently Z is co-meager. But Z ={x∈Y :(x−C)∩H =∅}⊆Y \(H +C). Thus, H +C is meager. (cid:3) One can show that if there are Hurewicz sets, then there are ones satisfying the property in Conjecture 1. There are several well-studied classes of weakly infinite dimensional subsets of RN which are not countable dimensional. In Example 1 we give, under CH, an example of a set S in a very restrictive class of weakly infinite dimensional spaces (but not countable dimensional), and a Hurewicz set H, such that S+H =RN. The following is an unresolvedweaker instance of Conjecture 1: Problem 1. Is it true that if C is a strongly countable dimensional set in RN and H is a Hurewicz set, then H +C 6=RN? 4. New classes of weakly infinite dimensional spaces. Manyexamplesininfinitarydimensiontheorybelongtoclassesofweaklyinfinite dimensionalspacesintroducedin[2]. OurexamplesinconnectionwithConjecture1 are also in these classes. These classes are defined by applying selection principles to specific types of open covers. We now introduce these covers and selection principles. Classes of open covers For a given space X the symbol O denotes the collection of all open covers of X. The following special types of open covers are relevant to our discussion: • Ofd: This is the collection of all open covers U such that there is for each finite dimensional F ⊂X a U ∈U with F ⊆U, but X is not a member of U. (Thus, we are assuming X is not finite dimensional.) • Ocfd: This is the collection of all open covers U such that there is for each closed finite dimensional F ⊂ X a U ∈ U with F ⊆ U, but X is not a member of U. (Thus, we are assuming X is not finite dimensional.) • Okfd: This is the collection of all open covers U such that there is for each compact finite dimensional F ⊂ X a U ∈ U with F ⊆ U, but X is not a member of U. (Thus, we are assuming X is not compact and finite dimensional.) • Ok: This is the collection of all open covers U such that there is for each compact F ⊂X aU ∈U withF ⊆U,but X is nota memberofU. (Thus, we are assuming X is not compact.) WEAKLY INFINITE DIMENSIONAL SUBSETS OF RN 5 • Ω: This is the collection of ω covers of X. An open cover U of X is an ω-cover if X is not a member of U, but for each finite subset F of X there is a U ∈U with F ⊆U. • Γ: Thisisthecollectionofγ coversofX. AnopencoverU ofX isaγ-cover if it is infinite and every infinite subset of U is a cover of X. • Ogp: This is the collection of groupable open covers: An open cover U of a space X is groupable if there is a partition U = F into finite sets Sn∈N n F that are disjoint from each other, such that for each x∈X there is an n N with x∈ F for all n≥N. S n We have the following inclusion relations among these classes of open covers: Ofd ⊂Ocfd ⊂Okfd ⊂Ω⊂O; Ok ⊂Okfd; Γ⊂Ω. It is also worth noting that if X is a separable metric space then each U ∈Okfd has a countable subset V ∈Okfd. The same is true about Ok. Three selection principles. Let A and B be families of sets. The selection principle, S (A,B) states: 1 Foreachsequence(A :n∈N)ofelementsofAthereisasequence n (B : n ∈ N) such that for all n we have B ∈ A , and {B : n ∈ n n n n N}∈B. The selection principle S (A,B) states: fin Thereisforeachsequence(A :n∈N)ofmembersofAasequence n (B : n ∈ N) of finite sets such that for each n we have B ⊂ A n n n and B ∈B. Sn∈N n The selection principle S (A,B) states: c Foreachsequence(A :n∈N)ofelementsofA,thereisasequence n (B :n∈N) whereeachB is a refinementofthe collectionofsets n n A ,eachB isapairwisedisjointfamily,and B isamember n n Sn∈N n of B. InourcontextthefamiliesAandB willbeclassesofopencoversofatopological space. The selection principle S (A,B) has the following monotonicity properties: 1 If A⊂C and B ⊂D, then we have the following implications: S (C,B)⇒S (A,B); S (A,B)⇒S (A,D). 1 1 1 1 Replacing S with S or S in these implications give corresponding facts for the 1 fin c other two selection principles. Specialinstancesofthesethreeselectionprinciplesappearinclassicalliterature. For example: S (O,O) is known as Rothberger’s property and was introduced in 1 the1938paper[24]. S (O,O)isknownasMenger’spropertyandwasintroduced fin in the 1925paper [7]. S (O,O) is known as property C, and was introduced in the c 1978paper[1]. S (O,O)isaselectiveversionofBing’spropertyofscreenability[3], c thus also known as selective screenability. Selective screenability is an important propertyininfinitarydimensiontheory. LetO denotethecollectionofopencovers 2 U with |U| ≤ 2. Then S (O ,O) is equivalent to Alexandroff’s notion of weakly c 2 infinite dimensional. A space whichis notweakly infinite dimensionalis saidto be strongly infinite dimensional. The class S1(Okfd,O) 6 LILJANABABINKOSTOVAANDMARIONSCHEEPERS MetrizablespaceswithpropertyS1(Okfd,O)seemimportant. First: Theorem17 of [2] shows that for separable metric spaces S1(Ofd,O)⇒Sc(O,O). Monotonicity propertiesofthe selectionprinciplesgive: S1(Okfd,O)⇒S1(Ocfd,O)⇒S1(Ofd,O). Thus for metrizable spaces S1(Okfd,O) implies the important property Sc(O,O). Second: The following lemma shows that these spaces have the classical Menger property S (O,O): fin Lemma 9. S1(Okfd,O)⇒S1(Ok,O)⇒Sfin(O,O). Third: Severalcentralexamplesfromthetheoryofinfinitedimensionalspacesare inthis class. Forexample: R.Pol’sexample in[22]whichshowsthata weaklyinfi- nitedimensionalspaceneednotbecountabledimensionalisintheclassS1(Okfd,O). The preservation of infinitary dimension properties by products is not well un- derstood yet. Whenever a new class of infinite dimensionality is defined it is of interest to know how this class behaves under various mathematical constructions, like products. We now make some remarks about products by spaces in the class S1(Okfd,O), using recent examples constructed by E. Pol and R. Pol in [21]: Proposition 10. Assume Martin’s Axiom. (1) Finite powers of sets in S1(Okfd,O) need not be in Sc(O,O). (2) The product of a space in S1(Okfd,O) with the space of irrational numbers need not be weakly infinite dimensional. To see this consider the following: In the proof of Proposition 4.1 of [21] E. Pol and R. Pol construct, using Martin’s Axiom, spaces E and E which have 0 1 S1(Okfd,O). To see that these two spaces have this property we consider elements oftheargumentgiveninpart(6)oftheproofof[21],Proposition4.1: Fori∈{0,1} the spaces E are of the form E∪S where i i • S is the union of countably many compact, finite dimensional subspaces, i and • For each open set W ⊃Si in Ei, |Ei\W|<2ℵ0. Consider Ei. Let (Un : n < ∞) be a sequence in Okfd. Write Si as a union of compactfinite dimensionalsubspacesSi ⊂Si ⊂···⊂Si ⊂···. Foreachn choose 1 2 n a U ∈ U with Si ⊆ U . Then W = U is an open set 2·n−1 2·n−1 n 2·n−1 Sn<∞ 2·n−1 containing Si, and so Ei \W has cardinality less than 2ℵ0. But Martin’s Axiom implies that a separable metric space of cardinality less than 2ℵ0 has Rothberger’s property S (O,O) (Theorem 5 of [13]). Thus, choose for eachn a U ∈U such 1 2·n 2·n that E \W ⊆ U . i Sn<∞ 2·n ItfollowsthatthespaceX constructedthereasthefreeunionofE andE still 0 1 has S1(Okfd,O). Recall that a metric space (X,d) has the Haver property if there is for eachsequence (ǫ :n<∞) ofpositive realnumbers a sequence (U :n<∞) n n such that each V is a pairwise disjoint family of open sets, each of d-diameter less n than ǫ , and U is a cover of X. Metrizable spaces with property S (O,O) n Sn<∞ n c havetheHaverpropertyinallequivalentmetrics. Since[21]showsthatX2 hasthe Menger property but under some metric X2 does not have the Haver property, it follows that X2 does not have property S (O,O). c Toseetheseconditem: InTheorem6.1of[21]itispointedoutthatthe product of the space E with the subspace of irrationalnumbers of the real line is not even i S (O ,O), that is, is strongly infinite dimensional. c 2 The class S1(Okfd,Ω) WEAKLY INFINITE DIMENSIONAL SUBSETS OF RN 7 S1(Ok,Ω) was considered in Section 2 of [18]. Theorem 2.1 of [18] implies: Lemma 11 (Pansera, Pavlovic). A space has S1(Ok,Ω) if, and only if, it has S1(Ok,O) in all finite powers. Proposition 12. IfaspacehaspropertyS1(Okfd,Ω)thenithaspropertySfin(Ω,Ω). Proof: Let X be a space with S1(Okfd,Ω). By monotonicity properties of S1(·,·) X then has S1(Ok,Ω). Then by Lemma 11 Xn has S1(Ok,O) for all finite n. By Corollary 9 Xn has S (O,O) for all n. By Theorem 3.9 of [11] this is equivalent fin to: X has S (Ω,Ω). (cid:3) fin Usingstandardtechniquesonecanprove: IfaspacehasthepropertyS1(Okfd,Ω), then it has the property S1(Okfd,O) in all finite powers. We don’t know if the converse is true: Problem 2. For metrizable space X is it true that if each Xn has property S1(Okfd,O) then X has S1(Okfd,Ω)? It islikely thatthe spacesE0 andE1 of[21] mentionedaboveareinS1(Okfd,Ω), but we have not examined this. The space X which is the free union of E and E 0 1 isanexampleofaspaceinS1(Okfd,O)butnotinS1(Okfd,Ω). Thus,underMartin’s Axiom, S1(Okfd,Ω) is a proper subclass of S1(Okfd,O). Below we construct under CH an example (Example 1) of a space which has the property S1(Okfd,Ω) and which is not countable dimensional. The class S1(Okfd,Γ) S1(Ok,Γ)wasexplicitlyconsideredin[4],Section3,andin[5],Section3,wherethe second implication of Corollary 13 is noted. Corollary 13. S1(Okfd,Γ)⇒S1(Ok,Γ)⇒Sfin(Ω,Ogp). Theorem 12 of [4] shows that each σ-compact space has the property S1(Ok,Γ). ThisisnottrueforS1(Okfd,Γ): TheHilbertcubeisnotweaklyinfinitedimensional, but is a σ-compact space. Combining [5] Theorem 10 and [5] Proposition 13 gives the following result: Lemma 14 (Di Maio, Kocinac, Meccarriello). If X has S1(Ok,Γ), then for all n also Xn has this property. The Hurewicz covering property is a strengthening of the Menger property S (O,O)andisdefinedasfollows: TopologicalspaceX hastheHurewiczcovering fin property if: Foreachsequence(U :n<∞)ofopencoversofX thereisasequence n (V :n <∞) of finite sets such that for each n, V ⊆ U , and for each x∈ X, for n n n all but finitely many n, x∈ V . It was shown in [12] that the Hurewicz covering S n proprty is equivalent to the selection principle S (Ω,Ogp). fin Corollary 15. If a space X has property S1(Okfd,Γ) then it has Sfin(Ω,Ogp) in all finite powers. Using the technique in the proof of [2] Theorem 17 and using the techniques in [2] Lemma 6 one obtains the following two results: Theorem 16. S1(Ofd,Ogp)⇒Sc(O,Ogp). Lemma 17. Ifaspace X has S1(Okfd,Γ), thenit has Sc(O,O) inall finitepowers. 8 LILJANABABINKOSTOVAANDMARIONSCHEEPERS Proof: Let X be a space with property S1(Okfd,Γ). By Lemma 13 it has property S1(Ok,Γ), and by Lemma 14 it has S1(Ok,Γ) in all finite powers, and thus the Hurewiczpropertyinallfinitepowers. ThenbyCorollary13of[2],allfinitepowers of X has the property S (O,O). (cid:3) c Let X be an infinite dimensional space which satisfies S1(Okfd,Γ). By [2] and Corollary13,allfinitepowersofX havetheHaverproperty,andindeed,theproduct of X with any space having the Haver property again has the Haver property. It followsthat X×P has the Haverproperty. We don’t know if this product must be selectively screenable: Problem 3. If X has S1(Okfd,Γ) then does X×P have Sc(O,O)? 5. Examples. We now describe two infinite dimensional examples that will be used to demon- strate a number of facts in connection with the preceding sections. In both cases we are interested in showing properties of Hurewicz sets, and thus we are confined to assuming the Continuum Hypothesis. A ZFC+CH example of S1(Okfd,Ω). Write: R := {x ∈ RN : (∀j > n)(x(n) = 0)} and R := ∞ R . Standard n ∞ Sn=1 n arguments prove Lemma 18 and Corollary 19 below. Lemma 18. Assume G⊃R is an open subset of RN and I , ··· , I are closed ∞ 1 n intervalsofpositivelength. Foreachf ∈RN thereareanm>nandclosedintervals I , ··· , I of positive length such that n+1 m I × ··· ×I × RN ⊆G−f. 1 m Corollary 19. If (G :n<∞) is a sequenceof G subset of RN suchthat each G n δ n containsR ,andif(f :n<∞)issequenceofelementsofRN,then ∞ (G −f ) ∞ n Tn=1 n n contains a Hilbert cube. Example 1 (CH). There are subsets X and Y of RN such that: (1) X and Y satisfy S1(Okfd,Ω); (2) X \R and Y \R are Hurewicz sets; ∞ ∞ (3) X and Y have S1(Okfd,Ω) and X ×Y not. (4) X ∪Y is S1(Okfd,O), and not S1(Okfd,Ω). Fix the following enumerations: • (fα :α<2ℵ0), the list of all elements of RN • (Fα :α<2ℵ0), the list of all finite dimensional Gδ subsets of RN • (Lα :α<2ℵ0), the list of all Gδ subsets of RN containing R∞ • ((Unα :n<ω):α<2ℵ0) where for each α each element of R∞ is in Unα for all but finitely many n, and each Uα is open in RN. n For each α<2ℵ0 the set Tα =Tn<∞(Sm≥nUmα) is a Gδ set containing R∞. Since G = T ∩L is a dense G set which contains R . Thus G ∩(G −f ) 0 0 0 δ ∞ 0 0 0 contains, by Corollay 19, a homeomorphic copy of the Hilbert cube. Choose x ∈(G ∩(G −f ))\((R ∪F )∪((R ∪F )−f )) 0 0 0 0 ∞ 0 ∞ 0 0 WEAKLY INFINITE DIMENSIONAL SUBSETS OF RN 9 and fix y ∈ G with x = y − f . Then define S1(0) = {n : x ∈ U0} and 0 0 0 0 0 0 0 n S2(0)={n:y ∈U0}. 0 0 n Assume we have 0 < α < ω , and that we have selected for each γ < α an x 1 γ and a y and defined sets S1(γ) and S2(γ), ν ≤γ such that: γ ν ν (1) δ <γ <α⇒S1(δ)=S2(δ)=ω; γ γ (2) γ ≤ν <δ <α⇒ (a) S1(δ)⊆∗ S1(δ); S2(δ)⊆∗ S2(δ) γ ν γ ν (b) S1(δ), S1(ν)⊆S1(0); S2(δ), S2(ν)⊆S2(0) are all infinite γ γ γ γ γ γ (3) γ ≤δ <α⇒{n:x ∈Uγ}⊇S1(δ); {n:y ∈Uγ}⊇S2(δ) δ n γ δ n γ (4) γ ≤ δ < α ⇒ {x , y } ⊆ G \(R (∪ F ) {x : ν < δ} {y : ν < δ δ γ ∞S ν≤δ ν S ν S ν δ}); (5) δ <α⇒x =y −f . δ δ δ Towards selecting x and y we consider two cases: α α Case 1: α=β+1 For δ <α define S1(δ)=ω =S2(δ). For γ ≤β we define α α ∞ Tγ1 = \( [ Umγ), n=1 m≥n,m∈S1(β) γ ∞ Tγ2 = \( [ Umγ), n=1 m≥n,m∈S2(β) γ and for α we define ∞ Tα1 =Tα2 = \( [ Umα). n=1 m≥n and Gα = \(Lγ ∩Tγ1∩Tγ2). γ≤α Then G is a G set containing R and so by Corollary19 the set G ∩(G −f ) α δ ∞ α α α contains a Hilbert cube. But the set Bα =(R∞[(∪γ≤αFγ)[{xγ :γ <α}[{yγ :γ <α})−({fγ :γ ≤α}∪{0}) is countable dimensional and thus does not contain a Hilbert cube. Choose x ∈ α (G (G −f ))\B and fix y ∈G with x =y −f . αT α α α α α α α α Then (4) holds at α. Since for γ < α we have x ∈ T1 and y ∈ T2 the sets α γ α γ S1(α) = {n ∈ S1(β) : x ∈ Uγ} and S2(α) = {n ∈ S2(β) : y ∈ Uγ} are infinite. γ γ α n γ γ α n Similarly the sets S1(α)={n:x ∈Uα} and S2(α)={n:y ∈Uα} are infinite. α α n α α n Case 2: α is a limit ordinal. For γ < α choose infinite sets F1 and F2 so that for all β < α, F1 ⊆∗ S1(β) and γ γ γ γ F2 ⊆∗ S2(β). γ γ For γ ≤β we define ∞ Tγ1 = \( [ Umγ), n=1 m≥n,m∈F1 γ ∞ Tγ2 = \( [ Umγ), n=1 m≥n,m∈F2 γ 10 LILJANABABINKOSTOVAANDMARIONSCHEEPERS and for α we define ∞ Tα1 =Tα2 = \( [ Umα). n=1 m≥n and Gα = \(Lγ ∩Tγ1∩Tγ2). γ≤α SinceG isaG setcontainingR ,Corollary19impliesthatthesetG ∩(G −f ) α δ ∞ α α α contains a Hilbert cube. The set Bα =(R∞[(∪γ≤αFγ)[{xγ :γ <α}[{yγ :γ <α})−({fγ :γ ≤α}∪{0}) is countable dimensional and does not contain a Hilbert cube. Choose x ∈(G (G −f ))\B and fix y ∈G with x =y −f . α αT α α α α α α α α Then (4) holds at α. Since for γ < α we have x ∈ T1 and y ∈ T2 the sets α γ α γ S1(α) = {n ∈ F1 : x ∈ Uγ} and S2(α) = {n ∈ F2 : y ∈ Uγ} are infinite. γ γ α n γ γ α n Similarly the sets S1(α)={n:x ∈Uα} and S2(α)={n:y ∈Uα} are infinite. α α n α α n Thiscompletesthe descriptionofthe processforchoosingx andy forα<ω . α α 1 Finally define X =R {x :α<ω } and Y =R {y :α<ω }. ∞S α 1 ∞S α 1 Claim 1: If U is a collection of open subsets of RN which contains an infinite subset V, each infinite subset of which covers R , then there are countable sets A ∞ X and A such that V is an ω cover of X\A and of Y \A . Y X Y Proof of Claim 1: For let V be such a subfamily of U. We may assume that V is countable. Thus, we may assume that for some α, fixed from now on, V is (Uα : n n<ω), asenumeratedbefore the constructionfX andY. PutA ={x :γ ≤α} X γ and A ={y :γ ≤α}. Y γ Consider any finite subset F ⊂ X \A . We may write F = {x ,··· ,x } X γ1 γn where we have α < γ < ··· < γ . From the construction of X and Y we have 1 n S1(γ )⊆∗ ···⊆∗ S1(γ ). Choose n large enough that α n α 1 S1(γ )\n⊆···⊆S1(γ ). α m α 1 Then for k ∈S1(γ ) we have F ⊂Uα. A similar argument applies to Y \A . α m m Y Claim 2: X and Y both satisfy the selection principle S1(Okfd,Ω). ProofofClaim2: WeproveitforX. TheproofforY issimilar. Let(U :n<∞) n be asequenceoffamiliesofopensetsofRN suchthateachisanelementofOkfd for X. Write N = Z where each Z is infinite and Z ∩Z = ∅ whenever we Sk<∞ k k k m have k 6=m. Starting with Z , choose for each n ∈ Z a U ∈ U such that every infinite 1 1 n n subset of {U : n ∈ Z } covers R . Choose a countable set X ⊂ X such that n 1 ∞ 1 {U :n∈Z } is an ω-cover of X \X . n 1 1 Continuing with Z , choose for each n ∈ Z a U ∈ U such that every infinite 2 2 n n subsetof{U :n∈Z }coversX ∪R . ChooseacountablesetX ⊂X suchthat n 2 1 ∞ 2 {U :n∈Z } is an ω-cover of X \X . Note that we may assume X ∩X =∅. n 2 2 2 1 With X , ··· , X selected countable subsets of X, and for each j ≤n, U ∈U 1 n k k selectedfor k ∈Z so that (U :k∈Z ) is anω-coverof X\X , and every infinite j k j j subset of(U :k ∈Z ) coversX ∪···∪X ∪R . Choose U ∈U for k ∈Z k j 1 j−1 ∞ k k n+1 so that each infinite subset of {U :k ∈Z } is a cover of R ∪X ∪···X and k n+1 ∞ 1 n byClaim1chooseacountableX ⊂RN suchthat{U :k ∈Z }is anω-cover n+1 k n+1 of X \X . n+1

Weakly infinite dimensional subsets of R^N

Similar Weakly infinite dimensional subsets of R^N