Weakly infinite dimensional subsets of R^N

WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N LILJANA BABINKOSTO V A AND MARION SCHEEPERS Abstract. The Con tinuum Hypothesis impli es an Erd¨ os-Sierpi ´ nski like dual- ity betw een the ideal of first catego ry s ubsets of R N , and the ideal of coun table dimensional subsets of R N . The algebraic sum of a Hurewicz subse t - a di- mension theoretic analogue of Sierpinski sets and Lusin sets - of R N with any compactly coun table dimensional subset of R N has first category . 1. Introduction. The main purpose of this paper is threefold: W e po int out stro ng analo g ies betw een the well-known class of countable dimensio nal subsets o f the Hilb ert cub e, and the classes of Lebes gue measur e zero subsets of the r eal line and the class o f first categor y subsets of complete s eparable metric spa ces. A classica l theorem of Erd¨ os a nd Sierpi ´ n ski giv es, under the Contin uum Hypothesis, some explana tion of this analogy . W e show also that the dimension theor e tic ana logue of Lusin s ets and Sierpinski sets has some of the deeper pr op erties shared by Lusin sets and Sierpi ´ nski sets. And we g ive some information ab out a c lass of weakly infinite dimensional spaces whic h is emer ging as an imp ortant class in dimension theo ry . In particula r we show, using the Co ntin uum Hyp othesis, that the most restr ictive of these c la sses contains metric spaces which are not coun table dimensional. R N denotes the Tyc honoff pro duct of coun tably ma n y copies of the real line, R . The s ymbol [0 , 1] denotes the unit interv a l a nd the subspa ce [0 , 1] N of R N denotes the Hilb ert cube. The symbol P denotes the set o f irra tional num b ers, view ed as a subspace of the real line. 2. The count able dimensional subsets of R N Hurewicz [9] defined a subset of a top olo gical spac e to be c ountable dimensional if it is a union of countably ma n y zero- dimensional subsets of the space, and prov ed that the Hilb ert cub e, thus a lso R N is not countable dimensiona l. Th us, the co llec- tion CD of co unt able dimensio nal subsets o f R N is a σ - ide a l. Since each metrizable space o f cardinality less than 2 ℵ 0 is zerodimensiona l, the cofinalit y o f CD , denoted cof ( CD ), is 2 ℵ 0 . Smirnov and independently Nagami also showed that every sepa- rable metric space is a union o f ℵ 1 zero-dimensio nal subsets, and thus the covering nu mber of CD , denoted cov ( CD ), is ℵ 1 . This is unlik e the σ - ideal M of first c ate- gory subsets of R , and the σ - ide a l N of Leb esg ue measur e zero subse ts of R , where the co finality and the cov ering num ber are not decided by ZFC. If one assumes the Contin uum Hyp othesis (CH), these differences v anish a nd one can giv e an explanation for the un usually str ong a nalogies b etw een the theory of the countable dimensiona l subspaces of R N and the theory of the first category subsets of R N : These t wo collections are dual in the sense of E r d¨ os a nd Sierpi ´ nski. Chapter 1 9 of [17] gives a nice exp osition of the f ollowing classica l theorem: 1 2 LILJANA BABINKOSTO V A AND MARION SCHE EPERS Theorem 1 (Er d¨ o s-Sierpi ´ nski Dualit y T he o rem) . Le t X b e a set of c ar di nality ℵ 1 . L et I and J b e σ -c omplete ide a ls on X such that: (1) X = A S B wher e A and B a disjoint sets with A ∈ I and B ∈ J . (2) cof ( I ) = ℵ 1 = cof ( J ) . (3) F or e ach I ∈ I ther e i s an S ⊂ X \ I such that | S | = ℵ 1 and S ∈ I . (4) F or e ach J ∈ J ther e is an S ⊂ X \ J su ch that | S | = ℵ 1 and S ∈ J . Then ther e is a bije ctive function f : X − → X which is its own inverse, such that for e ac h set E ⊆ X we have E ∈ I ⇔ f [ E ] ∈ J . A cla ssical theor em of T umarkin [2 8] is an imp or tant to o l in pr oving the duality betw een coun table dimensio nal sets and first c ategory s ets of R N under CH: Theorem 2 (T umarkin) . In a sep ar able met ric sp ac e e ach n -dimensional set is c ontaine d in an n -dimensional G δ -set. Corollary 3. Assume the Continuum Hyp othesis. Ther e is a bi je ction f : R N − → R N such that f = f − 1 and for e ach set E ⊂ R N , E is c ountable dime nsional if, and only if, f [ E ] is first c ate gory. Pro of: In Theorem 1 take I to b e CD , the ideal of c ountable dimensional s ubsets of R N , and J to be M , the idea l of first catego ry subs e ts of R N . By Hurewicz’s theorem that R N is not co un table dimensio nal, CD is a σ -co mplete ideal o n R N . By the Ba ire catego r y theorem M is a σ -co mplete ideal on R N . Since R N is separa ble, let D b e a countable dense set. The n D is zero-dimens ional. By Theorem 2 there is a zero-dimens ional G δ set A ⊃ D . But then B = R N \ A is a first category set, a nd s o (1) of Theorem 1 is satisfied. Theorem 2 also implies tha t each countable dimensional subset of R N is contained in a coun table union of zero -dimensional G δ sets, and so CD has a co final family of cardinality 2 ℵ 0 . Since M has a cofinal fa mily of c a rdinality 2 ℵ 0 , C.H. implies that (2) of Theorem 1 holds. E vidently (3) and (4) also hold.  3. Hurewicz sets. Some stro ng ana logies be t ween the theo ry o f the ideal o f coun table dimensio nal subsets o f R N and the theo r ies o f the idea l N of Leb esgue measure zero subsets of the real line, and the ideal M of fir st category subsets of uncountable complete separable metric space s emerge when we consider the following features: Call a subset of R N a Hur ewicz set if it is uncountable, but its intersection with every zero-dimensio nal subset of R N is coun table. Hurewicz sets ar e first category: T ak e a countable dense set D ⊂ R N . Since D is zero dimensional it is, by Theorem 2, contained in a zer o dimensional dense G δ set E . Since a Hurewicz s et meets E in only coun tably man y points the Hurewicz set is first ca tegory . W e shall see below that Hurewicz sets are first categor y in a stro ng sense. The notion of a Hurewicz set is a nalogous to the no tions of a Lusin set and of a Sierpinski s e t: A subset of R is a Lusin set if it is uncoun table but its in ters e c tion 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 in tersection with ea ch Leb esgue measure zero set is countable. Mahlo [15] and Lusin [14] indep endently show ed tha t the Contin uum Hyp othesis implies the existence of a Lus in set of car dina lit y 2 ℵ 0 , a nd Sier pinski [26] show ed WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N 3 that the Contin uum Hyp othesis implies the existence of a Sier pins k i set of c ardi- nality 2 ℵ 0 . It is known that the existence of a Lusin set of ca rdinality 2 ℵ 0 do es not imply the Co ntin uum Hyp othesis , a nd that the existence of a Sierpi´ nski set of cardinality 2 ℵ 0 do es not imply the Contin uum Hyp othesis. But Ro th b erg er [25] prov ed that the simultaneous existence of b oth a Lusin set o f ca rdinality 2 ℵ 0 and a Sierpinski set o f car dinality 2 ℵ 0 is eq uiv alent to 2 ℵ 0 = ℵ 1 . W. Hure w ic z [10] showed that the existence of a Hurewicz s et is equiv a lent to the Contin uum Hyp othesis. Some deep er prop er ties o f the Lusin sets and the Sier pi´ nsk i sets a re tied up with the a lgebraic prop erties of the real line viewed as a top ological g roup. Theorem 4 (Galvin-Mycielski-So lov ay) . If L ⊂ R is a Lus in set, then for e ach first c ate go ry set M ⊂ R , L + M 6 = R . Galvin, Mycielski and So lov ay [6] proved a significantly stronger theo rem: Strong measure zero sets are characterized as sets whose tra nslates by first categor y sets do not cov er the r eal line. Theor em 4 follows from this since Sierpi ´ nski proved that Lusin sets are str o ng meas ure zero. Pawlik owski [19] pr oved the counterpart for Sierpi ´ nski sets: Theorem 5 (Pa wlikowski) . If S ⊂ R is a Sierpi´ nski s et , then for e ach L eb esgue me asur e zer o set N ⊂ R , S + N 6 = R . W e exp ect that a s imilar s tatement is true a b o ut Hure wic z sets in the top ologica l group ( R N , +): Conjecture 1. If H ⊂ R N is a Hur ewicz set, then for e ach c ountable di mensional set C ⊂ R N , H + C 6 = R N . W e ha ve a partial result on this conjecture: Ca ll a subset of R N str ongly c ount- able dimensional if it is a union of coun tably man y closed, finite dimensiona l sub- sets. This notion was introduced by Naga ta [16] and Smirnov [2 7]. Every strong ly countable dimensional set is co un table dimensional, but no t con versely . Thus, t he σ -ideal SCD generated by the s trongly countable dimensional subsets o f R N is a prop er subideal of CD . Indeed, ev ery strongly coun table dimensional set is of first category in R N , and thus SCD is also a subideal of the σ -idea l of first categ ory subsets of R N . Ca ll a subset of R N c omp a ctly c ountable dimensio nal if it is a union of countably many compac t finite dimensiona l sets. The σ -deal KCD gener ated by the co mpact finite dimensional sets is a pro per subideal of SCD . W e s hall show Theorem 6. If H is a Hur ewicz subset o f R N , then for every c omp actly c ountable dimensional set C ⊂ R N , H + C is a first c ate gory s u bset of R N . Since the unio n of countably man y fir st catego r y se ts is first catego ry , Theorem 6 follows directly from Lemma 8 b elow. This Lemma uses another impor tant classical result: Theorem 7 (Hurewicz, T umar kin) . In a sep ar abl e metric sp ac e the un ion of c ountably many clo se d, n -dimensional sets, is an n -dimensional set. Lemma 8. If H ⊆ R N is a Hur ewicz set and C ⊂ R N is a c omp a ct n -dimensional set, then H + C is a first c ate gory subset of R N . Pro of: Let D ⊂ R N be a countable dense set. Then D − C is a union o f countably many closed n -dimensio nal sets , so by Theorem 7 it is n -dimensional. 4 LILJANA BABINKOSTO V A AND MARION SCHE EPERS By Theor em 2 cho ose op en s ets U 1 ⊃ U 2 ⊃ · · · ⊃ U k ⊃ · · · ⊃ D − C such that T ∞ k =1 U k is n -dimensional. Then put Y = { x ∈ R N : x − C ⊆ T ∞ k =1 U k } . Then Y is dense in R N . Note that Y is a G δ -set: Consider a k , and an x ∈ Y . Since C is compact there is an open neighbor ho o d V k ( x ) of x such that V k ( x ) − C ⊂ U k . But then V k = ∪ x ∈ Y V k ( x ) is an open set containing Y , a nd V k − C ⊂ U k . It follows that Y = ∩ k< ∞ V k . Next, put X = H ∩ ( ∞ \ k =1 U k ) . Since H is a Hurewicz s et and T ∞ k =1 U k is n - dimensional, X is a countable set. Define Z = Y \ ( X + C ). Since X + C is a union of countably many closed, n -dimensional sets, it is an n - dimens io nal F σ -set and th us first-categor y in R N . Consequently Z is co- meager. But Z = { x ∈ Y : ( x − C ) ∩ H = ∅} ⊆ Y \ ( H + C ) . Thu s, H + C is meager.  One can show that if there are Hurewicz s ets, then there are ones sa tisfying the prop erty in Co njecture 1. There are several w ell-studied classes of weakly infinite dimensional subsets of R N which are not countable dimensio nal. In Example 1 we give, under CH, a n example of a set S in a very r e strictive cla ss of weakly infinite dimensional s pa ces (but not c o unt able dimensional), and a Hurewicz set H , s uch that S + H = R N . The follo wing is an unresolved weaker instance o f Co njectur e 1: Problem 1. Is it true that if C is a str o ngly c o un t able dimensional set in R N and H is a Hur ewicz set, then H + C 6 = R N ? 4. New classes of weakl y infinite dimensional sp a ces. Many examples in infinitary dimens io n theory b e long to classes of w eak ly infinite dimensional space s introduced in [2]. Our examples in connection with Conjecture 1 are a lso in these classes. These classes ar e defined b y applying selection principles to sp ecific types of op en covers. W e now introduce these cov ers and sele ction principles. Classes of op en cov ers F or a given space X the symbol O denotes the collection of a ll o pen cov ers of X . The follo wing sp ecial t yp es of op en cov ers are r elev an t to our discussion: • O fd : This is the collection of all op en cov ers U such that there is for each finite dimensional F ⊂ X a U ∈ U with F ⊆ U , but X is not a mem b er of U . (Thus, we ar e ass uming X is not finite dimensional.) • O cfd : This is the collection o f a ll o pen cov ers U such that there is for eac h close d finite dimensional F ⊂ X a U ∈ U with F ⊆ U , but X is not a mem b er of U . (Thus, we a re a s suming X is not finite dimensional.) • O kfd : This is the collection of all op en covers U such that there is for each c omp act finite dimensio nal F ⊂ X a U ∈ U with F ⊆ U , but X is not a member of U . (Th us, we are assuming X is not compa c t and finite dimensional.) • O k : This is the c ollection o f all o pen cov ers U such that there is for ea ch c omp a ct F ⊂ X a U ∈ U with F ⊆ U , but X is not a member of U . (Thus, we are ass uming X is not compact.) WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N 5 • Ω: This is the collec tio n of ω cov ers of X . An o pen cov er U of X is an ω -cover if X is not a member of U , but for each finite subset F of X ther e is a U ∈ U with F ⊆ U . • Γ: This is the co llection of γ covers of X . An op en cov er U of X is a γ -cover if it is infinite and every infinite subs e t of U is a cov er o f X . • O gp : This is the collection of g roupable op en cov ers: An o pe n cov er U of a space X is gr oup abl e if there is a partition U = S n ∈ N F n int o finite sets F n that a re disjoin t fr om e ach other, suc h that for eac h x ∈ X there is an N with x ∈ S F n for all n ≥ N . W e ha ve the following inclusion relations among these classes of open cov ers: O fd ⊂ O cfd ⊂ O kfd ⊂ Ω ⊂ O ; O k ⊂ O kfd ; Γ ⊂ Ω . It is a lso worth noting tha t if X is a s e parable metric space then each U ∈ O kfd has a coun table s ubset V ∈ O kfd . The same is true a bo ut O k . Three se lection principle s. Let A and B be families of sets. The selection principle, S 1 ( A , B ) sta tes: F or each sequence ( A n : n ∈ N ) of elements of A there is a seq ue nc e ( B n : n ∈ N ) such that for all n we ha ve B n ∈ A n , and { B n : n ∈ N } ∈ B . The s election principle S f in ( A , B ) sta tes: There is for each sequence ( A n : n ∈ N ) of mem b ers of A a sequence ( B n : n ∈ N ) of finite sets such that for each n we ha ve B n ⊂ A n and S n ∈ N B n ∈ B . The s election principle S c ( A , B ) states: F or each sequence ( A n : n ∈ N ) of elements of A , there is a sequence ( B n : n ∈ N ) wher e each B n is a refinement of the collection of sets A n , ea ch B n is a pairwise disjoint family , and S n ∈ N B n is a member of B . In our context the families A and B will b e classes o f op en cov ers of a top ologica l space. The selection principle S 1 ( A , B ) has th e following monotonicit y proper ties: If A ⊂ C a nd B ⊂ D , then we hav e the follo wing implicatio ns: S 1 ( C , B ) ⇒ S 1 ( A , B ); S 1 ( A , B ) ⇒ S 1 ( A , D ) . Replacing S 1 with S f in or S c in these implications give co r resp onding facts for the other tw o selec tio n pr inciples. Spec ia l instances of these three selectio n principles app ear in c la ssical literature. F or example: S 1 ( O , O ) is known as Rothberge r ’s prop erty and was introduce d in the 1938 pap er [24]. S f in ( O , O ) is known as Menge r’s prop erty and was introduce d in the 1925 pap er [7]. S c ( O , O ) is known as pr op erty C, and was introduced in the 1978 pap er [1]. S c ( O , O ) is a selective version of Bing ’s prop erty of screenability [3], th us als o known a s sele ctive scr e enability . Selec tiv e scr eenability is an imp o rtant prop erty in infinitar y dimension theo ry . Let O 2 denote the collec tio n of op en covers U with |U | ≤ 2. Then S c ( O 2 , O ) is equiv alent to Alexa ndroff ’s no tion of we akly infinite dimensional . A spa ce which is not weakly infinite dimensio nal is sa id to b e str ongly i nfin ite dimensional . The class S 1 ( O kfd , O ) 6 LILJANA BABINKOSTO V A AND MARION SCHE EPERS Metrizable spaces with prop erty S 1 ( O kfd , O ) seem impo rtant. First: Theorem 17 of [2] shows that for separ able metric spaces S 1 ( O fd , O ) ⇒ S c ( O , O ). Monotonicity prop erties of the selection principles give: S 1 ( O kfd , O ) ⇒ S 1 ( O cfd , O ) ⇒ S 1 ( O fd , O ). Thu s for metr izable spaces S 1 ( O kfd , O ) implies the imp ortant prop erty S c ( O , O ). Second: The following lemma shows that these s paces hav e the classic a l Me ng er prop erty S f in ( O , O ): Lemma 9. S 1 ( O kfd , O ) ⇒ S 1 ( O k , O ) ⇒ S f in ( O , O ) . Third: Sev era l central examples from the theory of infinite dimensio nal spaces a re in this c la ss. F or e x ample: R. Pol’s ex a mple in [22] which shows that a weakly infi- nite dimensional space need not be countable dimensional is in the cla ss S 1 ( O kfd , O ). The preser v ation of infinitary dimension prop erties by pro ducts is not well un- dersto o d yet. Whenever a new class of infinite dimensio nality is defined it is of int eres t to k now how this class b ehaves under v arious mathema tica l constructions, like products. W e now make some remarks abo ut pro ducts by spaces in the class S 1 ( O kfd , O ), using recent examples constructed by E . Pol and R. Pol in [21]: Prop ositio n 10. Assume Martin ’s Axiom. (1) Finite p owers of sets in S 1 ( O kfd , O ) ne e d not b e in S c ( O , O ) . (2) The pr o duct of a sp ac e in S 1 ( O kfd , O ) with the sp ac e of irr ational numb ers ne e d not b e we akly infinite dimensional. T o see this conside r the following: In the pro of of P rop osition 4.1 of [21] E. Pol and R. Pol construct, using Mar tin’s Axiom, spaces E 0 and E 1 which have S 1 ( O kfd , O ). T o see that these tw o spaces hav e this prop erty we co nsider elemen ts of the a rgument g iven in pa rt (6) of the pro of o f [21], Prop os ition 4.1: F or i ∈ { 0 , 1 } the spa ces E i are of the form E ∪ S i where • S i is the unio n of countably many c ompact, finite dimensional subspace s , and • F or each op en set W ⊃ S i in E i , | E i \ W | < 2 ℵ 0 . Consider E i . Let ( U n : n < ∞ ) be a sequence in O kfd . W rite S i as a union of compact finite dimensional subs pa ces S i 1 ⊂ S i 2 ⊂ · · · ⊂ S i n ⊂ · · · . F or each n choose a U 2 · n − 1 ∈ U 2 · n − 1 with S i n ⊆ U 2 · n − 1 . Then W = S n< ∞ U 2 · n − 1 is a n op en set containing S i , and so E i \ W has car dinality less tha n 2 ℵ 0 . But Mar tin’s Axiom implies that a separable metr ic space of c a rdinality less than 2 ℵ 0 has Rothber ger’s prop erty S 1 ( O , O ) (Theorem 5 of [13]). Thus, choose for each n a U 2 · n ∈ U 2 · n such that E i \ W ⊆ S n< ∞ U 2 · n . It follows that the spac e X cons tr ucted there a s the free union of E 0 and E 1 still has S 1 ( O kfd , O ). Recall that a metric space ( X, d ) has the Hav er proper t y if there is for each sequence ( ǫ n : n < ∞ ) o f p ositive real num ber s a sequence ( U n : n < ∞ ) such that each V n is a pairwise disjo int family o f op en sets, each o f d -dia meter less than ǫ n , and S n< ∞ U n is a co ver of X . Metr izable spaces with pro per ty S c ( O , O ) hav e the Hav er prop erty in all equiv alent metrics. Since [21] shows that X 2 has the Menger prop erty but under some metric X 2 do es not hav e the Hav er prop erty , it follows that X 2 do es not have pro pe rty S c ( O , O ). T o see the second item: In Theor em 6.1 of [21] it is p ointed out that the pro duct of the spa ce E i with the subspace of irr ational nu mbers o f the real line is no t even S c ( O 2 , O ), that is, is s trongly infinit e dimensional. The class S 1 ( O kfd , Ω) WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N 7 S 1 ( O k , Ω) w as considered in Section 2 of [18]. Theorem 2.1 of [18] implies: Lemma 11 (Pansera, Pa vlovic) . A sp ac e has S 1 ( O k , Ω) if, and only if, it has S 1 ( O k , O ) in al l finite p owers. Prop ositio n 12. If a sp ac e has pr op erty S 1 ( O kfd , Ω) then it has pr op erty S f in (Ω , Ω) . Pro of: Let X b e a spa ce with S 1 ( O kfd , Ω). By mo no tonicity pro pe r ties of S 1 ( · , · ) X then has S 1 ( O k , Ω). Then by Lemma 1 1 X n has S 1 ( O k , O ) for all finite n . By Corollar y 9 X n has S f in ( O , O ) for all n . By Theorem 3.9 of [11] this is equiv alen t to: X has S f in (Ω , Ω).  Using standard techniques one can prov e: If a space has the prop erty S 1 ( O kfd , Ω), then it has the prop erty S 1 ( O kfd , O ) in all finite p owers. W e don’t know if the conv erse is true: Problem 2. F or metrizable sp ac e X is it tru e that if e ach X n has pr op erty S 1 ( O kfd , O ) then X has S 1 ( O kfd , Ω) ? It is likely that the spa ces E 0 and E 1 of [21] mentioned ab ov e are in S 1 ( O kfd , Ω), but we hav e no t e x amined this. The spa ce X which is the free unio n o f E 0 and E 1 is an example of a space in S 1 ( O kfd , O ) but not in S 1 ( O kfd , Ω). Thus, under Martin’s Axiom, S 1 ( O kfd , Ω) is a prop er subcla ss of S 1 ( O kfd , O ). Below w e construct under CH an exa mple (E xample 1) of a space which ha s the pro pe r ty S 1 ( O kfd , Ω) and which is not countable dimensional. The class S 1 ( O kfd , Γ) S 1 ( O k , Γ) was explicitly considered in [4], Section 3, and in [5], Section 3, where the second implica tio n of Corollary 13 is noted. Corollary 13. S 1 ( O kfd , Γ) ⇒ S 1 ( O k , Γ) ⇒ S f in (Ω , O gp ) . Theorem 12 o f [4] shows tha t eac h σ -compa ct s pace has the proper t y S 1 ( O k , Γ). This is not true for S 1 ( O kfd , Γ): The Hilb e r t cub e is not weakly infinite dimensional, but is a σ -co mpact space. Co m bining [5] Theo rem 10 and [5] Pro po sition 13 gives the following result: Lemma 14 (Di Maio, Ko cinac, Mecca rriello) . If X has S 1 ( O k , Γ) , t hen for al l n also X n has this pr op erty. The Hurewicz covering pro p e rty is a s trengthening of the Menger pr op erty S f in ( O , O ) a nd is defined a s follows: T op ologica l space X has the Hu r ewicz c overing pr op ert y if: F or each seq ue nc e ( U n : n < ∞ ) of o pen covers o f X there is a s equence ( V n : n < ∞ ) of finit e sets such that for eac h n , V n ⊆ U n , and for each x ∈ X , for all but finitely man y n , x ∈ S V n . It w as shown in [1 2] that the Hur e wicz covering proprty is equiv a lent to the selection principle S f in (Ω , O gp ). Corollary 15. If a sp ac e X has pr op erty S 1 ( O kfd , Γ) then it has S f in (Ω , O gp ) in al l finite p owers. Using the technique in the pro of of [2] Theorem 17 and using the techniques in [2] Lemma 6 one obtains the following tw o res ults: Theorem 16. S 1 ( O fd , O gp ) ⇒ S c ( O , O gp ) . Lemma 17. If a sp ac e X has S 1 ( O kfd , Γ) , then it has S c ( O , O ) in al l finite p owers. 8 LILJANA BABINKOSTO V A AND MARION SCHE EPERS Pro of: Let X b e a space with pro p er t y S 1 ( O kfd , Γ). By Le mma 13 it has pro per ty S 1 ( O k , Γ), and by Lemma 14 it has S 1 ( O k , Γ) in all finite powers, a nd thus the Hurewicz pro p er t y in all finite p owers. Then by Corolla r y 13 of [2], all finite p ow ers of X has the prope r ty S c ( O , O ).  Let X be an infinite dimensio na l s pace which sa tisfies S 1 ( O kfd , Γ). By [2] and Corollar y 13, all finite p ow ers of X hav e the Hav er prop erty , and indeed, the pro duct of X with any space having the Hav er prop erty again has the Hav er prop erty . It follows that X × P ha s the Haver pr o p e rty . W e don’t know if this pro duct must b e selectively scr e enable: Problem 3. If X has S 1 ( O kfd , Γ) then do es X × P have S c ( O , O ) ? 5. Examples. W e now describ e tw o infinite dimensional exa mples that will b e used to demon- strate a num b er of facts in co nnection with the preceding sections. In bo th ca ses we ar e in teres ted in showing prop erties of Hur e wicz sets, and thus we a re co nfined to ass uming the Con tinuum Hyp othesis. A ZFC+CH example o f S 1 ( O kfd , Ω) . W rite: R n := { x ∈ R N : ( ∀ j > n )( x ( n ) = 0 ) } and R ∞ := S ∞ n =1 R n . Standard arguments prove Lemma 18 and Cor ollary 19 below. Lemma 18. Assume G ⊃ R ∞ is an op e n su bset of R N and I 1 , · · · , I n ar e close d intervals of p ositive length. F or e ach f ∈ R N ther e ar e an m > n and close d intervals I n +1 , · · · , I m of p o sitive length such that I 1 × · · · × I m × R N ⊆ G − f . Corollary 19. If ( G n : n < ∞ ) is a se quenc e of G δ subset of R N such that e ach G n c ontains R ∞ , and if ( f n : n < ∞ ) is se quenc e of elements of R N , t hen T ∞ n =1 ( G n − f n ) c ontains a Hilb ert cub e. Example 1 (CH) . Ther e ar e subsets X and Y of R N such that: (1) X and Y satisfy S 1 ( O kfd , Ω) ; (2) X \ R ∞ and Y \ R ∞ ar e Hur ewicz sets; (3) X and Y have S 1 ( O kfd , Ω) and X × Y not. (4) X ∪ Y is S 1 ( O kfd , O ) , and not S 1 ( O kfd , Ω) . Fix the following enumerations: • ( f α : α < 2 ℵ 0 ), the list of all elements o f R N • ( F α : α < 2 ℵ 0 ), the list of all finite dimensional G δ subsets of R N • ( L α : α < 2 ℵ 0 ), the list of all G δ subsets of R N containing R ∞ • (( U α n : n < ω ) : α < 2 ℵ 0 ) where for each α each element of R ∞ is in U α n for all but finitely many n , and each U α n is open in R N . F or eac h α < 2 ℵ 0 the se t T α = T n< ∞ ( S m ≥ n U α m ) is a G δ set c ontaining R ∞ . Since G 0 = T 0 ∩ L 0 is a dense G δ set whic h cont ains R ∞ . Thus G 0 ∩ ( G 0 − f 0 ) contains, by Cor ollay 19, a homeomorphic copy of the Hilbert cub e . Cho o se x 0 ∈ ( G 0 ∩ ( G 0 − f 0 )) \ (( R ∞ ∪ F 0 ) ∪ (( R ∞ ∪ F 0 ) − f 0 )) WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N 9 and fix y 0 ∈ G 0 with x 0 = y 0 − f 0 . Then define S 1 0 (0) = { n : x 0 ∈ U 0 n } and S 2 0 (0) = { n : y 0 ∈ U 0 n } . Assume we hav e 0 < α < ω 1 , a nd that w e ha ve selected for each γ < α an x γ and a y γ and defined sets S 1 ν ( γ ) and S 2 ν ( γ ), ν ≤ γ such that: (1) δ < γ < α ⇒ S 1 γ ( δ ) = S 2 γ ( δ ) = ω ; (2) γ ≤ ν < δ < α ⇒ (a) S 1 γ ( δ ) ⊆ ∗ S 1 ν ( δ ); S 2 γ ( δ ) ⊆ ∗ S 2 ν ( δ ) (b) S 1 γ ( δ ) , S 1 γ ( ν ) ⊆ S 1 γ (0); S 2 γ ( δ ) , S 2 γ ( ν ) ⊆ S 2 γ (0) a re a ll infinite (3) γ ≤ δ < α ⇒ { n : x δ ∈ U γ n } ⊇ S 1 γ ( δ ); { n : y δ ∈ U γ n } ⊇ S 2 γ ( δ ) (4) γ ≤ δ < α ⇒ { x δ , y δ } ⊆ G γ \ ( R ∞ S ( ∪ ν ≤ δ F ν ) S { x ν : ν < δ } S { y ν : ν < δ } ); (5) δ < α ⇒ x δ = y δ − f δ . T ow ards selecting x α and y α we cons ider t wo cas es: Case 1: α = β + 1 F or δ < α define S 1 α ( δ ) = ω = S 2 α ( δ ). F or γ ≤ β we define T 1 γ = ∞ \ n =1 ( [ m ≥ n, m ∈ S 1 γ ( β ) U γ m ) , T 2 γ = ∞ \ n =1 ( [ m ≥ n, m ∈ S 2 γ ( β ) U γ m ) , and for α we define T 1 α = T 2 α = ∞ \ n =1 ( [ m ≥ n U α m ) . and G α = \ γ ≤ α ( L γ ∩ T 1 γ ∩ T 2 γ ) . Then G α is a G δ set containing R ∞ and so by Corollar y 19 the set G α ∩ ( G α − f α ) contains a Hilbert cube. But the set B α = ( R ∞ [ ( ∪ γ ≤ α F γ ) [ { x γ : γ < α } [ { y γ : γ < α } ) − ( { f γ : γ ≤ α } ∪ { 0 } ) is countable dimensional a nd th us does no t co nt ain a Hilber t cub e. Cho ose x α ∈ ( G α T ( G α − f α )) \ B α and fix y α ∈ G α with x α = y α − f α . Then (4) holds at α . Since for γ < α w e have x α ∈ T 1 γ and y α ∈ T 2 γ the sets S 1 γ ( α ) = { n ∈ S 1 γ ( β ) : x α ∈ U γ n } and S 2 γ ( α ) = { n ∈ S 2 γ ( β ) : y α ∈ U γ n } are infinite. Similarly the sets S 1 α ( α ) = { n : x α ∈ U α n } and S 2 α ( α ) = { n : y α ∈ U α n } are infinite. Case 2: α is a limit ordinal. F or γ < α choose infinite sets F 1 γ and F 2 γ so that for all β < α , F 1 γ ⊆ ∗ S 1 γ ( β ) and F 2 γ ⊆ ∗ S 2 γ ( β ). F or γ ≤ β we define T 1 γ = ∞ \ n =1 ( [ m ≥ n, m ∈ F 1 γ U γ m ) , T 2 γ = ∞ \ n =1 ( [ m ≥ n, m ∈ F 2 γ U γ m ) , 10 LILJANA BABINKOSTO V A AND MARION SCHE EPERS and for α we define T 1 α = T 2 α = ∞ \ n =1 ( [ m ≥ n U α m ) . and G α = \ γ ≤ α ( L γ ∩ T 1 γ ∩ T 2 γ ) . Since G α is a G δ set containing R ∞ , Cor ollary 19 implies that the set G α ∩ ( G α − f α ) contains a Hilbert cube. The set B α = ( R ∞ [ ( ∪ γ ≤ α F γ ) [ { x γ : γ < α } [ { y γ : γ < α } ) − ( { f γ : γ ≤ α } ∪ { 0 } ) is countable dimensional and do es not contain a Hilbert cube. Cho ose x α ∈ ( G α T ( G α − f α )) \ B α and fix y α ∈ G α with x α = y α − f α . Then (4) holds at α . Since for γ < α w e have x α ∈ T 1 γ and y α ∈ T 2 γ the sets S 1 γ ( α ) = { n ∈ F 1 γ : x α ∈ U γ n } a nd S 2 γ ( α ) = { n ∈ F 2 γ : y α ∈ U γ n } a re infinite. Similarly the sets S 1 α ( α ) = { n : x α ∈ U α n } and S 2 α ( α ) = { n : y α ∈ U α n } are infinite. This completes the des cription of the pro ces s for choo sing x α and y α for α < ω 1 . Finally define X = R ∞ S { x α : α < ω 1 } and Y = R ∞ S { y α : α < ω 1 } . Claim 1: If U is a c ol le ction of op en subsets of R N which c ontains an infinite subset V , e ach infinite su bset of wh ich c overs R ∞ , then ther e ar e c ountable sets A X and A Y such that V is an ω c over of X \ A X and of Y \ A Y . Pro of of Claim 1 : F or let V b e such a subfamily of U . W e ma y ass ume that V is countable. Thus, we may a ssume that for so me α , fixe d from now o n, V is ( U α n : n < ω ), as enumerated b efore the c o nstruction f X and Y . Put A X = { x γ : γ ≤ α } and A Y = { y γ : γ ≤ α } . Consider a ny finite subset F ⊂ X \ A X . W e may write F = { x γ 1 , · · · , x γ n } where we have α < γ 1 < · · · < γ n . F rom the constr uction of X and Y we hav e S 1 α ( γ n ) ⊆ ∗ · · · ⊆ ∗ S 1 α ( γ 1 ). Cho os e n la rge enough that S 1 α ( γ m ) \ n ⊆ · · · ⊆ S 1 α ( γ 1 ) . Then for k ∈ S 1 α ( γ m ) we have F ⊂ U α m . A similar ar gument applies to Y \ A Y . Claim 2: X and Y b oth satisfy the sele ction p rinciple S 1 ( O kfd , Ω) . Pro of o f Claim 2: W e prove it for X . The pr o of for Y is similar. Le t ( U n : n < ∞ ) be a seq ue nce of families o f op en s ets of R N such that ea ch is an element of O kfd for X . W r ite N = S k< ∞ Z k where each Z k is infinite and Z k ∩ Z m = ∅ whenever we hav e k 6 = m . Starting with Z 1 , cho o se for e a ch n ∈ Z 1 a U n ∈ U n such that every infinite subset of { U n : n ∈ Z 1 } covers R ∞ . Cho ose a countable set X 1 ⊂ X such that { U n : n ∈ Z 1 } is a n ω -cov er of X \ X 1 . Contin uing with Z 2 , choos e for each n ∈ Z 2 a U n ∈ U n such that every infinite subset of { U n : n ∈ Z 2 } cov ers X 1 ∪ R ∞ . Cho ose a countable set X 2 ⊂ X s uch that { U n : n ∈ Z 2 } is a n ω -cov er of X \ X 2 . Note that we may ass ume X 2 ∩ X 1 = ∅ . With X 1 , · · · , X n selected countable subsets of X , and for each j ≤ n , U k ∈ U k selected for k ∈ Z j so that ( U k : k ∈ Z j ) is an ω -cover of X \ X j , a nd every infinite subset of ( U k : k ∈ Z j ) cov ers X 1 ∪ · · · ∪ X j − 1 ∪ R ∞ . Cho ose U k ∈ U k for k ∈ Z n +1 so that each infinite subset of { U k : k ∈ Z n +1 } is a co ver o f R ∞ ∪ X 1 ∪ · · · X n and by Claim 1 cho ose a countable X n +1 ⊂ R N such that { U k : k ∈ Z n +1 } is a n ω -cover of X \ X n +1 . WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N 11 Thu s w e obtain U k : k ∈ N and coun table subsets X n of X , disjoint from each other, such that for each n , { U k : k ∈ N } is an ω -cov er of X \ X n . Since fo r each finite s ubs e t F of X there is an n with F ∩ X n = ∅ , it follows that { U k : k < ∞} is an ω -cov er for X . Claim 3: The sets X \ R ∞ and Y \ R ∞ b oth ar e Hu r ewicz sets. Pro of of Claim 3: F o r consider a ny zero dimensiona l subset of R N . It is contained in a zer o dimensional set o f form F γ , and F γ ∩ ( X \ R ∞ ) ⊂ { x ν : ν < γ } , a co un table set. Claim 4: X \ R ∞ − Y \ R ∞ = R N . Pro of of Claim 4: This follo ws fro m (5) in the c o nstruction o f X a nd Y . Remarks: (1) X and Y ar e Menger in al l fin ite p owers: Pro of: Cla im 2 and Corolla ry 1 2. (2) Ther e is a Hur ewicz set H and an S 1 ( O kfd , Ω) set S with H + S = R N . Pro of: P ut H = − ( Y \ R ∞ ) a nd S = X . Apply Claims 2, 3 and 4. (3) S 1 ( O kfd , Ω) is not pr eserve d by finite pr o d uct s : Pro of: Since the map ( x, y ) 7→ x − y is contin uous, if X × Y is Menger , so is X − Y = R N . But the latter is no t Menger . Thus X × Y is not Meng er. By Corollar y 9 X × Y is not S 1 ( O kfd , O ) . (4) S 1 ( O kfd , Ω) is not pr eserve d by finite unions: Pro of: F or the same reaso n, X ∪ Y is not Menger in all finite p ow ers, even though X ∪ Y is Menger. It follo ws tha t X ∪ Y has the prop er t y S 1 ( O kfd , O ), but no t the prop erty S 1 ( O kfd , Ω), and also not the prop e rty S 1 ( O k , Ω). A ZFC+CH example of S 1 ( O kfd , Γ) The po int with the following example is to demons trate that spaces in this cla s s need no t b e coun table dimensional. W e do not know if non-count able dimensional examples of metric spaces in this clas s can be o btained without reso r ting to hy- po theses b eyond the standard axioms o f Zer melo-F ra enkel set theory . Put J = { 0 } ∪ [ 1 2 , 1]. Define J m = { x ∈ J N : ( ∀ n > m )( x ( n ) = 0) } and J ∞ = S m ∈ N J m . Since each J m is compac t and m - dimensional, J ∞ is compac tly countable dimensio nal and is a de ns e subset of J N . F or X any subset of N , de fine: J ( X ) = { x ∈ J N : { n : x ( n ) 6 = 0 } ⊆ X } . When X is infinite, define X ∗ = { Y ⊂ N : Y is infinite and Y \ X is finite } , and J ∗ ( X ) = S { J ( Y ) : Y ∈ X ∗ } . Lemma 20. L et an infinite set X ⊂ N , a c ountable set C ⊂ J N , as wel l as a family J of op en subsets of J N b e given so that r elative to C ∪ J ∞ , J ∈ O kfd . Then ther e ar e an infinite Y ⊂ X , and a se quenc e D 1 , D 2 , · · · , D n , · · · in J su ch that J ∞ ∪ C ∪ J ∗ ( Y ) ⊆ S m ∈ N ( T m ≥ n D m ) . 12 LILJANA BABINKOSTO V A AND MARION SCHE EPERS Pro of: Enumerate C bijectively as ( c n : n < ∞ ). Cho ose k 0 = min( X ). Then J ( { 1 , · · · , k 0 } ) = J k 0 × { 0 } N ⊂ J ∞ is compact and finite dimensional. Since J is in O kfd , choos e D 1 ∈ J with J 0 ∪ { c 0 } ∪ J ( { 1 , · · · , k 0 } ) ⊂ D 1 . F or ea ch x ∈ J ( { 1 , · · · , k 0 } ) choose a basic o p e n set B x of the form I 1 ( x ) × · · · × I k 0 ( x ) × { 0 } k ( x ) × J N ⊂ D 1 . Since J ( { 1 , · · · , k 0 } is compa ct, choose a finite set { x 1 , · · · , x m } ∈ J ( { 1 , · · · , k 0 } ) f or whic h J ( { 1 , · · · , k 0 } ) ⊆ m [ i =1 B x i ⊆ D 1 . Cho ose k 1 ∈ X with k 1 > ma x { k 0 + k ( x i ) : 1 ≤ i ≤ m } . Then w e have J k 0 × { 0 } k 1 − k 0 × J N ⊆ D 1 . This sp ecifies k 0 , k 1 and D 1 . Now J ( { 1 , · · · , k 1 } ) = J k 1 × { 0 } N ⊂ J ∞ is a c o mpact and finite dimensional subset of J ∞ . Since J is in O kfd for C ∪ J ∞ , cho ose a D 2 ∈ J with J 1 ∪ { c 0 , c 1 } ∪ J ( { 1 , · · · , k 1 } ) ⊆ D 2 . F or each x ∈ J ( { 1 , · · · , k 1 } ) c ho ose a basic op en set B x of the for m I 1 ( x ) × · · · × I k 1 ( x ) × { 0 } k ( x ) × J N ⊂ D 2 . Since J ( { 1 , · · · , k 1 } is co mpact, choose a finite set { x 1 , · · · , x m } ∈ J ( { 1 , · · · , k 1 } ) for whic h J ( { 1 , · · · , k 1 } ) ⊆ m [ i =1 B x i ⊆ D 2 . Cho ose k 2 ∈ X with k 2 > ma x { k 1 + k ( x i ) : 1 ≤ i ≤ m } . Then w e have J k 1 × { 0 } k 2 − k 1 × J N ⊆ D 2 . This specifies k 0 , k 1 and k 2 , as w ell a s D 1 and D 2 . Contin uing in this wa y we find sequences (1) k 0 < k 1 < · · · < k n < · · · in X and (2) D 1 , D 2 , · · · , D n , · · · ∈ J such that for n ≥ 1 w e have: J n ∪ { c 0 , · · · , c n } ∪ J ( { 1 , · · · , k n } ) ⊆ D n +1 and J k n × { 0 } k n +1 − k n × J N ⊆ D n +1 . Put Y = { k n : 0 ≤ n < ∞} , a n infinite subset o f X . Consider an y infinite set Z ⊂ N with Z \ Y finite. C ho ose k n so la r ge that Z \ { 1 , · · · , k n } ⊆ Y . Now J ( Z ) ⊆ J k m × { 0 } k m +1 − k m × J N ⊆ D m for all m > n , and so J ( Z ) ⊆ \ m>n D m . It follows that J ∞ ∪ C ∪ J ∗ ( Y ) ⊆ S n< ∞ ( T m>n D m ).  F or each Z ∈ Y ∗ the space J ( Z ) has a subspace homeomor phic to a Hilb ert cube , and so J ( Z ) is strongly infinite dimensional. Mo reov er, J ∗ ( Z ) ⊆ J ∗ ( Y ). Now we cons truct the example: Example 2 (CH) . Ther e is a set X ⊂ [0 , 1 ] N which is not c ountable dimensional, but it ha s t he pr op erty S 1 ( O kfd , Γ) . Pro of: Let ( H α : α < ω 1 ) enumerate the finite dimensional G δ -subsets o f J N . Also, let ( J α : α < ω 1 ) enumerate all countable families of o pen sets, and let ( f α : α < ω 1 ) enumerate N N . Now using CH recursively choose X α ⊂ N infinite and x α ∈ J ( X α ) such that: WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N 13 (1) α < β ⇒ X β ⊆ ∗ X α , (2) If J α is an O kfd -cov er o f J ∞ S { x β : β < α } , then X α ∈ [ N ] ℵ 0 is chosen such that for each β < α , X α ⊂ ∗ X β and for some sequenc e ( D n : n ∈ N ) in J α we hav e J ∞ ∪ { x β : β < α } ∪ J ∗ ( X α ) ⊆ S n ∈ N ( T m ≥ n D m ), a nd (3) F or eac h β , x β 6∈ ∪ γ ≤ β H γ . Here is how we accomplish (2): Fir st, choo s e a n infinite set X ⊂ N such that for all β < α , X ⊂ ∗ X β . Since J α is an O kfd -cov er for J ∞ ∪ { x β : β < α } , cho ose by Lemma 20 an X α ∈ [ X ] ℵ 0 and a sequence ( D n : n ∈ N ) fro m J α such that J ∗ ( X α ) ⊆ S n ∈ N ( T m ≥ n D m )). Then we hav e J ∞ ∪ J ∗ ( X α ) ∪ { x β : β < α } ⊆ [ n ∈ N ( \ m ≥ n D m ) . Then as J ( X α ) contains a homeomorphic copy of the Hilb ert cub e, choos e x α ∈ J ( X α ) \ S β ≤ α H β . Put Y = { x α : α < ω 1 } and put X = J ∞ S Y . Claim 1: Y is a Hur ewicz set . F or let F b e a finite dimensional subset of the Hilb ert cub e. Cho o se α < ω 1 so that F ⊂ H α , Then, as Y ∩ H α ⊂ { x β : β < α } , it follows that Y ∩ F is countable. Thus Y has a coun table in tersection with coun table dimensional sets. Claim 2: Each op en c over of X which is in O kfd c ontains a subset which is a γ -c over fo r X . F or let U be in O kfd for X . Cho ose an α with U = J α . Now at stag e α w e c hose X α and a sequence ( D n : n ∈ N ) from J α so that J ∗ ( X α ) S J ∞ S { x β : β < α } ⊆ S n ∈ N ( T m ≥ n D m ). Since for all γ > α we hav e X γ ⊂ ∗ X α , it follows that for all γ > α w e hav e J ∗ ( X γ ) ⊆ J ∗ ( X α ). But then X ⊆ S n ∈ N ( T m ≥ n D m ). Claim 3: X has S 1 ( O kfd , Γ). F or let ( U n : n < ∞ ) b e a sequence fro m O kfd for X . Cho ose an infinite discrete subset ( d n : n < ∞ ) from X . F or each n , put V n = { U \ { d n } : U = V 1 ∩ · · · ∩ V n and for i ≤ n, V i ∈ U i } \ {∅} . Then put U = S n< ∞ V n . Still U is in O kfd for X . By Claim 2 choo se D m ∈ U , m < ∞ , suc h that X ⊆ S m ∈ N T n ≥ m D n . Choo se n 1 so la r ge that d 1 ∈ D m for all m ≥ n 1 . Then cho ose n 2 > n 1 so larg e tha t for an m 1 < n 2 , D n 1 ∈ V m 1 and { d 1 , · · · , d max( { n 1 ,m 1 } } ⊆ D k for all k ≥ n 2 . Cho ose n 3 > n 2 so larg e that for an m 2 < n 3 we ha ve D n 2 ∈ V m 2 , and fo r all k ≥ n 3 we hav e { d j : j ≤ max( { n 2 , m 2 } ) } ⊂ D k , and so on. W e ha ve m 1 < m 2 < · · · < m k , a nd X ⊆ S k< ∞ T j ≥ k D n j , a nd each D n j is of the form T i ≤ m j V j i . F or i ≤ m 1 put U i = V 1 i , and for k ≥ 1 and for m k < i ≤ m k +1 put U i = V k +1 i . Then we hav e for each i that U i ∈ U i , and { U i : i < ∞} is a γ -cov er of X . By Claim 1 X is not co un table dimensional.  Remarks: When we choos e X α we may ass ume the enumeration function of X α dominates f α . This guar a ntees that the subspa ce Y o f X do es not ha ve the Meng er pr op erty . By Corollary 15 X has the Hurewicz co vering pro per ty in all finit e p ow ers. By Corollar y 1 3 of [2] X has the prop erty S c ( O , O ) in a ll finite pow ers. 14 LILJANA BABINKOSTO V A AND MARION SCHE EPERS In Theo rem 15 of [4] it was sho wn that a Tych ono ff space X has the prop er t y S 1 ( O k , Γ) if, a nd only if, the space C ( X ) consisting of contin uous real-v alued func- tions on X has the following prop erty: F or every sequence ( A n : n < ∞ ) of subsets of X , each having the function f ∈ A n in the compact-op en top olo gy on C ( X ), there is a sequence ( f n : n ∈ N ) in C ( X ) such that for ea ch n , f n ∈ A n , a nd the sequence ( f n : n < ∞ ) conv erg es to f in the point-op en top ology on C ( X ). Since S 1 ( O kfd , Γ) implies S 1 ( O k , Γ), it follows that for the spac e X constructed in example 2, the function space C ( X ) has the corres po nding pro per ty . 6. Added in proof: Shortly after w e submitted our pap er Roman Pol informed us that a certain mo dification of our cons truction of Ex ample 2 a nswers our Problem 3. With Pol’s kind p ermissio n w e include his rema rks here: With CH assumed and with the notation established in Example 2 do the follow- ing: In a ddition to listing the H α , J α and f α in the beginning of t he construction, also list all op en sets in J N as ( G α : α < ω 1 ). A t stage α , when pic king the p oint x α we pro ce ed as follo ws : if J ( X α ) \ ( G α ∪ ( ∪ β ≤ α H β )) is nonempty , x α is c hosen from this set. O therwise, x α is an ar bitrary po int in J ( X α ) \ ( ∪ β ≤ α H β ) if this is a subse t of G α . Claim 4: Y = { x α : α < ω 1 } is not in S c ( O 2 , O ). T o this end r epe a t an argument from pa ges 9 0, 91 of [2 3]: F o r x ∈ J N write supp ( x ) = { n : x ( n ) 6 = 0 } . F or ea ch i, let U i, 0 ( resp ectively , U i, 1 ) b e an op en set in J N consisting of po ints x such that for some j , | supp ( x ) ∩ { 1 , 2 , ..., j } | = i , and 0 < x ( j ) < 5 / 6 ( r esp ectively , x ( j ) > 4 / 6 ). A key observ ation ( which justifies (18) on pag e 90 ) is the following . Let X = { n 1 , n 2 , ... } with n 1 < n 2 < ... . Then the tr ace of U i, 0 ( r esp ectively , U i, 1 ) o n J ( X ) co ns ists of the po ints x in J ( X ) with 0 < x ( n i ) < 5 / 6 ( r esp ectively , x ( n i ) > 4 / 6 ). Then for each i set U i = { U i, 0 , U i, 1 } . Sub claim 4.1: If X is an y infinite subset of N , the tr aces of U i on J ( X ) form a sequence in J ( X ) witnessing that J ( X ) does not hav e S c ( O 2 , O ). F or consider the close d subspace C of J ( X ) defined as follows: F or n ∈ X put S n = [ 1 2 , 1], and for n 6∈ X put S n = { 0 } . Then C = Q ∞ n =1 S n is homeomorphic to [ 1 2 , 1] X . F or eac h i , define: A i 0 = { x ∈ J ( X ) : | su pp ( x ) ∩ { 1 , · · · , j }| = i and x ( j ) = 1 2 } and A i 1 = { x ∈ J ( X ) : | su pp ( x ) ∩ { 1 , · · · , j } | = i and x ( j ) = 1 } . Then A i 0 and A i 1 are disjoint clo sed subsets of J ( X ) and are e s sentially o ppo site faces of the Hilb ert cube [ 1 2 , 1] X . Moreover, A ij ⊂ U ij and A ij ∩ U i (1 − j ) = ∅ for j ∈ { 0 , 1 } . Then the arg umen t used for the Hilbe r t cub e can b e adapted and applied to the sequenc e o f o pp os ite pairs of faces A i 0 and A i 1 to show that the sequence ( U n : n < ∞ ) does not have a sequence of disjoin t r efinements that co ver J ( X ). O bs erve that if C is a countable dimensiona l subset of J ( X ), then the same is true abo ut J ( X ) \ C . Now, re tur n to the set Y : Assume, aiming at a contradiction, that Y is an S c ( O 2 , O ) - space. Let V i be a disjoint open collection in J N such t hat V i refines WEAKL Y INFINITE DIMENSIONAL SUBSETS OF R N 15 U i and the unio n G of all e le men ts of all V i contains Y . Then G = G α for so me α < ω 1 and since x α is in G α , b y the rules for choo sing x α we hav e that fo r some countable dimensional set C , J ( X α ) \ C is contained in G α . Therefore , the traces of V i on J ( X α ) \ C provide a for bidden sequence of op en disjoint r efinements of the traces of U i on J ( X α ) \ C , a contradiction. Claim 5: With X = J ∞ ∪ Y , the pr o duct X × P is not in S c ( O 2 , O ). Indeed, let u : J − → { 0 , 1 } take 0 to 0 and the interv al [ 1 2 ,1] to 1, and let f : J N − → { 0 , 1 } N be the pr o duct map. Then u and f ar e contin uous maps, implying f (the graph of f ) is a clo sed subset of J N × { 0 , 1 } N . Since the set P = { x ∈ { 0 , 1 } N : |{ n : x ( n ) = 1 }| = ℵ 0 } is homeo morphic to the irrationa ls, and f − 1 ( P ) = J N \ J ∞ ⊇ Y , we hav e f ∩ ( X × P ) = f ∩ ( Y × P ) = { ( y , f ( y )) : y ∈ Y } is a clo sed subset of X × P . The ma p g : Y − → f ∩ ( Y × P ) defined by g ( y ) = ( y , f ( y )) is one-to-one, contin uous, and it s in verse is the pro jection onto the fir st co ordinate, which also is contin uous, a nd therefore g shows that Y is homeomorphic to f ∩ ( Y × P ). By Claim 4 the c lo sed subset f ∩ ( X × P ) of X × P is not S c ( O 2 , O ), and thus X × P is no t S c ( O 2 , O ). 7. Acknow ledgements W e would like to thank Elzbieta Pol and Roman Pol for communicating their results to us, for a careful r eading of our pap er, a nd for remarks that improv ed the quality o f the paper. References [1] D.F. Addis and J.H. 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