SPM Bulletin 22

S P M BULLETIN ISSUE NUMBER 22: Septem b er 2007 C E Contents 1. Editor’s note 1 2. In vited con tribution: Ultrafilters and small sets 2 3. Researc h announcemen ts 3 3.1. In v erse Systems and I-F a v orable Spaces 3 3.2. Com binatorial and hy brid principles for σ - directed families o f coun table sets mo dulo finite 3 3.3. A dic hotom y c haracterizing analytic dig r a phs of uncoun table Borel c hromatic n umber in an y dimension 3 3.4. A dic hotom y c haracterizing analytic dig r a phs of uncoun table Borel c hromatic n umber in an y dimension 4 3.5. Large con tin uum, oracles 4 3.6. Borel hierarc hies in infinite pro ducts of P olish spaces 4 3.7. A game for the Borel functions 4 3.8. On some problems in general top o logy 5 4. Problem of the Issue 5 References 6 5. Unsolv ed problems from earlier issues 7 1. Editor ’s note W e are no w after the F irst Eur op e an Set The ory Me eting , a historically imp ortant and w ell orga nized ev en t. The t a lks g a v e the right blend of theory a nd applicatio ns of set theory . Thanks to Benedikt Lo e we , Grzegorz Plebanek, Jouk o V¨ a¨ an¨ anen, and Boban V elic k o vic for the organization. F ollo wing Jana Fla ˇ sk o v a’s talk at this meeting, w e ha v e invited her to con tribute a section to this issue. W e thank her for her inte resting con tribution in Second 2 and in the Pr oblem of the Issue section. The list of problems at the end of the bulletin b ecame longer than one pag e. W e therefore remo v ed the first few, and will con tinue this w ay unless some more problems are solv ed and their space b ecomes av ailable. . . 1 2 S P M BULLETIN 22 (SEPTEMBER 2007) A m uc h b etter v ersion o f Shelah’s pap er sho wing that g ≤ b + is no w a v ailable at arxiv.org/a bs/math/0612353 Enjo y , Bo az Tsab an , b oa z.tsaban@w eizmann.ac.il http://www. cs.biu.ac.il/~tsaban 2. Invited contribution: Ul trafil ters and small se ts There hav e b een sev eral attempts to connect ultr a filters with families of “small” sets. Tw o of them — 0 - p oin ts and I -ultrafilters — w ere imp o r tan t for m y Ph.D. thesis. The first one is due to Gryzlov [8]: an ultrafilter U ∈ N ∗ is called a 0 -p oint if for ev ery one-t o -one function f : N → N there exists a set U ∈ U suc h that f [ U ] has asymptotic densit y zero. The second term was in tro duced b y Baumgartner [1]: Let I b e a f a mily of subsets of a set X such that I con tains all singletons and is closed under subsets. Giv en a f r ee ultrafilter U o n N , w e sa y that U is an I -ultra filter if for an y F : N → X there is A ∈ U suc h that F [ A ] ∈ I . In my Ph.D. thesis [3] (on whic h a ll my pap ers a r e more less based) I studied I - ultrafilters in the setting X = N and I is an ideal on N or another family of “small” subsets of N that con tains finite se ts and is closed under subsets. As I we re considered the ideal o f sets with a symptotic densit y zero Z 0 = { A ⊆ N : lim sup n →∞ | A ∩ n | n = 0 } , the summable ideal I 1 /n = { A ⊆ N : P a ∈ A 1 a < ∞} or the family o f (almost) thin sets and ( S C )-sets. Here are the (probably not common) definitions: W e sa y that A ⊆ N with an increasing en umeration A = { a n : n ∈ N } is thin : if lim n →∞ a n a n +1 = 0 ; almost thin : if lim n →∞ a n a n +1 < 1; ( S C ) -set : if lim n →∞ a n +1 − a n = ∞ . In the thesis v arious examples of I -ultrafilters for a ll these (and also some other) families I are constructed under additional set theoretic assumptions. In my first pap er [4] it is sho wn that thin sets and a lmost thin sets actually deter- mine the same class of I -ultra filters and there is a pro of that the existence of these ultrafilters is indep enden t of ZFC. The relation b etw ee n this class of ultrafilters and selectiv e ultra filters or Q -p oin ts is studied. Some construction made in the pap er under CH were prov ed in the thesis assuming MA ctble . The next pap er [5] fo cuses on I -ultr afilters where I is the summable ideal I 1 /n or the densit y ideal Z 0 . The relation b et w een these tw o classes of ultra filt ers is sho wn a nd also the relat io n to the class of P - p oin ts. Assuming CH or MA ctble sev eral examples of these ultrafilters are constructed. Again, stronger v ersions of some of the results can b e found in the thesis. One of the few Z FC results in my thesis is the following: There exists an ultra- filter U ∈ N ∗ suc h that fo r ev ery one-to-o ne function f : N → N there exists a set S P M BULLETIN 22 (Septemb er 2007) 3 U ∈ U with f [ U ] in the summable ideal. This theorem strengthens Gryzlo v’s result concerning the existence of 0-p oints and it w as published also separately in [6]. Connections b et w een v arious I -ultrafilters a nd some w ell-know n ultrafilt ers such as P -p oin ts were studied in t w o sections o f m y thesis. It is known that P - p oin ts can b e described as I -ultra filters in t w o differen t w ay s: If X = 2 N then P -p oints are precisely the I -ultrafilters for I consisting of all finite and con v erging sequences, if X = ω 1 then P -p oin ts are precisely the I -ultr a filters f or I = { A ⊆ ω 1 : A has order ty p e ≤ ω } . My latest pap er [7] deals with the question whether there is a family I of subsets of natural n um b ers suc h that P -p oints are precisely the I -ultrafilters. Ho w ev er, only some partial answ ers are given. During the 1 st Europ ean Set Theory Meeting in B¸ edlew o I gav e a talk “On sums and pro ducts of certain I - ultrafilters”. As the title suggests it w as a summary of m y kno wledge ab out sums and pro duc ts of some I -ultrafilt ers. The slides and notes with pro ofs on whic h the t alk w as based (as we ll as m y Ph.D. t hesis) are av ailable online on my w ebpage: http://home .zcu.cz/~flaskova Jana F la ˇ sko v ´ a Departmen t of Mathematics, Univ ersit y of W est Bohemia 3. Research announcements 3.1. Inv erse Systems and I-F a v orable Spaces. Let X b e a compact space. Play er I has a winning strategy in the o p e n-o p en game pla y ed on X if, and only if X can b e represen ted as a limit of σ -complete in v erse system of compact metrizable spaces with sk eletal b o nding maps. http://arxi v.org/abs/0706.3815 A ndrzej Kucharski and Szymo n Plewik 3.2. Combi natorial and h ybrid principles for σ -d irect ed families of coun t- able sets mo dulo finite. W e cons ider strong combinatorial principles for σ -directed families o f coun table sets in the ordering by inclusion mo dulo finite, e.g. P -ideals of coun table sets. W e tr y for principles as strong as p ossible while remaining compatible with CH, a nd we also consider principles compatible with the existence of nonsp ecial Aronsza j n trees. The main thrust is tow a rds a bstra ct principles with game t heoretic form ulations. Some of these principles are purely comb inato r ial, while the ultimate principles are primarily combinatorial but also hav e asp ects of forcing axioms. http://arxi v.org/abs/0706.3729 James Hirsc h o rn 3.3. A dic hotomy c haracterizing analytic digraphs of uncountab le B orel c hromatic num ber in an y dimension. W e study the extension of the Kec hris- Solec ki-T o dorcevic dic hotomy on analytic graphs to dimensions higher than 2. W e pro v e that the extension is p ossible in an y dimension, finite or infinite. The original 4 S P M BULLETIN 22 (SEPTEMBER 2007) pro of w orks in the case of t he finite dimension. W e first pro ve that the natural exten- sion do es not w ork in t he case of the infinite dimension, f o r the notio n of con tin uous homomorphism used in the o riginal t heorem. Then w e solv e the problem in the case of the infinite dimension. Finally , we pro v e that the natur a l extension w orks in the case of t he infinite dimension, but for the notio n of Baire-measurable homomorphism. http://arxi v.org/abs/0707.1313 Dominique L e c omte 3.4. A dic hotomy c haracterizing analytic digraphs of uncountab le B orel c hromatic num ber in an y dimension. W e study the extension of the Kec hris- Solec ki-T o dorcevic dic hotomy on analytic graphs to dimensions higher than 2. W e pro v e that the extension is p ossible in an y dimension, finite or infinite. The original pro of w orks in the case of t he finite dimension. W e first pro ve that the natural exten- sion do es not w ork in t he case of the infinite dimension, f o r the notio n of con tin uous homomorphism used in the o riginal t heorem. Then w e solv e the problem in the case of the infinite dimension. Finally , we pro v e that the natur a l extension w orks in the case of t he infinite dimension, but for the notio n of Baire-measurable homomorphism. http://arxi v.org/abs/0707.1313 Dominique L e c omte 3.5. Large con tinuum , or acles. Our main theorem is ab out iterated forcing for making the contin uum la r g er than ℵ 2 . W e presen t a generalization of math.LO/030 32 94 whic h is dealing with oracles for random, etc., replacing ℵ 1 , ℵ 2 b y λ, λ + (starting with λ = λ <λ > ℵ 1 ). W ell, instead of prop erness w e demand absolute c.c.c. So w e get, e.g. the contin uum is λ + but we can get cov ( M ) = λ . W e giv e some applications. As in math.LO/030 3294 , it is a “pa rtial” coun table supp ort iterat ion but it is c.c.c. http://arxi v.org/abs/0707.1818 Sahar on Shelah 3.6. Bor el hierarc hies in infinite pro ducts of Polish spaces. Let H b e a pro d- uct of coun tably infinite n um b er of copies o f an uncoun table P olish space X . Let Σ ξ ( ¯ Σ ξ ) b e the class of Borel sets of additiv e class ξ for the pro duct o f copies of the discrete top olo gy on X (the P olish top olog y o n X ), and let B = ∪ ξ <ω 1 ¯ Σ ξ . W e prov e in the L ´ evy–Solo v a y mo del that ¯ Σ ξ = Σ ξ ∩ B for 1 ≤ ξ < ω 1 . http://arxi v.org/abs/0707.1967 R ana Ba rua and Ashok Maitr a 3.7. A game for the Borel functions. Abstract. W e presen t a n infinite g ame that c haracterizes the Bor el functions o n Baire Space. www.illc.uv a.nl/Publications/ResearchReports/PP-2006-24.text.pdf Brian Semmes S P M BULLETIN 22 (Septemb er 2007) 5 3.8. On some problems in general t op ology . W e prov e that Arhangel’ski ˘ ı’s prob- lem has a consisten t p ositiv e answ er: If V is a model of CH, then for some ℵ 1 -complete ℵ 2 -c.c. forcing notion P o f cardinalit y ℵ 2 , w e hav e that P fo rces “CH and there is a Lindel¨ of regula r top ological space of size ℵ 2 with clop en basis with ev ery p oin t of pseudo-c haracter ℵ 0 (i.e., each singleton is the in tersection of countably man y o p e n sets)”. Also, we prov e the consistency o f: CH + 2 ℵ 1 > ℵ 2 + “there is no space as ab o ve with ℵ 2 p oin ts” (starting with a weakly compact cardinal). App eared in: Set The ory , Boise ID , 19 92–1994, Contemporary Mathematics, v ol. 192, 91–101. http://arxi v.org/abs/0708.1981 Sahar on Shelah 4. Problem of the Issue Definitions not stated below can be found in Section 2 ab ov e. Some mor e definitions concerning ultrafilters can b e found in [2]. The problem of this issue concerns pro ducts of I -ultrafilters for the case X = N and I is an ideal on N . Definition 4.1. If U and V are ultra filters on N then U · V is the ultrafilter on N × N defined b y M ∈ U · V if and only if { n : { m : h n, m i ∈ M } ∈ V } ∈ U . Since isomorphic ultrafilters can b e iden tified we ma y regard U · V as an ultr afilter on N . The ultra filter U · V is called the pr o duct of ultr afilters U and V . Definition 4.2. L et I b e an ideal on N . W e say that I -ultrafilters are cl o se d under pr o ducts if the pro duc t of tw o a rbitrary I -ultrafilters is a gain an I -ultra filter. F or a P -ideal I the class of I - ultrafilters is closed under pro ducts [3]. Ho w ev er, not m uc h is kno wn for other ideals. F or example, if I is the ideal generated by thin sets or the ideal generated b y ( S C )-sets then I -ultrafilters are not closed under pro ducts [3]. In fact, ev en mor e is t r ue. Theorem 4.3 ([3]) . F or every U ∈ N ∗ the ultr afi lter U · U is n ot an ( S C ) -ultr afi lter (thin ultr afilter). This prop ert y shares the class of all P -p oints (the partitio n {{ n } × N : n ∈ N } witnesses the fact that no pro du ct of t w o f r ee ultrafilters is a P -p oint), but it is consisten t with Z F C that there exist thin ultrafilters (and hence ( S C )-ultrafilters) whic h are not P -p oints [3]. Another example of an ideal whic h is no t a P -ideal is the followin g. Definition 4.4. The van der Waer den ide al W is the family of all A ⊆ N suc h that A do es not contain a rithmetic pro g ressions o f arbitr a ry length. Problem 4.5. A r e W -ultr afilters close d under pr o ducts? 6 S P M BULLETIN 22 (SEPTEMBER 2007) A p ositiv e answ er w ould pro vide (consisten t) examples o f W - ultr a filters tha t are neither P -p oin ts nor (SC)-ultrafilters (and thin ultrafilters). But I exp ect rather a negativ e answ er. Jana F la ˇ sko v ´ a Reference s [1] J. B aumgartner, Ultra filters on ω , Jour nal of Symbolic Log ic 60 (1995), 624 –639. [2] W. Comfort and S. Negrep ontis, The Theory of Ultrafilters , Springe r , B erlin, New Y ork , 1974. [3] J. Flaˇ sko v´ a, Ultrafilters and sm all sets , Ph.D. Thesis, Char les Universit y , Pra gue, 200 6. [4] J. Fla ˇ sko v´ a , Thi n u ltr afilters , Acta Univ ersitatis Caro linae – Mathematica et Ph ysica 46 (2005), 13 – 19. [5] J. Flaˇ sk ov´ a, Ultr afilters and two ide als on ω , in: W DS’05 Pro ceedings of Con tributed P ap ers: Par t I – Mathematics and Comp ute r Sciences (ed. Jana Sa franko v a), P rague, Matfyzpress, pp. 78 – 83, 200 5. [6] J. Flaˇ sk ov´ a, Mor e than a 0 -p oint , Commen tatio nes Mathematica e Universitatis Car o linae 47 (2006), 617 –621. [7] J. Fla ˇ sko v´ a, A note on I -ultr afilters and P -p oints , submitted to Pr o c e e dings of the Winter Scho ol in Ab stra ct Analysi s 2007 . [8] A. Gryzlov, On the ory of the sp ac e β N , Gener al top olog y 166 , Mo sko v. Gos. Univ., Mo scow, 20–34 , 1986 (in Rus s ian). S P M BULLETIN 22 (Septemb er 2007) 7 5. Unsol ved p r oblems from earlier iss ues Issue 4. Do es S 1 (Ω , T) imply U f in (Γ , Γ) ? Issue 5. Is p = p ∗ ? (Se e the definition of p ∗ in that issue.) Issue 6. Do es ther e exist (in ZFC) an unc ountable se t sa tisfying S 1 ( B Γ , B ) ? Issue 8. Do es X 6∈ NON ( M ) and Y 6∈ D imply that X ∪ Y 6∈ COF ( M ) ? Issue 9 (CH) . I s Split (Λ , Λ) pr eserve d under finite unions? Issue 10. Is cov ( M ) = od ? (Se e the definition of od in that i s sue.) Issue 11. Do es S 1 (Γ , Γ) always c ontain an elem ent of c ar din a lity b ? Issue 12. Could ther e b e a Bair e me tric sp ac e M of weight ℵ 1 and a p artition U of M into ℵ 1 me ager sets wher e fo r e ach U ′ ⊂ U , S U ′ has the B air e pr op erty in M ? Issue 14. Do es ther e exist (in ZFC) a set of r e als X of c ar d inality d such that al l finite p owers of X have Menger’s pr op e rty S f in ( O , O ) ? Issue 15. Can a B or el non- σ -c omp act gr oup b e gen e r ate d b y a Hur ewic z subsp ac e? Issue 16 (MA) . I s ther e an unc ountable X ⊆ R s atisfying S 1 ( B Ω , B Γ ) ? Issue 17 (CH) . Is ther e a total ly imp erfe ct X satisfying U f in ( O , Γ) that c an b e mapp e d c ontinuously onto { 0 , 1 } N ? Issue 18 (CH) . I s ther e a Hur e w icz X such that X 2 is Menger but not Hur ewicz? Issue 19. Do es the Pytke ev pr op erty of C p ( X ) imply the Menger pr op erty of X ? Issue 20. Do es every h er e ditarily Hur ewicz s p ac e satisfy S 1 ( B Γ , B Γ ) ? Issue 21 (CH) . Is ther e a R othb er ger-b ounde d G ≤ Z N such that G 2 is not Menger- b ounde d? Issue 22. L et W b e the van der Waer den i d e a l. A r e W -ultr afilters clo s e d under pr o ducts? Previous issues. The p revious issu es of this b ulletin, whic h con tain general in f ormation (first issue), basic definitions, researc h announ cemen ts, and op en problems (all issues) are a v aila ble online, at http://fr ont.math.u cdavis.edu/search?&t=%22SPM+Bulletin%22 Con tributions. P lease su bmit y our con tribu tions (announcemen ts, d iscussions, and op en problems) by e-mailing us. I t is p referred to write them in L A T E X. The authors are urged to use a s standard n otatio n as p ossible, or otherw ise giv e the defi nitions or a referen ce to where the notation is explained. Contributions to this b ulletin would not require any transfer of cop yrigh t, and material pr esen ted here can b e pub lished elsewhere. Subscription. T o receiv e this bulletin (free) to your e-mailb o x, e-mail us : b oaz.tsaban@w eizmann.ac.il