SPM Bulletin 23

S P M BULLETIN ISSUE NUMBER 23: Decem b er 2007 CE Contents 1. Editor’s note 2 2. Researc h announcemen ts 2 2.1. Cardinal inv arian ts o f the con tin uum and comb inatorics on uncoun table cardinals 2 2.2. Selection principles and countable dimension 2 2.3. Pro ducts and selection principles 2 2.4. On completely don ut (doughnut) sets 2 2.5. On minimal non-p otentially closed subse ts of the plane 3 2.6. One Dimensional Lo cally Connected S -spaces 3 2.7. Co v ering an u ncoun table square b y c ountably man y con tin uous functions 3 2.8. Ramsey degrees o f finite ultrametric spaces, ultrametric Urysohn spaces and dynamics of their isometry groups 3 2.9. Big Ramsey degrees and divisibilit y in classes o f ultra metric spaces 4 2.10. Definable Davie s’ Theorem 4 2.11. Selection Principles and Baire spaces 4 2.12. Selectiv e screenabilit y in top ological groups 4 2.13. Selectiv e screenabilit y and the Hurewicz prop ert y 4 2.14. P artitioning tr iples and pa r t ially ordered sets 4 2.15. A p olarized partition relation for car dina ls o f countable cofinality 4 2.16. On top ological spaces of singular densit y and minimal w eigh t 5 2.17. There is a v an Do uw en MAD family 5 2.18. Cardinal sequences of LCS spaces under GCH 5 2.19. Dirich let sets and Erdos-K unen-Mauldin theorem 5 2.20. Lo cal Ramsey theory: An abstract a pproac h 5 2.21. There are no hereditary pro ductiv e γ -spaces 6 3. Problem of the Issue 6 References 6 4. Unsolv ed problems from earlier issues 7 1 2 S P M BULLETIN 23 (DECEMBER 2007) 1. Editor ’s note A surprising n um b er of new results and directions in “core” SPM by Babinkosto v a and Sc heep ers in the last quarter of the y ear! See §§ 2.2–2.3 and 2.11 –2.13. Ha v e a go o d 2 008, Bo az Tsab an , tsaban@math.biu.ac.il http://www. cs.biu.ac.il/~tsaban 2. Research announcements 2.1. Cardinal in v arian ts of the con tin uum and combina torics on uncoun t- able cardinals. W e explore the connection b et w een com binatoria l principles on un- coun table cardinals, like stick and club, on the one hand, and the com binatorics of sets of reals and, in particular, cardinal inv arian ts of the con tin uum, on the other hand. F or example, w e prov e that additivity of measure implies that Martins axiom holds for an y Cohen algebra. W e construct a mo del in whic h club holds, y et the co v ering num ber of the n ull ideal cov ( N ) is large. W e sho w that for uncountable cardinals κ ≤ λ and F ⊆ [ λ ] κ , if all subsets of λ either contain, or are disjoin t f rom, a member of F , then F has size at least cov ( N ) etc. As an application, w e solv e the Gross space pro blem under c = ℵ 2 b y sho wing that there is suc h a space ov er an y coun table field. In t w o app endices, w e solv e problems o f F uc hino, Shelah and Soukup, and of Kraszewski, resp ectiv ely . J¨ or g B r end le 2.2. Selection principles and coun table dimension. W e characterize coun table dimensionalit y and stro ng countable dimensionalit y by means of an infinite game. http://arxi v.org/abs/0709.2893 Liljana Babinkostova and Marion Sche ep ers 2.3. Pr o du cts and selection principles. The pro duct of a Sierpinski set and a Lusin set has Menger’s prop ert y . The pro duct of a gamma set a nd a Lusin set has Roth b erger’s prop erty . http://arxi v.org/abs/0709.2895 Liljana Babinkostova and Marion Sche ep ers 2.4. On completely donut (doughnu t) sets. A set h A, B i = { X ∈ [ ω ] ω : A ⊆ X ⊆ B } is a don ut, whene v er A ⊆ B ⊆ ω and B \ A is infinite. A subset S ⊆ [ ω ] ω is completely donut, whenev er for eac h don ut h A, B i there exis ts a don ut h C, D i ⊆ h A, B i suc h that h C , D i ⊆ S or h C , D i ∩ S = ∅ . If alwa ys holds h C , D i ∩ S = ∅ , then S is no where don ut. W e examine families of completely don ut and no where don ut sets. The results corresp ond to completely R amsey and no where Ramsey sets. http://arxi v.org/abs/0709.3016 Piotr Kalemb a, S z ymon Plewik and Anna Wojcie chowska S P M BULLETIN 23 (December 2007) 3 2.5. On minimal non-p otentially closed subsets of the plane. W e study the Borel subsets of the plane that can b e made closed b y refining the Polish t o p ology on the real line. These sets a re called p oten tially closed. W e first compare Borel subsets of the plane using pro ducts of contin uous functions. W e show the existence of a perfect antic hain made of minimal sets among non-p oten tially closed sets. W e apply this result to graphs, quasi-orders and partia l orders. W e a lso giv e a non- p oten tially closed set minim um f or ano t her notion of comparison. Finally , w e sho w that w e cannot hav e injectivit y in the Kec hris-Solec ki-T o dorcevic dic hotomy ab out analytic graphs. T op olog y and its Applications 154 (200 7), 24 1–262. http://arxi v.org/abs/0710.0152 Dominique L e c omte 1 2.6. One Dimensional Lo cally Connected S -spaces. W e construct, assuming Jensen’s principle ♦ , a one-dimensional lo cally connected hereditarily separable con- tin uum without conv ergent sequences. The construction is a n in v erse limit in ω 1 steps, a nd is patterned aft er the original F edorc h uk construction of a compact S - space. T o mak e it one-dimensional, eac h space in the inv erse limit is a cop y of the Menger sp onge. http://arxi v.org/abs/0710.1085 Jo an E. Hart Kenneth Kunen 2.7. Cov ering an uncoun table square b y coun tably man y con tin uous func- tions. W e prov e that there exists a countable family of contin uous real functions whose graphs together with their in v erses cov er an uncoun table square, i.e. a set of the for m X × X , where X is uncoun table. This is mo t iv ated b y an old r esult of Sierpi ´ nski, sa ying that ℵ 1 × ℵ 1 is co v ered b y coun tably many graphs o f functions and in v erses of functions. Another motiv ation comes from Shelah’s study of plana r Borel sets without p erfect rectangles. http://arxi v.org/abs/0710.1402 Wieslaw K ubis 2.8. Ramsey degrees of finite ultrametric spaces, ultrametric Urysohn spa- ces and dynamics of their isometry groups. W e study Ramsey-theoretic pr o p- erties of sev eral natura l classes of finite ultrametric spaces, describ e the corresp onding Urysohn spaces a nd compute a dynamical in v ariant a ttac hed to their isometry groups. http://arxi v.org/abs/0710.2347 L. Nguyen V an Th´ e 1 Lecomte has rece ntly uplo aded q uite a few interesting pap ers to the ArXiv. Check there. 4 S P M BULLETIN 23 (DECEMBER 2007) 2.9. Big Ramsey degrees and divisibilit y in classes of ultrametric spaces. Giv en a coun table set S of p o sitive r eals, we study finite- dimensional Ramsey-theoretic prop erties of the coun table ultrametric Urysohn space with distances in S . http://arxi v.org/abs/0710.2352 L. Nguyen V an Th´ e 2.10. Definable Da vies’ Theorem. W e pro v e the fo llo wing analogue of a Theorem of R.O. Da vies: Ev ery Σ 1 2 function f : R × R → R can b e represen ted as a sum of rectangular Σ 1 2 functions if and only if all reals are constructible. http://arxi v.org/abs/0711.0162 Asger T ornquist and Wil liam Weis s 2.11. Selection P rinciples and Baire spaces. W e prov e tha t if X is a separable metric space with the Hurewicz co v ering prop ert y , then the Banach-Mazur game pla y ed on X is determined. The implication is not true whe n “Hurewicz co v ering prop ert y” is replaced with “Menger cov ering prop erty”. http://arxi v.org/abs/0711.1104 Marion Sche ep ers 2.12. Selectiv e screenabilit y in top ological gr oups. W e examine the selec tiv e screenabilit y prop ert y in top ological groups. In the metrizable case we also giv e ch ar- acterizations in terms of the Hav er prop ert y and finitary Hav er prop ert y resp ectiv ely relativ e t o left-in v arian t metrics. W e prov e theorems stating conditions under whic h the pr o p erties are preserv ed by pro ducts. Among metrizable groups we c haracterize the ones of countable co v ering dimension by a natur a l game. http://arxi v.org/abs/0711.1322 Liljana Babinkostova 2.13. Selectiv e screenabilit y and the Hurewicz prop erty . W e c haracterize the Hurewicz co v ering prop ert y in metrizable spaces in terms of prop erties o f the metrics of the space. Then w e sho w that a w eak v ersion of selectiv e screenabilit y , w hen com bined with the Hurewicz prop erty , implies selectiv e screenabilit y . http://arxi v.org/abs/0711.1516 Liljana Babinkostova 2.14. Pa rtit ioning triples and partially ordered sets. http://www. ams.org/journal-getitem?pii=S0002-9939-07-09170-8 A lbin Jones 2.15. A p olarized partition relation for cardinals of countable cofinality. http://www. ams.org/journal-getitem?pii=S0002-9939-07-09143-5 A lbin Jones S P M BULLETIN 23 (December 2007) 5 2.16. On top ological spaces of singular densit y and minimal weigh t. In a recen t pap er, Juh´ asz and Shelah establish t he consistency of a regular hereditarily Lindel¨ of space o f densit y ℵ ω 1 . It is natural to a sk what is t he minimal p ossible v alue for the w eigh t of suc h space; more sp ecifically , can the we ight b e ℵ ω 1 ? In this pap er, we isolate a certain consequence of the Generalized Con tin uum Hy- p othesis, whic h w e will refer to as the Pr ev a lent Singular Car din als Hyp othesis , and sho w it implies t ha t ev ery top ological space of density and w eigh t ℵ ω 1 is not heredi- tarily Lindel¨ of. The a ssertion PSH is v ery w eak, and in fact holds in all currently kno wn mo dels of ZF C. dx.doi.org/ 10.1016/j.topol.2007.09.013 Assaf Rinot 2.17. There is a v an D ouw en MAD family . W e pro v e in ZF C that there is a MAD family of functions in N N whic h is also maximal with resp ect to infinite partial functions. This solves a long standing question of v an Dou w en. W e also pro v e that suc h families cannot be analytic. This strengthens Steprans’ result that strongly MAD families cannot b e analytic. http://arxi v.org/abs/0711.4400 Dilip R agha van 2.18. Cardinal sequences of LCS spaces under GCH. W e give full character- ization of the sequences of regular cardinals that may arise as cardinal sequence s of lo cally compact scattered spaces under GCH. The pro ofs are based on constructions of univ ersal lo cally compact scattered spaces. http://arxi v.org/abs/0712.0584 Juan Carlos Martinez, L ajos S o ukup 2.19. Dirich let sets and Erdos-Kunen-Mauldin theorem. By a theorem prov ed b y Erdos, Kunen and Mauldin, f or an y nonempty p erfect set P on the real line t here exists a p erfect set M of Leb esgue measure zero suc h that P + M = R . W e pro v e a stro ng er v ersion of this theorem in whic h t he obtained p erfect set M is a Dirichlet set. Using this result w e show that f o r a wide range of families of subsets of the reals, all additiv e sets are p erfectly meager in transitiv e sense. W e also prov e that ev ery prop er a nalytic subgroup G of the reals is con tained in an F σ set F suc h that F + G is a meager n ull set. http://arxi v.org/abs/0712.2112 Peter Elias 2.20. Lo cal Ramsey theory: A n abstract approac h. It is sho wn t hat the know n notion of selectiv e coideal can b e extended to a family H of subsets of R , where ( R , ≤ , r ) is a top ological Ramsey space in the sense of T o dor cevic. Then it is pro v en 6 S P M BULLETIN 23 (DECEMBER 2007) that, if H selectiv e, the H -Ramsey and H -Baire subsets of R are equiv alen t. This extends results of F arah for semiselectiv e coideals o f N . Also, it is prov en that the family of H -Ramsey subsets of R is closed under the Souslin op eration. http://arxi v.org/abs/0712.2393 Jos´ e Mijar es and Jes´ us Nieto 2.21. There are no hereditary productive γ -spaces. W e sho w that if X is an uncoun table pro ductiv e γ -set [F. Jordan, Pro ductiv e lo cal pr o p erties of function spaces, T op olo g y Appl. 154 , 870–883, 2007], then there is a coun table Y ⊆ X such that X \ Y is not Hurewicz. Along the wa y w e will pro v e a general result ab out F r´ ec het- α 2 filters to gain infor- mation ab out countable unions of γ -spaces and pro ductiv e γ -spaces. In pa r t icular, w e answ er a question of A. Miller b y sho wing that a n increasing coun table union of γ -spaces is ag ain a γ -space. W e also use recen t methods of B. Tsaban and L. Zdomskyy to sho w that λ - spaces with Hurewicz prop erty a re precisely those spaces for whic h ev ery co- coun table set is Hurewicz. F r a ncis Jor dan 3. Problem of the Issue F or a sequence { X n } n ∈ N of subsets of X , define lim inf X n = S m T n ≥ m X n . F or a family F of subsets of X , L ( F ) denotes its closure under the op eration lim inf . X has the δ -pr op erty if for eac h op en ω -cov er U of X , X ∈ L ( U ). This prop erty w as in tro duced by Gerlits and Nagy in their seminal pap er [1]. Clearly , the γ -prop ert y  Ω Γ  implies the δ -pro p ert y . S 1 (Ω , Γ) =  Ω Γ  [1]. Problem 3.1 (Gerlits-Nag y [1 ]) . Is the δ -pr op erty e quivale nt to  Ω Γ  ? Miller suggested tha t, as a union of an increasing sequence o f sets with t he δ - prop ert y ha s again the δ -property , one can obta in a negativ e answ er b y finding an increasing sequence { X n } n ∈ N of sets, eac h satisfying  Ω Γ  , suc h that their union do es not satisfy  Ω Γ  . Recen tly , F ra ncis Jordan solv ed Miller’s question by pro ving that this is imp ossible (Section 2.21 ab ov e). A closely-related problem, due to Sak ai, concerns the Pytk eev prop erty in function spaces. This problem asks whether, for eac h set of reals X , if C p ( X ) has the Pytk eev prop ert y then C p ( X ) is F r ´ ec het ( i.e., X satisfies  Ω Γ  ) – see [2] for details. Bo az Tsab an Reference s [1] J. Gerlits and Zs. Nagy , Some pr op erties of C ( X ) , I , T o polo gy and its Applications 14 (1982 ), 151–1 61. [2] M. Sak ai, Sp e cial subsets of re als char acterizing lo c al pr op erties of function sp ac es , in: Selectio n Principles and Co v ering Propertie s in T op ology (Lj. D.R. Koˇ cinac, e d.), Quaderni di Matematica 18 (20 0 7), 195 –225. S P M BULLETIN 23 (December 2007) 7 4. Unsol ved problems from e arlier iss ues Issue 1. Is ` Ω Γ ´ = ` Ω T ´ ? Issue 2. Is U f in ( O , Ω) = S f in (Γ , Ω) ? And if not, do es U f in ( O , Γ) imply S f in (Γ , Ω) ? 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 ZF C) an unc ountable set satisfying S f in ( B , B ) ? Issue 8. Do es X 6∈ NON ( M ) and Y 6∈ D i mply that X ∪ Y 6∈ COF ( M ) ? Issue 9 ( CH) . Is Split (Λ , Λ) pr eserve d under finite uni ons? Issue 10. Is cov ( M ) = o d ? (Se e the definition of o d in that i ssue.) Issue 11. Do es S 1 (Γ , Γ) always c ontain an element of c ar dinality b ? Issue 12. Could ther e b e a Bair e m etric sp ac e M of weight ℵ 1 and a p artition U of M into ℵ 1 me ager sets wher e for e ach U ′ ⊂ U , S U ′ has the Bair e pr op erty in M ? Issue 14. Do es ther e exist (in ZF C) a set of r e als X of c ar dinali ty d such that al l finite p owers of X have Menger’s pr op erty S f in ( O , O ) ? Issue 15. Can a Bor el non- σ -c omp act gr oup b e gener ate d by a Hur ewicz subsp ac e? Issue 16 (MA ) . Is ther e an unc ountable X ⊆ R satisfying 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 m app e d c ontinuously onto { 0 , 1 } N ? Issue 18 (CH) . Is ther e a Hur ewicz 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 ) impl y that X has Menger’s pr op erty? Issue 20. Do es every her e ditarily Hur ewicz sp 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 W aer den ide al. Ar e W -ultr afilters close d under pr o ducts? Issue 23. Is the δ -pr op erty e quivalent to the γ -pr op erty ` Ω Γ ´ ? Previous issues. T he previous issues of this bu lletin are a v ailable online at http://f ront.math. ucdavis.edu/search?&t=%22SPM+Bulletin%22 Con tributions. Announcements, discussions, and op en problems should b e emailed to tsaban@m ath.biu.ac .il Subscription. 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