SPM Bulletin 27
Contents: 1. Editor's note 2. Research announcements 2.1. Pseudocompact group topologies with no infinite compact subsets 2.2. Selective coideals on (FIN[1]k) 2.3. Entire functions mapping uncountable dense sets of reals onto each other monotonically…
Authors: Boaz Tsaban
S P M BULLETIN ISSUE NUMBER 27: April 2009 CE Contents 1. Editor’s note 2 2. Researc h announcemen ts 2 2.1. Pseudo compact group to p ologies with no infinite compact subsets 2 2.2. Selectiv e coideals on ( F I N [ ∞ ] k , ≤ ) 2 2.3. En tire functions mapping uncountable dense sets of reals onto eac h other monotonically 2 2.4. Effectiv e refining o f Borel co v erings 3 2.5. Symmetry and colorings: Some results and op en problems 3 2.6. Man y par t it ion relations b elo w densit y 3 2.7. Lindelof indestructibilit y , top ological games and selection principles 3 2.8. Lo cally precompact groups: (Lo cal) realcompactness a nd connectedness 3 2.9. The group Aut( µ ) is Ro elck e precompact 4 2.10. An infinite com binatorial statemen t with a p ose t par a meter 4 2.11. The Sc h ur ℓ 1 Theorem for filters 4 2.12. On uniform a symptotic upp er densit y in lo cally compact abelian gr o ups 4 2.13. A c 0 -saturated Banac h sp ace with no long unconditional basic sequences 5 2.14. Zero subspaces of p olynomials on ℓ 1 (Γ) 5 2.15. MAD F amilies and SANE Pla y er 5 2.16. On the consistency of d λ > cov λ ( M ) 5 2.17. o -Boundedness of free top olo gical groups 5 2.18. Com binatorial and mo del-theoretical principles related to regularit y of ultrafilters and compactness of top ological spaces. V 6 2.19. Menger subsets of the Sorgenfrey line 6 2.20. Com binatorial and mo del-theoretical principles related to regularit y of ultrafilters and compactness of top ological spaces. VI 6 3. Unsolv ed pro blems from earlier issues 7 1 2 S P M BULLETIN 27 (APRIL 2009 ) 1. Editor ’s note A hard-disk (more precisely , disk-on-k ey) crash I ha v e exp erienced recen tly led to loss of some of the announcemen ts, a nd some mess in the c hronological o rder of the remaining ones. Apolo gy for those. The sp ecial issue of T op ology and its Applications, dedicated to SPM, did ve ry w ell in the do wnloads statistics: Go to http://top25.scien cedirect.com/ and choose the journal. Greetings, Bo az Tsab an , tsaban@math.biu.ac.il http://www. cs.biu.ac.il/~tsaban 2. Research announcements 2.1. Pseudo c o mpact group top ologies with no infinite comp act subsets. W e sho w that eve ry Ab elian group satisfying a mild cardinal inequality admits a pseu- do compact gr o up top ology from whic h all coun table subgroups inherit the maximal totally b ounded top o lo gy (we say that suc h a top ology satisfies prop ert y ♯ ). This criterion is used in conjunction with an analysis of the algebraic structure o f pseu- do compact gr o ups to obtain, under the Generalized Con tinuum Hyp othesis (GCH), a c haracterization of those pseudo compact g r o ups that admit suc h a to p ology . W e pro v e in particular that eac h of the follo wing groups admits a pseudo compact group top ology with prop ert y ♯ : (a) pseudo compact groups o f cardinality not greater than 2 2 c ; (b) (GCH) connected pseudocompact groups; (c) (GCH) pseudo compact gr o ups whose tor sion- free rank has uncountable cofinality . W e also observ e that pseudocom- pact groups with pro p ert y ♯ con tain no infinite compact sub sets and are examples of P on tryagin reflexiv e precompact groups that are not compact. http://arxi v.org/abs/0812.5033 Jor ge Galindo and Ser gio Mac ario 2.2. Selective coideals on ( F I N [ ∞ ] k , ≤ ) . A notion of selectiv e coideal on ( F I N [ ∞ ] k , ≤ ) is giv en. The natural v ersions of the lo cal Ramsey property and the abstract Baire prop erty relativ e to this con text are prov en t o b e equiv alen t, and it is also sho wn that the family o f subsets of F I N [ ∞ ] k ha ving t he lo cal Ramsey prop ert y relativ e to a selectiv e coideal on ( F I N [ ∞ ] k , ≤ ) is closed under the Souslin op eratio n. Finally , it is pro v en that suc h selectiv e coideals satisfy a sort o f canonical partitio n prop ert y , in the sense of T aylor. http://arxi v.org/abs/0901.1688 Jos´ e G. Mijar es an d Jes´ us Nieto 2.3. Entire functions mapping uncoun table dense sets of reals onto eac h other monotonically. http://www. ams.org/journal-getitem?pii=S0002-9947-09-04924-1 S P M BULLETIN 27 (April 2009 ) 3 Maxim R. B urke 2.4. Effect ive refining of Borel cov erings. http://www. ams.org/journal-getitem?pii=S0002-9947-09-04930-7 Gabriel D e bs; Je an Saint R aymond 2.5. Symmetry and colorings: Some results and op en problems. W e sur- v ey some principal results and op en problems related to color ings of algebraic and geometric ob jects endow ed with symmetries. http://arxi v.org/abs/0901.3356 T. Banakh, I. V. Pr otasov 2.6. Man y partition relations b elow densit y. W e fo r ce 2 λ to b e large and for man y pairs in the interv al ( λ, 2 λ ) a strong er v ersion of the p o la rized partition relations hold. W e apply this to pro blems in general top olo gy . E.g. consisten tly , ev ery 2 λ is success or of singular and for ev ery Hausdorff regular space X , hd( X ) ≤ s ( X ) +3 , hL( X ) ≤ s ( X ) +3 and b etter for s ( X ) regular, via a half-graph partition relation. F or the case s ( X ) = ℵ 0 w e get hd( X ), hL( X ) ≤ ℵ 2 (w e can get ≤ ℵ 1 < 2 ℵ 0 but in a subseque nce work). http://arxi v.org/abs/0902.0440 Sahar on Shelah 2.7. Lindelof indestructibilit y , top ological games and selection principles. Arhangel’skii prov ed that if a first coun table Hausdorff space is Lindel¨ of, then its cardinalit y is at most 2 ℵ 0 . Suc h a clean up p er b o und for Lindel¨ of spaces in the lar g er class of spaces whose points are G δ has b een more elusiv e. In this paper w e contin ue the agenda started in F.D. T all, On the cardinality of Lindel¨ of spaces with p oin ts G δ , T op ology and its Applications 6 3 (1995), 21 - 38, of considering the cardinality problem for spaces satisfying stronger versions o f the L indel¨ of prop ert y . Infinite games and selection principles , esp ecially the Rothberger property , are essen tial to o ls in our in v estigations. http://arxi v.org/abs/0902.1944 Marion Sch e ep ers and F r ank lin D. T al l 2.8. Lo c ally precompact groups: (Lo cal) realcompactness and connected- ness. A theorem of A. W eil asserts that a top ological group em b eds as a (dense) subgroup of a lo cally compact g r oup if and only if it contains a non- empt y precom- pact op en set; such groups are called lo cally precompact. Within the class of lo cally precompact gro ups, t he authors classify tho se groups with the fo llo wing top olo gical prop erties: Dieudonne completeness; lo cal realcompactness; realcompactness; hered- itary realcompactness; connectedness; lo cal connectedness. They also prov e that an ab elian lo cally precompact group o ccurs as the quasi-comp onent of a top ological group if and only if it is precompactly generated, that is, it is gene ra t ed algebraically b y a precompact subset. 4 S P M BULLETIN 27 (APRIL 2009 ) http://arxi v.org/abs/0902.2258 W. W. C omfort and G. Luk´ acs 2.9. The group Aut( µ ) is Ro elck e precompact. F ollo wing a similar result of Us - p enskij on the unitary group of a separable Hilb ert space w e sho w that with resp ect to the low er (or Ro elck e) uniform structure the P olish group G = Aut( µ ), of auto- morphisms of an atomless standard Borel probability space ( X , µ ), is precompact. W e identify the corresp onding compactification as the space of Mark ov op erators on L 2 ( µ ) and deduce that the algebra of right and left uniformly con tin uous functions, the algebra of w eakly almost p erio dic functions, and the algebra of Hilb ert functions on G , all coincide. Again follow ing Usp enskij w e also conclude that G is totally minimal. http://arxi v.org/abs/0902.3786 Eli Glasner 2.10. An infinite com binatorial st atement with a p oset parameter. W e in tro- duce an extension, indexed b y a partially ordered set P and cardinal num bers κ , λ , de- noted by ( κ, <λ ) P , of the classical relation ( κ, n, λ ) → ρ in infinite com binatorics. By definition, ( κ, n, λ ) → ρ holds, if ev ery map F : [ κ ] n → [ κ ] <λ has a ρ -elemen t free set. F or example, Kurato wski’s F ree Set Theorem states tha t ( κ, n, λ ) → n + 1 ho lds iff κ ≥ λ + n , where λ + n denotes the n -th cardinal success or of an infinite cardinal λ . By using the ( κ, <λ ) P framew ork, we presen t a self-contained pro of of the first author’s result that ( λ + n , n, λ ) → n + 2, fo r eac h infinite cardinal λ and eac h p ositiv e in teger n , whic h solv es a problem stated in the 1985 mono graph of Erd˝ os, Ha jnal, M´ at´ e, and Rado. F urthermore, b y using an order-dimension estimate established in 1971 by Ha jnal and Sp encer, w e pro v e the relation ( λ +( n − 1) , r , λ ) → 2 ⌊ 1 2 (1 − 2 − r ) − n/r ⌋ , for ev ery infinite cardinal λ and all p ositive in tegers n and r with 2 ≤ r < n . F or example, ( ℵ 210 , 4 , ℵ 0 ) → 32 , 768. Other order-dimension estimates yield relations suc h as ( ℵ 109 , 4 , ℵ 0 ) → 257 (using an estimate by F ¨ uredi and Kahn) and ( ℵ 7 , 4 , ℵ 0 ) → 10 (using an exact estimate b y Dushnik). http://arxi v.org/abs/0902.4448 Pierr e Gil lib ert, F ri e drich Wehrung 2.11. The Sc h ur ℓ 1 Theorem for filters. W e study classes o f filters F on N such that w eak and strong F -conv ergenc e of seque nces in ℓ 1 coincide. W e study also analogue o f ℓ 1 w eak sequen tial completeness theorem fo r filter conv ergence. http://arxi v.org/abs/0903.0659 A ntonio Avil ´ es, Bernar do Casc ales, Vladimir Kadets, Alexander L e onov 2.12. On uniform asym ptot ic upp er densit y in locally compact abelian groups. Starting out from r esults kno wn for the most classical cases of N , Z d , R d or for sigma-finite a b elian groups, here w e define the notion of asymptotic uniform upp er density in general lo cally compact ab elian groups. Ev en if a bit surprising, the S P M BULLETIN 27 (April 2009) 5 new notio n prov es to b e t he right extension of the classical cases o f Z d , R d . The new notion is used to extend some analogous r esults previously obtained o nly for classical cases or sigma- finite ab elian g r o ups. In particular, we show the follo wing extension of a w ell-know n result for Z of F urstenberg: if in a general lo cally compact Ab elian group G a subset S of G has p o sitive uniform asymptotic upp er densit y , then S-S is syndetic. http://arxi v.org/abs/0904.1567 Szilar d Gy. R evesz 2.13. A c 0 -saturated Banac h space with no long unconditional basic se- quences. http://www. ams.org/journal-getitem?pii=S0002-9947-09-04858-2 J. L op e z - Ab ad, S. T o dor c e vic 2.14. Zero subspaces of p olynomials on ℓ 1 (Γ) . W e provide t wo examples of com- plex homogeneous quadratic p o lynomials P on Banach spaces of the for m ℓ 1 (Γ). The first p olynomial P has b oth separable and nonseparable maximal zero subspaces. The second p olynomial P has the prop erty that while the index-set Γ is not countable, all zero subspaces of P are separable. http://arxi v.org/abs/0903.2374 A ntonio Avil´ e s , Stevo T o dor c e v ic 2.15. MAD F amilies and SANE Pla y er. W e throw some ligh t on the question: is there a MAD family (= a fa mily of infinite subsets of N , the interse ction of an y t w o is finite) whic h is completely separable (i.e. an y X ⊆ N is included in a finite union of mem b ers o f t he family or includes a mem b er (and ev en contin uum man y mem b ers) of t he f a mily). W e prov e that it is har d to prov e the consistency of the negation: (1) If 2 ℵ 0 < ℵ ω , then there is suc h a fa mily . (2) If there is no suc h families then some situation related to p cf holds whose consistency is large; a nd if a > ℵ 1 ev en unknown. http://arxi v.org/abs/0904.0816 Sahar on Shelah 2.16. On the consistency of d λ > cov λ ( M ) . W e pro ve the consistency of: for suit- able strongly inaccessible cardinal lambda the dominating n umber, i.e. the cofinality of λ λ is strictly bigger than cov λ ( M ), i.e. the minimal n um b er of now here dense subsets of 2 λ needed to co ve r it. This answe rs a question of Matet. http://arxi v.org/abs/0904.0817 Sahar on Shelah 2.17. o -Boundedness of free top ological groups. Assuming the absence of Q - p oin ts (which is consisten t with ZFC) w e prov e that the free top olog ical gro up F ( X ) o v er a T yc honov space X is o - b ounded if and only if ev ery contin uous metrizable image T of X satisfies the selection principle U f in ( O , Ω) (the latter means that for 6 S P M BULLETIN 27 (APRIL 2009) ev ery sequence < u n > n ∈ w of op en cov ers o f T there exists a sequence < v n > n ∈ w suc h that v n ∈ [ u n ]
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