SPM Bulletin 26

S P M BULLETIN ISSUE NUMBER 26: Decem b er 2008 CE Contents 1. Editor’s note 1 2. Pro ceedings of the Third W orkshop on Co v erings, Selections and Games in T op ology 2 3. Researc h announcemen ts 3 3.1. The comm utan t of L ( H ) in its ultrap ow er ma y or ma y not b e trivial 3 3.2. P artitions of trees and ACA’ 4 3.3. Pro ducts of straigh t spaces 4 3.4. Eac h second coun table ab elian group is a subgroup o f a second coun table divisible group 4 3.5. A dic hotom y f o r Borel f unctions 4 3.6. More Results on Regular Ultrafilters in ZFC 5 3.7. Minimal pseudo compact group top ologies on free ab elian gro ups 5 3.8. W eakly infinite dimensional subsets of R N 5 3.9. Selections, Extensions and Collection wise Normalit y 6 3.10. Op enly factorizable spaces and compact extensions of top olo g ical semigroups 6 3.11. Em b edding the bicyclic semigroup into coun tably compact t o p ological semigroups 6 3.12. Building suitable se ts for lo cally compact groups b y means of contin uous selections 6 3.13. The near coherence of filters principle do es not imply the filter dic hotom y principle 6 4. Unsolv ed problems from earlier issues 7 1. Editor ’s note This festiv e issue concludes the civilian y ear 2008 with details on a special issue of T op ology and its Applications dedicated to SPM, and with a quite large list of researc h announcemen ts. Ha v e a fruitful 2009, Bo az Tsab an , tsaban@math.biu.ac.il 1 2 S P M BULLETIN 26 (DECEMBER 2008) http://www. cs.biu.ac.il/~tsaban 2. Proceedings of the Third Work shop on Coverings, Selections and Games in Topology Sp ecial Issue 15 6 o f T op olo gy and its Applic ations . Guest editor: Ljubi ˇ sa D.R. Ko ˇ cinac. F rom t he preface: “The Third W orkshop on Cov erings, Selections and Games in T op o lo gy was held in V rnja ˇ ck a Banja , Serbia, fro m April 25 to April 29, 20 07, and organized by F acult y of Sciences and Mathematics, Univ ersit y of Ni ˇ s, and T ec hnical F acult y in ˇ Ca ˇ cak, Univ ersit y o f Kra gujev ac. “The previous t w o w orkshops under the same name w ere held in Lecce, Italy (June 27–29, 2002 a nd D ecem ber 19– 2 2, 2005 ). The main theme of this w orkshop w as Se- lection Principles Theory and v ariet y of its relations with other fields of T op ology and Mathematics (g ame theory , Ramsey theory , set theory , com binatorics, function spaces, hyperspaces, uniform spaces, top ological algebras, analysis, dimension t heory , lattice theory , Bo olean algebras). This sp ecial issue of T o p ology and its Applica- tions contains 19 pap ers presen ted at the meeting a nd ev aluated following the usual editorial pro cedure of the journal.” Con ten ts and links to abstracts and f ull texts: (1) Ljubi ˇ sa D .R. Koˇ cinac, Pr efac e , p. 1. http://dx.d oi.org/10.1016/ j.to pol.2008.05.022 (2) Lilja na Babinkosto v a, Sele ctive scr e ena b i l i ty in top olo gic al gr oups , pp. 2 –9. http://dx.d oi.org/10.1016/ j.to pol.2008.02.014 (3) T aras Banakh, Lyub omyr Zdomskyy , Sep ar at ion pr op erties b etwe en the σ - c omp actness and Hur ewic z p r op erty , pp. 10– 15. http://dx.d oi.org/10.1016/ j.to pol.2007.12.017 (4) Pa vle V.M . Blago jevic, Aleksandra S. Dimitrijevic Blago jev ic, Jo hn McCleary , Equilater al triangles on a Jor dan curve and a ge n er a lization of a the or em of Dold , pp. 1 6–23. http://dx.d oi.org/10.1016/ j.to pol.2008.04.008 (5) Lev Buk ov sk´ y, On wQN ∗ and w Q N ∗ sp ac es , pp. 24–27. http://dx.d oi.org/10.1016/ j.to pol.2007.10.010 (6) Giusepp e Di Maio, Ljubi ˇ sa D .R. Koˇ cinac, Statistic al c onver genc e in top olo gy , pp. 28–45. http://dx.d oi.org/10.1016/ j.to pol.2008.01.015 (7) Dr agan Djurcic, Ljubi ˇ sa D.R . Ko cinac, Mali ˇ sa R. ˇ Zi ˇ zo vi ´ c, C lasses of se quenc es of r e al numb ers, games and sele ction pr op erties , pp. 46–55. http://dx.d oi.org/10.1016/ j.to pol.2008.02.013 (8) Jakub Duda, Boa z Tsaban, Nul l s e ts and games i n Ba nach sp ac es , pp. 56–60. http://dx.d oi.org/10.1016/ j.to pol.2008.04.009 S P M BULLETIN 26 (December 2008) 3 (9) Vitaly V. F edorc h uk, Evgenij V. Osip o v Certain classes of we akly infinite- dimensional sp ac es a n d top olo gic al games , pp. 61–69. http://dx.d oi.org/10.1016/ j.to pol.2007.11.007 (10) Dimitris N. Georgiou, Stavros D. Iliadis, On the gr e a test s p l i tting top ol o gy , pp. 70–75. http://dx.d oi.org/10.1016/ j.to pol.2007.11.008 (11) Stav ros D. Iliadis, Universa l elemen ts in some c lasses of mappings and classes of G -sp ac es , pp. 76– 82. http://dx.d oi.org/10.1016/ j.to pol.2008.04.010 (12) Maria Joita, On fr ames in Hilb ert mo dules over p r o- C ∗ -algebr as , pp. 83 – 92. http://dx.d oi.org/10.1016/ j.to pol.2007.12.015 (13) Marion Sc heep ers, R othb e r ge r’ s pr op erty in a l l fi n ite p owers , pp. 9 3–103. http://dx.d oi.org/10.1016/ j.to pol.2007.10.011 (14) Du ˇ san Rep ov ˇ s, Bo az Tsaban, Lyubomyr Zdomskyy , Continuous sele ctions and σ -sp ac es , pp. 10 4–109. http://dx.d oi.org/10.1016/ j.to pol.2008.03.025 (15) Andrzej Kucharski, Szymon Ple wik, Inverse systems and I -favor able sp ac e s , pp. 110–116 . http://dx.d oi.org/10.1016/ j.to pol.2007.12.016 (16) Masami Sak ai, F unction sp ac es with a c ountable cs ∗ -network at a p oint , pp. 117– 123. http://dx.d oi.org/10.1016/ j.to pol.2007.10.012 (17) Mila Mr ˇ sevi ´ c, Milena Jeli ´ c, Sele ction principles in hyp ersp ac es with gener alize d Vietoris top olo gies , pp. 124–129 . http://dx.d oi.org/10.1016/ j.to pol.2008.01.016 (18) Heik e Milden berg er, Car dinal char acteristics for Menger-b ounde d sub gr oups , pp. 130–137 . http://dx.d oi.org/10.1016/ j.to pol.2008.04.011 (19) T omasz W eiss, A n ote o n unb ounde d str ongly m e a sur e zer o sub gr oups of the BaerSp e c k er gr oup , pp. 138–14 1 . http://dx.d oi.org/10.1016/ j.to pol.2008.03.026 (20) Sophia Zafiridou, Dendrites with a c ountable closur e of the set of end p oints , pp. 142–149 . http://dx.d oi.org/10.1016/ j.to pol.2008.04.012 3. Research announcements 3.1. The comm utan t of L ( H ) in its ultrap ow er ma y or ma y not b e trivial. Kirc h b erg aske d in 2004 whether the comm utan t of L ( H ) in its (norm) ultrap ow er is trivial. Assuming the Con tinn uum Hyp othesis, w e prov e tha t the answ er depends on the c hoice of the ultrafilter. http://arxi v.org/abs/0808.3763 4 S P M BULLETIN 26 (DECEMBER 2008) Ilijas F ar ah, N. Christopher Phil lips, Juris Stepr¯ ans 3.2. Partitions of t rees and ACA’. W e sho w t ha t a v ersion of Ramsey’s theo- rem for trees for arbitra ry exponents is equiv alent to the subsystem ACA’ of reve rse mathematics. http://arxi v.org/abs/0809.2267 Bernar d A. Anderson, Jeffry L. Hirst 3.3. Pr o du cts of straigh t spaces. A metric spac e X is straight if for eac h finite co v er o f X b y closed sets, and fo r eac h real v alued function f on X , if f is uniformly con tin uous on eac h set of the co v er, then f is uniformly con tin uous on the whole of X . A lo cally connected space is straight iff it is uniformly lo cally connected (ULC). It is easily seen that ULC spaces are stable under finite pro ducts. On the other hand the pro duct of tw o straight spaces is not necessarily straight. W e prov e that the pro duc t X × Y of tw o metric spaces is straight if a nd only if b oth X and Y a re straigh t and one of the follo wing conditions holds: (a) b oth X and Y are precompact; (b) b oth X and Y are lo cally connected; (c) one of the spaces is b oth precompact and lo cally connected. In particular, when X satisfies (c), t he pro duct X × Z is straigh t for eve ry straight space Z . Finally , w e c haracterize when infinite pro ducts of metric spaces are ULC and we completely solv e the problem o f straightness of infinite pro ducts of ULC spaces. http://arxi v.org/abs/0809.5080 A lessandr o B er a r duc ci, D ikr an Dikr anjan, Jan Pelant 3.4. Each second coun table ab elian gr oup is a subgroup of a second count- able divisible group. It is sho wn that eac h pseudonorm defined on a subgroup H of an ab elian group G can b e extended to a pseudonorm on G such that the densities of the obtained pseudometrizable top ological g roups coincide. W e deriv e fr om this that an y Hausdorff ω -b o unded group top ology on H can b e extended to a Hausdorff ω -b ounde d group top ology on G . In its turn this result implies tha t eac h separable metrizable ab elian gr o up H is a subgroup o f a separable metrizable divisible group G . This result essen tially relies on the Axiom o f Choice and is not true under the Axiom of Determinacy (whic h contradicts to the Axiom of Choice but implies the Coun table Axiom of Choice). http://arxi v.org/abs/0810.3030 T. Banakh, L. Zdomskyy 3.5. A dic hotom y for Borel fun ctions. The dic hotom y discov ered by Soleck i states that an y Baire class 1 function is either σ - con tin uous or ”includes” the P a w- lik o wski function P . The aim of this pa p e r is to g iv e an argumen t whic h is simpler S P M BULLETIN 26 (December 2008) 5 than the o r iginal pro of of Solec ki and giv es a stronger statemen t: a dic hotomy for all Borel functions. http://arxi v.org/abs/0810.1391 Mar cin Sab ok 3.6. More Results on Regular Ult rafilters in ZF C. W e pro v e, in ZFC alone, some new results on regularity and decomp osability o f ultrafilters. W e also list some problems, and furnish applications to top ological spaces and to extended logics. http://arxi v.org/abs/0810.5587 Paolo Lip p arini 3.7. Minimal pseudo compact group top ologies on free ab elian groups. A Hausdorff top o logical group G is minimal if ev ery contin uous isomorphism f : G → H b et w een G and a Hausdorff top olo gical gr o up H is op en. Significantly strengthening a 1981 result of Sto y anov w e pro v e the follow ing theorem: F or ev ery infinite minimal group G there exists a sequence { σ n : n ∈ N } of cardinals suc h that w ( G ) = sup { σ n : n ∈ N } and sup { 2 σ n : n ∈ N } ≤ | G | ≤ 2 w ( G ) , where w ( G ) is the weigh t of G . If G is an infinite minimal a b elian group, then either | G | = 2 σ for some cardinal σ , or w ( G ) = min { σ : | G | ≤ 2 σ } . Moreo v er, w e sho w t hat the equalit y | G | = 2 w ( G ) holds whenev er cf ( w ( G )) > ω . F or a cardinal κ we denote by F κ the f ree ab elian gr o up with κ man y g enerators. If F κ admits a pseudocompact g roup top ology , then κ ≥ c , where c is the cardinality of the con tin uum. W e sho w that the existen ce of a minimal pseudo compact group top ology on F c is equiv alen t to the Lusin’s Hyp othesis 2 ω 1 = c . F or κ > c , we prov e that F κ admits a (zero-dimensional) minimal pseudo compact group top ology if and only if F κ has b oth a minimal group top olo gy and a pseudo compact group top ology . If κ > c , then F κ admits a connected minimal pseudo compact gr o up top ology of w eigh t σ if and only if κ = 2 σ . Finally , we establish that no infinite torsion-fr ee ab elian group can b e equipp ed with a lo c ally connected minimal gro up top o lo gy . http://arxi v.org/abs/0811.0914 Dikr an Dikr anjan, Anna Gior dano B runo, Dmitri Sh a khmatov 3.8. W eakly infinite dimensional subsets of R N . The Con tin uum Hyp othesis implies an Erd¨ os-Sierpi ´ nski lik e duality b e tw e en the ideal of first category subsets of R N , and the ideal of coun table dimensional subsets of R N . The algebraic sum o f a Hurewicz subset - a dimension theoretic analog ue of Sierpinski sets and Lusin sets - of R N with an y strongly coun table dimensional subset of R N has first category . http://arxi v.org/abs/0811.3661 Liljana Babinkostova a nd Marion Sch e ep ers 6 S P M BULLETIN 26 (DECEMBER 2008) 3.9. Selections, Extensions and Collection wise Normality . W e demonstrate that t he classical Mic hael’s selection theorem f or l.s.c. mappings with a collec tion- wise normal domain can b e reduced only to compact-v alued mappings mo dulo the Do wk er’s extension theorem for suc h spaces. The tec hnique dev elop ed t o ac hiev e this result is applied to construct selections for set-v alued mappings whose p oint images are in completely metrizable absolute retracts. http://arxi v.org/abs/0811.3945 V alentin Gutev and Nar cisse R oland L oufouma Makala 3.10. Op enly factorizable spaces and compact extensions of top ological semigroups. W e prov e that the semigroup op eration of a top olog ical semigroup S extends to a contin uous semigroup op eration on its the Stone- ˇ Cec h compactification β S pro vided S is a pseudo compact op enly factorizable space, whic h means that eac h map f : S → Y to a second countable space Y can b e written as the comp osition f = g ◦ p of an op en map p : X → Z on to a second coun table space Z a nd a map g : Z → Y . W e presen t a sp ectral c haracterization of op enly factorizable spaces and establish some prop erties of suc h spaces. http://arxi v.org/abs/0811.4272 T ar as Banakh , Svetlana Dimitr ova 3.11. Em b edding t he bicyclic semigroup in to coun tably compact top ologi- cal semigroups. W e study algebraic and top o lo gical prop erties of top olog ical semi- groups containing a copy of the bicyclic semigroup C ( p, q ). W e pro v e that eac h t o p o- logical semigroup S with pseudo compact square con tains no dense copy of C ( p, q ). On the other hand, we construct a consisten t example of a T yc hono v coun tably compact semigroup con taining a copy o f C ( p, q ). http://arxi v.org/abs/0811.4276 T ar as Banakh , Svetlana Dimitr ova, Ole g Gutik 3.12. Building suitable sets for lo cally compact groups by means of con tin- uous selections. If a discrete subset S of a top o logical group G with the iden tity 1 generates a dense subgroup of G and S ∪ 1 is closed in G , then S is called a suit- able set for G . W e apply Mic hael’s selection theorem to offer a direct, self-con tained, purely t o p ological pro of of the result of Ho f mann and Morris on the existence of suit- able sets in lo cally compact gro ups. Our approac h uses o nly ele men tary facts fro m (top ological) group theory . http://arxi v.org/abs/0812.0489 Dmitri Shakhmatov 3.13. The near coherence of filters principle do es not imply t he filt er di- c hotom y principle. http://www. ams.org/journal-getitem?pii=S0002-9947-08-04806-X Heike Milden b er ger and Sahar on Sh e lah S P M BULLETIN 26 (December 2008) 7 4. Unsol ved p r oblems from e arlier issues Issue 1. Is  Ω Γ  =  Ω T  ? Issue 2. Is U fin ( O , Ω) = S fin (Γ , Ω) ? And if not, do es U fin ( O , Γ) imply S fin (Γ , Ω) ? Issue 4. Do es S 1 (Ω , T) imply U fin (Γ , Γ) ? Issue 5. Is p = p ∗ ? (Se e the de fi nition of p ∗ in that issue.) Issue 6. Do es ther e exist (in ZFC) an unc ountable s e t satisfying S fin ( B , B ) ? Issue 8. Do es X 6∈ NON ( M ) a n d Y 6∈ D imply that X ∪ Y 6∈ COF ( M ) ? Issue 9 (CH) . Is Split (Λ , Λ ) pr eserve d under finite unio n s? Issue 10. Is cov ( M ) = od ? (Se e the definition of od in that issue.) Issue 11. Do es S 1 (Γ , Γ) always c ontain an eleme nt of c ar dinality b ? Issue 12. Could ther e b e a Bair e metric sp ac e M of weight ℵ 1 and a p artition U of M in to ℵ 1 me ager sets w her e for e ach U ′ ⊂ U , S U ′ has the B air 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 dinality d such that al l finite p owers of X have Menger’ s pr op erty S fin ( O , O ) ? Issue 15. Can a Bor el non - σ -c om p act gr oup b e gene r ate d by a Hur ewicz subsp ac e? Issue 16 (MA) . Is ther e a n 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 fin ( O , Γ) that c an b e ma pp e d c ontinuously onto { 0 , 1 } N ? Issue 18 (CH) . Is ther e a Hur ewicz X such that X 2 is Menger b ut not Hur ewicz? Issue 19. D o es the Pytke ev pr op erty o f C p ( X ) imply that X has Menger’s pr op ert y? Issue 20. Do es eve ry her e ditarily Hur ewicz sp ac e satisfy S 1 ( B Γ , B Γ ) ? Issue 21 (CH) . Is ther e a R o thb 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 d e r Waer den ide al. A r e W -ultr afilters clos e d under pr o ducts? Issue 23. Is the δ - p r op erty e quivalent to the γ -pr op erty  Ω Γ  ? Previous issues. The previous issues of this bulletin are a v aila ble online at http://f ront.math. ucdavis.edu/search?&t=%22SPM+Bulletin%22 Con tributions. Ann ouncemen ts, d iscussions, and op en problems sh ould b e emailed to tsaban@m ath.biu.ac .il Subscription. T o receiv e this b ulletin (free) to your e-mailb ox, e-mail us.