SPM Bulletin 32

S P M BULLETIN ISSUE NUMBER 32: Octob er 2011 CE 1. Editor ’s note Dear F riends , 1. By no w probably most of y ou kno w that Misha Matv eev has passed a w ay recen t ly . I quote Ronnie Levy’s concise description, distributed via T op olo gy News : Misha Matv eev of George Mason Univ ersit y died of an apparent heart attac k. He was found in his office at a ppro ximately 1 AM on March 17. Misha w as a prolific researc her in general to po logy . His in terests included, but w ere not limited to, star-co v ering prop erties, selection principles (suc h as the Roth b erger and Menger pro perties), and mono- tonic co v ering prop erties. F rom Maddalena (Milena) Bonanzinga, I learned that Misha w a s thinking then on a researc h topic for his forthcoming visit to the Univ ersit y of Mes sina. It is sym bolic that suc h an excellen t mathematician passes a w a y while doing mathematics. I met Misha only once in p erson, in the 4 4th annual Spring T op ology and Dy- namics Conference, Mississippi State Univ ersity , Mississippi Stat e, USA, 2010, and quic kly noticed his h um ble and kind c haracter. Misha injected many fresh ideas and p erspectiv es in to the field, and I alw a ys had in mind the hop e that one day , I will collab orate with him on some of his new ideas concerning SPM. I ha v e recen tly visited the Univers it y of Messina, one of Misha’s fav orite places to visit. I was fortunate to collab orate, for m y fir st time, with Filipp o Cammaroto , Milena Bonanzinga, Bruno An tonio P a nsera, and Andrei Cataliato — all former collab orators of Mis ha. I w as also g iv en the opp ortunit y to make commen ts and suggestions f o r a nearly finis hed pap er of Misha with Bonanzinga. This is of some consolation for me. I w ould lik e to use this opp o rtunit y to thank m y friends f r om Messina for giving me this o ppor tunit y . 2. With a sharp c hange from bad news to go o d news, I am glad to inform y ou that the F ourth SPM W orkshop will tak e place on the coming June (2012). A preliminary announcemen t is give n b elo w. Please circulate this information among y our friends, studen t s, and colleagues. Bo az Tsab an , tsaban@math.biu.ac.il http://www. cs.biu.ac.il /~tsaban 1 2 S P M BULLETIN 32 (OCTOBER 2011) 2. IV Work shop on Co verings, Selections and Games in Topology Dear colleague, Next ye ar, our m utual friend L j ubi ˇ sa Koˇ cinac turns 65. F or t his o ccasion, I am organizing the IV Workshop on Cove rings, Sele ctions and Games in T op olo gy . Ljubisa Ko cinac initiated and started this series of conferences in 200 2, Lecce, Italy . The w orkshop will take place at the D epart ment of Mathematics, Seconda Univ ersit di Nap oli, Caserta, Italy . T en tativ e time ta ble: 25–30 June 2012 . (Arriv al: 25, w ork: 26–28/29 , excursion: 29, departure: 30.) 2.1. Organizing Committee. Agata Caserta, Giuseppe Di Maio (c hair), Dragan Djurcic. 2.2. Scien tific Committee. Alex ander V. Arhangelskii, Giusepp e Di Maio, Lju- bisa D.R. Ko cinac, Masami Sak ai, Marion Sc heepers, Boaz Tsaban, C. Guido, R. Lucc hetti. 2.3. F urther information. Eac h talk will last ab out 30 min utes. Of course, in a perio d of financial cuts w e do not kno w up to no w the suppo rt that w e can offer to participan ts, the amoun t of registration fee, etc. This is indeed a v ery preliminary rep ort, written to circulate this import a n t and happ y news. W e w ould appreciate y our forw arding this information to any one who ma y b e in t erested in attending this conference. On the b ehalf of the organizing committee, I hop e to see y ou in Caserta. Giusepp e Di Maio 3. Long announcements 3.1. Constructing univ ersally small subsets of a giv en pac king index in P olish groups. A subset of a P olish space X is called univers ally small if it belongs to eac h ccc σ - ideal with Borel base on X . Under CH in each uncoun table Ab elian Polish group G w e construct a univ ersally small subset A 0 ⊂ G suc h that | A 0 ∩ g A 0 | = c for eac h g ∈ G . F or eac h cardinal n um ber κ ∈ [5 , c + ] the set A 0 con tains a unive rsally small subset A of G with sharp pack ing index sup {|D | + : D ⊂ { g A } g ∈ G is disjoin t } equal to κ . http://arxi v.org/abs/11 06.2235 T ar as B anakh and Nadya Lyaskovska 3.2. Amenabili t y and Ramsey Theory. The purp ose of this article is to connect the notion of the amenability o f a discrete group with a new for m of structural R amsey theory . The Ramsey theoretic reform ulation of amenabilit y constitutes a considerable w eak ening of the Følner criterion. As a b y-pro duct, it will b e sho wn that in an y non S P M BULLETIN 32 (Octob er 2011) 3 amenable group G, there is a subset E of G suc h that no finitely additive probability measure on G measures all translates of E equally . http://arxi v.org/abs/11 06.3127 Justin T atch Mo or e 3.3. Hindman ’s Theorem, E llis’s Lemma, and Thompson’s group F . The purp ose of this article is to form ulate generalizations of Hindman’s Theorem and Ellis’s Lemma for non asso ciativ e group oids. These conjectures will be prov en to b e eq uiv alen t and it will b e sho wn that they imply the amenabilit y o f Thompson’s group F. In fact the amenabilit y o f F is equiv alen t to a finite form of the conjectured extension of Hindman’s Theorem. http://arxi v.org/abs/11 06.4735 Justin T atch Mo or e 3.4. A coun terexample in t he theory of D -spaces. Assuming ♦ , we construct a T 2 example of a hereditarily Lindel¨ of space of size ω 1 whic h is not a D -space. The example has the prop ert y that all finite p o w ers are also Lindel¨ of. http://arxi v.org/abs/11 06.5116 Daniel T. So ukup, Paul J. Szeptycki 3.5. Borel’s Conjecture in T op ological Groups. W e introduce a natural gener- alization of Borel’s Conjecture. F or each infinite cardinal n um b er κ , let BC κ denote this generalization. Then BC ℵ 0 is equiv alent to the classical Borel conjecture. W e obtain the follo wing consistency results: (1) If it is consisten t that there is a 1 -inaccess ible cardinal then it is consisten t that BC ℵ 1 holds. (2) If it is consisten t that BC ℵ 1 holds, then it is consisten t that there is an inac- cessible cardinal. (3) If it is consisten t that there is a 1- ina cce ssible cardinal with ω inaccessible cardinals ab o v e it, then ¬ BC ℵ ω + ( ∀ n < ω ) BC ℵ n is consisten t. (4) If it is consisten t that there is a 2-h uge cardinal, then it is consisten t that BC ℵ ω holds. (5) If it is consisten t that there is a 3-h uge cardinal, then it is consisten t that BC κ holds for a prop er class of cardinals κ of coun table cofinalit y . http://arxi v.org/abs/11 07.5383 F r e d Galvin and Mario n Sche ep ers 3.6. The top ology of ultrafilters as subspaces of 2 ω . Using the pro perty of b eing completely Ba ire, coun table dense homogeneit y and the p erfect set prop ert y we will b e able, under Mart in’s Axiom for coun table posets, to distinguish non- principal ultrafilters on ω up to ho meomorphism. Here, w e identify ultra filters with subpaces of 2 ω in the ob vious w a y . Using the same metho ds, still under Martin’s Axiom for coun table p osets, w e will construct a non-principal ultrafilter U ⊆ 2 ω suc h that U ω 4 S P M BULLETIN 32 (OCTOBER 2011) is coun table dense homogeneous. This consisten tly answ ers a question o f Hru ˇ s´ ak and Zamora Avil ´ es. Finally , w e will giv e some partial r esults ab out the relation of suc h top ological prop erties with the com binatorial prop erty of b eing a P-p oin t. http://arxi v.org/abs/11 08.2533 A ndr e a Me dini and David Milovich 3.7. Another not e on the c lass of paracompact spa ces whose pro duct with ev ery paracompact space is paracompact. Abstract. The pap er con tains the follo wing t w o results: (1) Let X b e a paracompact space and M b e a metric space suc h that X can b e em bedded in M ℵ 1 in suc h a wa y that the pro jections o f X on to initial coun t- ably man y co ordinates are closed. Th en the pro duct X × Y is paracompact for ev ery paracompact space Y if and only if the first play er of the G ( D C , X ) game, in tro duced b y T elgarsky , has a winning strategy . (2) If X is paracompact space , Y is a closed image of X and t he first pla y er of the G ( D C , X ) g ame has a winning strategy , then also the first pla y er of the G ( D C , Y ) game has a winning strategy . K. Alster 3.8. On paracompactness in the Cartesian pro ducts and the T elgarsky’s game. L et X b e a paracompact space and M b e a metric space suc h that X can b e em b edded in M ℵ 1 in suc h a wa y that the pro j ections p α : X → M α are closed at ev ery x ∈ X , a nd p − 1 α p α ( x ) is clop en for all x ∈ X . Then the pro duct X × Y is paracompact for ev ery paracompact space Y if and only if the first pla yer o f the G ( D C , X ) ga me, intro duced b y T elgarsky , has a winning strategy . K. Alster 3.9. Elemen t ary c hains and compact spaces wit h a small diagonal. It is a w ell known op en problem if, in ZFC, eac h compact space with a small diagonal is metrizable. W e explore prop erties of compact spaces with a small diagonal using elemen ta r y c hains o f submo dels. W e pro v e that ccc subspaces of suc h spaces hav e coun table π -we igh t. W e generalize a r esult of Gr uenhage a bo ut spaces whic h are metrizably fib ered. Finally w e discov er that if there is a Luzin set of reals, then ev ery compact space with a small diagonal will hav e many p oin ts of coun ta ble c haracter. http://arxi v.org/abs/11 09.1736 A lan D ow and Klaa s Pieter Hart 4. Shor t announcements 4.1. On large indecomposable B ana c h spaces. http://arxi v.org/abs/11 06.2916 Piotr Koszmide r S P M BULLETIN 32 (Octob er 2011) 5 4.2. A C ( K ) Banac h space whic h do es not ha v e the Sc hro eder-Bernstein prop ert y. http://arxi v.org/abs/11 06.2917 Piotr Koszmide r 4.3. Linearly Ordered F amilies of Baire 1 F unctions. http://arxi v.org/abs/11 09.5281 M´ arton Ele kes 4.4. Chains of Baire class 1 functions and v arious notions of sp ecial trees. http://arxi v.org/abs/11 09.5283 M´ arton Ele kes and Juris Stepr ans 4.5. T ransfinite Sequences of Con tinuous and Baire Class 1 F unctions. http://arxi v.org/abs/11 09.5284 M´ arton Ele kes and Kenne th Kunen 6 S P M BULLETIN 32 (OCTOBER 2011) 5. Unsol ved problems from earlier is sues 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 . D o es S 1 (Ω , T) imply U fin (Γ , Γ ) ? Issue 5 . Is p = p ∗ ? (See the definition of p ∗ in that issue.) Issue 6 . D o es there exist (in ZFC) an uncountable set satisfying S fin ( B , B )? Issue 8 . D o es X 6∈ NON ( M ) and Y 6∈ D imply that X ∪ Y 6∈ COF ( M )? Issue 9 (CH) . Is Split (Λ , Λ) preserv ed under finite unions? Issue 10 . Is cov ( M ) = od ? (See t he definition of o d in that issue.) Issue 12 . Could there b e a Baire metric space M of w eigh t ℵ 1 and a partitio n U of M into ℵ 1 meager sets where for eac h U ′ ⊂ U , S U ′ has the Baire prop ert y in M ? Issue 14 . Do es there exist (in ZF C) a set of reals X of car dinality d suc h that all finite p o w ers of X ha v e Menger’s prop ert y S fin ( O , O )? Issue 15 . Can a Borel no n- σ -compact group be generated b y a Hurewicz subspace? Issue 16 (MA) . Is there X ⊆ R of cardinalit y con tin uum, satisfying S 1 ( B Ω , B Γ )? Issue 17 (CH) . Is there a t o tally imp erfect X satisfying U fin ( O , Γ) that can b e mapp ed con tin uously on to { 0 , 1 } N ? Issue 18 (CH) . Is there a Hurewicz X suc h that X 2 is Menger but not Hurewicz? Issue 19 . D o es the Pytk eev prop ert y of C p ( X ) imply that X has Menger’s prop erty ? Issue 20 . D o es ev ery hereditarily Hurewicz space satisfy S 1 ( B Γ , B Γ )? Issue 21 (CH) . Is there a R oth berger- b ounded G ≤ Z N suc h that G 2 is not Menger- b ounded? Issue 22 . Let W b e the v an der W aerden ideal. Are W - ultrafilters closed under pro ducts? Issue 23 . Is the δ -prop ert y equiv alent to the γ - prop ert y  Ω Γ  ? Previous issues. T he previous issues of this bu lletin are a v ailable online at http://f ront.math.uc davis.edu/search?&t=%22SPM+Bulletin%22 Con tributions. Announcemen ts, d iscussions, and op en problems should b e emailed to tsaban@m ath.biu.ac.i l Subscription. T o r ece iv e this bulletin (fr ee) to your e-mailbox, e-mail us.