Problems from Topology Proceedings Edited by Elliott Pearl Topology Atlas, Toronto, 2003 Topology Atlas Toronto, Ontario, Canada http://at.yorku.ca/topology/ [email protected] Cataloguing in Publication Data Problems from topology proceedings / edited by Elliott Pearl. vi, 216 p. Includes bibliographical references. ISBN 0-9730867-1-8 1. Topology—Problems, exercises, etc. I. Pearl, Elliott. II. Title. Dewey 514 20 LC QA611 MSC (2000) 54-06 Copyright (cid:13)c 2003 Topology Atlas. All rights reserved. Users of this publication are permitted to make fair use of the material in teaching, research and reviewing. No part of this publication may be distributed for commercial purposes without the prior permission of the publisher. ISBN 0-9730867-1-8 ProducedNovember2003. Preliminaryversionsofthispublicationweredistributed on the Topology Atlas website. This publication is available in several electronic formats on the Topology Atlas website. Produced in Canada Contents Preface ........................................................................v Contributed Problems in Topology Proceedings .................................1 Edited by Peter J. Nyikos and Elliott Pearl. Classic Problems .............................................................69 By Peter J. Nyikos. New Classic Problems ........................................................91 ContributionsbyZ.T.Balogh,S.W.Davis,A.Dow,G.Gruenhage,P.J.Nyikos, M.E. Rudin, F.D. Tall, S. Watson. Problems from M.E. Rudin’s Lecture notes in set-theoretic topology ..........103 By Elliott Pearl. ProblemsfromA.V.Arhangel(cid:48)ski˘ı’sStructureandclassificationoftopologicalspaces and cardinal invariants ......................................................123 By A.V. Arhangel(cid:48)ski˘ı and Elliott Pearl. A note on P. Nyikos’s A survey of two problems in topology ..................135 By Elliott Pearl. A note on Open problems in infinite-dimensional topology ....................139 By Elliott Pearl. Non-uniformlycontinuoushomeomorphismswithuniformlycontinuousiterates141 By W.R. Utz. Questions on homeomorphism groups of chainable and homogeneous continua 143 By Beverly L. Brechner. Some problems in applied knot theory and geometric topology ...............145 Contibutions by D.W. Sumners, J.L. Bryant, R.C. Lacher, R.F. Williams, J. Vieitez. Problems from Chattanooga, 1996 ...........................................153 Contributions by W.W. Comfort, F.D. Tall, D.J. Lutzer, C. Pan, G. Gruenhage, S. Purisch, P.J. Nyikos. iii iv CONTENTS Problems from Oxford, 2000 .................................................155 Contributions by A.V. Arhangel(cid:48)ski˘ı, S. Antonyan, K.P. Hart, L. Ludwig, M. Matveev, J.T. Moore, P.J. Nyikos, S.A. Peregudov, R. Pol, J.T. Rogers, M.E. Rudin, K. Shankar. Continuum theory problems .................................................165 By Wayne Lewis. Problems in continuum theory ...............................................183 By Janusz R. Prajs. The plane fixed-point problem ...............................................191 By Charles L. Hagopian. On an old problem of Knaster ...............................................195 By Janusz J. Charatonik. Means on arc-like continua ..................................................197 By Janusz J. Charatonik. Classification of homogeneous continua ......................................201 By James T. Rogers, Jr. Preface Ihopethatthiscollectionofproblemswillbeaninterestingandusefulresource for researchers. ThisvolumeconsistsofmaterialfromtheProblemSection ofthejournalTopol- ogy Proceedings originally collected and edited by Peter Nyikos and subsequently edited by Elliott Pearl for this publication. This volume also contains some other well-known problems lists that have appeared in Topology Proceedings. Some warnings and acknowledgments are in order. Ihavemadesomechangestotheoriginalsourcematerial. Theoriginalwording of the problems is mostly intact. I have rewritten many of the solutions, originally contributed by Peter Nyikos, in order to give a more uniform current presenta- tion. I have contributed some new reports of solutions. I have often taken wording from abstracts of articles and from reviews (Mathematical Reviews and Zentral- blatt MATH) without specific attribution. In cases where the person submitting a problem was not responsible for first asking the problem, I have tried to provide a reference to the original source of the problem. Regrettably,Icannotguaranteethatallassumptionsregardinglowerseparation axioms have been reported accurately from the original sources. Some portions of this volume have been checked by experts for accuracy of updates and transcription. I have corrected some typographical errors from the original source material. I have surely introduced new typographical errors during the process of typesetting the original documents. The large bibliography sections were prepared using some of the features of MathSciNet and Zentralblatt MATH. No index has been prepared for this volume. This volume is distributed in several electronic formats some of which are searchable with viewing applications. IthankYorkUniversityforaccess toonlineresources. IthankYorkUniversity and the University of Toronto for access to their libraries. IthankDmitriShakhmatovandStephenWatsonfordevelopingTopologyAtlas as a research tool for the community of topologists. IthankGaryGruenhage,JohnC.Mayer,PeterNyikos,MuratTuncaliandthe editorialboardofTopology Proceedings forpermissiontoreprintthismaterialfrom Topology Proceedings and to proceed with this publishing project. I thank Peter Nyikos for maintaining the problem section for twenty years. The material from Mary Ellen Rudin’s Lecture notes in set-theoretic topology are distributed with the permission of the American Mathematical Society. A.V. Arhangel(cid:48)ski˘ı has given his permission to include in this volume the ma- terial from his survey article Structure and classification of topological spaces and cardinal invariants. v vi PREFACE The chapter Problems in continuum theory consists of material from the ar- ticle Several old and new problems in continuum theory by Janusz J. Charatonik and Janusz R. Prajs and from the website that they maintain. This material is distributed with the permission of the authors. The original essay The plane fixed-point problem by Charles Hagopian is dis- tributed with the permission of the author. The original essays On an old problem of Knaster and Means on arc-like cont- inua by Janusz J. Charatonik are distributed with the permission of the author. The original essay Expansive diffeomorphisms on 3-manifolds by Jos´e Vieitez is distributed with the permission of the author. I thank many people for contributing solutions and checking portions (small and large) of this edition: A.V. Arhangel(cid:48)ski˘ı, Christoph Bandt, Paul Bankston, Carlos Borges, Raushan Buzyakova, Dennis Burke, Max Burke, Janusz Chara- tonik, Chris Ciesielski, Sheldon Davis, Alan Dow, Alexander Dranishnikov, Todd Eisworth, Gary Gruenhage, Charles Hagopian, K.P. Hart, Oleg Okunev, Piotr Koszmider, Paul Latiolais, Arkady Leiderman, Ronnie Levy, Wayne Lewis, Lew Ludwig,DavidLutzer,MikhailMatveev,JustinMoore,GrzegorzPlebanek,Janusz Prajs, Jim Rogers, Andrzej Roslanowski, Mary Ellen Rudin, Masami Sakai, John Schommer, Dmitri Shakhmatov, Weixiao Shen, Alex Shibakov, Petr Simon, Greg Swiatek, Paul Szeptycki, Frank Tall, Gino Tironi, Artur Tomita, Vassilis Tzannes, Vladimir Uspenskij, W.R. Utz, Stephen Watson, Bob Williams, Scott Williams. I welcome any corrections or new information on solutions. Indeed, I hope to use your contributions to prepare a revised edition of this volume. Elliott Pearl November, 2003 Toronto, ON, Canada [email protected] Contributed Problems in Topology Proceedings Editor’s notes. This is a collection of problems and solutions that appeared in the problem section of the journal Topology Proceedings. The problem section was edited by Peter J. Nyikos for twenty years from the journal’s founding in 1976. John C. Mayer began editing the problem section with volume 21 in 1996. In this version, thenotesandsolutionscollectedthroughoutthetwenty-sevenyearhistory of the problem section have been updated with current information. Conventions and notation. The person who contributed each problem is men- tioned in parentheses after the respective problem number. This is not necessarily the person who first asked the problem. Usually there is a reference to a relevant article in Topology Proceedings. Sometimes there is a reference to other relevant articles. There are a few discontinuities in the numbering of the problems. Some problems have been omitted. A. Cardinal invariants A1. (K. Kunen [226]) Does MA+¬CH imply that there are no L-spaces? Notes. Kunen [226] showed that MA+¬CH implies that there are no Luzin spaces (hence there are no Souslin lines either). A Luzin space is an uncountable Hausdorff space in which every nowhere dense subset is countable and which has at most countably many isolated points. Solution. U. Abraham and S. Todorˇcevi´c [2] showed that the existence of an L-space is consistent with MA+¬CH. A3. (E. van Douwen [98]) Is every point-finite open family in a c.c.c. space σ- centered (i.e., the union of countably many centered families)? Solution. No (Ortwin F¨orster). J. Stepra¯ns and S. Watson [348] described a subspace of the Pixley-Roy space on the irrationals that is a first countable c.c.c. space which does not have a σ-linked base. A4. (E. van Douwen [228, Problem 391]) For which κ > ω is there a compact homogeneous Hausdorff space X with c(X)=κ? Notes. This is known as van Douwen’s problem. Here c(X) denotes cellularity, i.e., the supremum of all possible cardinalities of collections of disjoint open sets. There is an example with c(X)=2ℵ0. A5. (A.V. Arhangel(cid:48)ski˘ı) Let c(X) denote the cellularity of X. Does there exist a space X such that c(X2)>c(X)? Solution. Yes (S. Todorˇcevi´c [363]). Peter J. Nyikos and Elliott Pearl, Contributed Problems in Topology Proceedings, Problems from Topology Proceedings, Topology Atlas, 2003, pp. 1–68. 1 2 contributed problems A6.(A.V.Arhangel(cid:48)ski˘ı)Letd(X)denotethedensityofX andlett(X)denotethe tightness of X, ω·min{κ : (∀A ⊂ X)(∀x ∈ clA)(∃B ⊂ A)x∈clB,|B|≤κ}. Does there exist a compact space X such that c(X)=t(X)<d(X). Yes, if CH or there exists a Souslin line. A7. (T. Przymusin´ski) Does there exist for every cardinal λ an isometrically uni- versal metric space of weight λ? Yes, if GCH. A8.(V.Saks[325])AsetC ⊂βω\ω isacluster set ifthereexistx∈βω\ω anda sequence {x :n∈ω} in βω such that C ={D ∈βω\ω :x=D\limx ,{n:x (cid:54)= n n n x} ∈ D}. Here a point of βω is identified with the ultrafilter on ω that converges to it. Is it a theorem of ZFC that βω\ω is not the union of fewer than 2c cluster sets? Notes. See especially [325, Theorem 3.1]. A9. (E. van Douwen [102]) If G is an infinite countably compact group, is |G|ω = |G|? Yes, if GCH. Solution. No is consistent. A. Tomita [365] showed that there is a model of CH in which there is a countably compact group of cardinality ℵ . ω A10. (E. van Douwen [103]) Is the character, or hereditary Lindelo¨f degree, or spread, equaltotheweightforacompactF-space? foracompactbasicallydiscon- nected space? Notes. Yes,forcompactextremallydisconnectedspacesbyaresultofB.Balcar and F. Franˇek [16]. A11. (G. Grabner [152]) Suppose that X is a wrb space. Does χ(X)=t(X)? Notes. A space is wrb if each point has a local base which is the countable union of Noetherian collections of subinfinite rank. A12. (P. Nyikos [280]) Does there exist, for each cardinal κ, a first countable, locally compact, countably compact space of cardinality ≥κ? Notes. Yes if (cid:3) and cf[κ]ω = κ+ for all singular cardinals of countable cofi- κ nality (P. Nyikos), hence yes if the Covering Lemma holds over the Core Model. A negativeanswerinsomemodelwouldthusimplythepresenceofinnermodelswith a proper class of measurable cardinals. An affirmative answer is compatible with any possible cardinal arithmetic (S. Shelah). A13. (E. van Douwen) Let exp X stand for the least cardinal κ (if it exists) such Y that X can be embedded as a closed subspace in a product of κ copies of Y. Does there exist an N-compact space X such that exp X (cid:54)=exp X? N R Notes. Such a space cannot be strongly zero-dimensional. A14.(E.vanDouwen)IseverycompactHausdorffspaceacontinuousimageofsome zero-dimensional compact space of the same cardinality? of the same character? The answer is well-known to be yes for weight. A15. (E. van Douwen) Is there for each κ ≥ ω a (preferably homogeneous, or even groupable) hereditarily paracompact (or hereditarily normal) space X with w(X)=κ and |X|=2κ? Notes. Yes to all questions if 2κ = κ+. Also, w(X) ≤ κ < |X| is always possible. B. GENERALIZED METRIC SPACES AND METRIZATION 3 A16. (E. van Douwen) Is there for each κ≥ω a homogeneous compact Hausdorff space X with χ(X)=κ and w(X)=2κ? Or is ω the only value of κ for which this is true? A17. (E. van Douwen [97]) Is there always a regular space without a Noetherian base? (Noetherian: no infinite ascending chains.) Notes. For any ordinal α, the space α has a Noetherian base if and only if α+1 does not contain a strongly inaccessible cardinal. A. Tamariz-Mascaru´a and R.G. Wilson [358] showed that there is a T space without a Noetherian base. 1 A19. (E. van Douwen) Is a first countable T space normal if every two disjoint 1 closed sets of size ≤c can be put into disjoint open sets? Solution. If there is no counterexample then there is an inner model with a proper class of measurable cardinals. But if the consistency of a supercompact cardinal is assumed, then an affirmative answer is consistent (I. Juha´sz). A20. (A. Garc´ıa-Maynez [139]) Let X be a T -space and let X be an infinite 3 cardinal. Assume the pluming degree of X is ≤ λ. Is it true that every compact subsetofX liesinacompactsetwhichhasalocalbasisforitsneighborhoodsystem consisting of at most λ elements? A21. (B. Shapirovski˘ı [311]) Let A be a subset of a space X and let x ∈ A(cid:48). A(cid:48) denotes the derived set. Define the accessibility number a(x,A) to be min{|B| : B ⊂A,x∈B(cid:48)}. Define t (x,X) to be sup{a(x,F):F is closed,x∈F(cid:48)}. As usual, c define t(x,X) as sup{a(x,A) : x ∈ A(cid:48)}. Can we ever have t (x,X) < t(x,X) in a c compact Hausdorff space? Notes. No, for c.c.c. compact spaces under GCH [337]. A22.(D.Shakhmatov[336])Assumethatτ isaTychonoff[resp.Hausdorff,regular, T etc.] homogeneous topology on a set X. Are there Tychonoff [resp. Hausdorff, 1 regular, T1 etc.] homogeneous topologies τ∗ and τ∗ on X such that τ∗ ⊂ τ ⊂ τ∗, w(X,τ∗)≤nw(X,τ) and w(X,τ∗)≤nw(X,τ)? Notes. For background on this problem for the case of topological groups and other topological algebras, see papers by A.V. Arhangel(cid:48)ski˘ı in [9] where the “left half” is achieved in the category of topological groups and continuous homeomor- phisms. In (D. Shakhmatov [334]) this is extended to many other categories. In (V. Pestov and D. Shakhmatov [294]), the right half is shown to fail in the cate- goriesoftopologicalgroupsandtopologicalvectorspaces,forcountablenetweight; in the latter case, R∞ provides a counterexample. B. Generalized metric spaces and metrization B1. (T. Przymusin´ski [307]) Can each normal (or metacompact) Moore space of weight ≤c be embedded into a separable Moore space? Notes. Under CH, the answer is yes even if “normal” and “metacompact” are completely dropped (E. van Douwen and T. Przymusin´ski). Solution. B. Fitzpatrick, J.W. Ott and G.M. Reed asked “Can each Moore spacewithweightatmostcbeembeddedinaseparableMoorespace?” Theanswer tothisquestionisindependentofZFC(E.vanDouwenandT.Przymusin´ski[110]). B2. (D. Burke [54]) Is the perfect image of a quasi-developable space also quasi- developable? Solution. Yes (D. Burke [56]). 4 contributed problems B3.(K.AlsterandP.Zenor[6])Iseverylocallyconnectedandlocallyrim-compact normal Moore space metrizable? Solution. Yes (P. Zenor [72]). B4. (D. Burke and D. Lutzer [61]) Must a strict p-space with a G -diagonal be δ developable (equivalently, θ-refinable (=submetacompact))? Notes. It was erroneously announced in [61] that J. Chaber had given an affirmative answer; however, Chaber did not claim to settle the question except in the cases where the space is locally compact or locally second countable [69, 71]. Solution. Yes,becauseeverystrictp-spaceissubmetacompact(S.L.Jiang[202]). B5.(H.Wicke[381])Iseverymonotonicallysemi-stratifiablehereditarilysubmeta- compact space semi-stratifiable? B6. (H. Wicke [381]) Is every monotonic β-space which is hereditarily submeta- compact a β-space? B7. (H. Wicke [381]) Does every primitive q-space with a θ-diagonal have a prim- itive base? Notes. R. Ruth [323] proved that a space has a primitive base if and only if it is both a θ-space and a primitive σ-space. Also, a primitive σ-space with a θ-diagonal has a primitive diagonal. B8. (C.E. Aull [13]) For all base axioms such that countably compact regular + base axiom ⇒ metrizable, is it true that regular + β + collectionwise normal + base axiom ⇒ metrizable? In particular, what about quasi-developable spaces, or those with δθ-bases or point-countable bases? B9. (C.E. Aull [32, 13]) Is every space in the class MOBI quasi-developable? B10.(C.E.Aull[13])Iseveryspacewithaσ-locallycountablebasequasi-develop- able? Notes. D.Burke[55, p.25]showedthatasubmetacompact(=θ-refinable)reg- ular space with a σ-locally countable base is developable. Thus Problems B10 and B11haveaffirmativeanswerswheresubmetacompactregularspacesareconcerned. D. Burke [56] showed that the class of spaces with primitive bases is closed under perfect maps. J. Kofner [221] showed that the class of quasi-metrizable spaces is also closed under perfect maps. Also, H.R. Bennett’s example of a paracompact, nonmetrizable space in MOBI [32] shows that the class of spaces with σ-locally countable bases is not preserved under compact open mappings. B11. (C.E. Aull [13]) Is every collectionwise normal space with a σ-locally count- able base metrizable (equivalently, paracompact)? B12. (C.E. Aull [13]) Is every first countable space with a weak uniform base (WUB) quasi-developable? Notes. A base B for a space X is a (weakly) uniform base if for each x ∈ X and each infinite subcollection H of B, each member of which contains X, H is a local base for X (resp. H = {x}). A T space has a uniform base iff it is a 3 (cid:84) metacompact Moore space (P.S. Alexandroff, R.W. Heath). B13. (C.E. Aull [13]) Does every developable space with a WUB and without isolated points have a uniform base? Equivalently, is it metacompact?