Open Problems in Topology

OPEN PROBLEMS IN TOPOLOGY Edited by Jan van Mill Free University Amsterdam, The Netherlands George M. Reed St. Edmund Hall Oxford University Oxford, United Kingdom 1990 NORTH-HOLLAND AMSTERDAM • NEW YORK • OXFORD • TOKYO Introduction This volume grew from a discussion by the editors on the difficulty of finding good thesis problems for graduate students in topology. Although at any given time we each had our own favorite problems, we acknowledged the need to offer students a wider selection from which to choose a topic peculiar to their interests. One of us remarked, “Wouldn’t it be nice to have a book of current unsolved problems always available to pull down from the shelf?” The other replied, “Why don’t we simply produce such a book?” Two years later and not so simply, here is the resulting volume. The intent is to provide not only a source book for thesis-level problems but also a chal- lenge to the best researchers in the field. Of course, the presented problems still reflect to some extent our own prejudices. However, as editors we have tried to represent as broad a perspective of topological research as possible. The topics range over algebraic topology, analytic set theory, continua theory, digital topology, dimension theory, domain theory, function spaces, generalized metric spaces, geometric topology, homogeneity, infinite-dimensional topology, knot theory, ordered spaces, set-theoretic topology, topological dynamics, and topological groups. Application areas include computer science, differential systems, functional analysis, and set theory. The authors are among the world leaders in their respective research areas. A key component in our specification for the volume was to provide current problems. Problems become quickly outdated, and any list soon loses its value if the status of the individual problems is uncertain. We have addressed this issue by arranging a running update on such status in each volume of the journal TOPOLOGY AND ITS APPLICATIONS. This will be useful only if the reader takes the trouble of informing one of the editors about solutions of problems posed in this book. Of course, it will also be sufficient to inform the author(s) of the paper in which the solved problem is stated. We plan a complete revision to the volume with the addition of new topics and authors within five years. To keep bookkeeping simple, each problem has two different labels. First, the label that was originally assigned to it by the author of the paper in which it is listed. The second label, the one in the outer margin, is a global one and is added by the editors; its main purpose is to draw the reader’s attention to the problems. A word on the indexes: there are two of them. The first index contains terms that are mentioned outside the problems, one may consult this index to find information on a particular subject. The second index contains terms that are mentioned in the problems, one may consult this index to locate problems concerning ones favorite subject. Although there is considerable overlap between the indexes, we think this is the best service we can offer the reader. v vi Introduction The editors would like to note that the volume has already been a suc- cess in the fact that its preparation has inspired the solution to several long- outstanding problems by the authors. We now look forward to reporting solutions by the readers. Good luck! Finally, the editors would like to thank Klaas Pieter Hart for his valu- able advice on TEXandMETAFONT. They also express their gratitude to Eva Coplakova for composing the indexes, and to Eva Coplakova and Geertje van Mill for typing the manuscript. Jan van Mill George M. Reed Table of Contents Introduction................................... v Contents..................................... vii I Set Theoretic Topology 1 Dow’s Questions by A. Dow ................................. 5 Stepr¯ans’ Problems by J. Steprans ............................... 13 1.TheTorontoProblem.......................... 15 2.Continuouscolouringsofclosedgraphs................ 16 3. Autohomeomorphisms of the Cech-Stoneˇ Compactification on the Integers.................................. 17 References................................... 20 Tall’s Problems by F. D. Tall ................................ 21 A.NormalMooreSpaceProblems..................... 23 B. Locally Compact Normal Non-collectionwise Normal Problems . 24 C.CollectionwiseHausdorffProblems................... 25 D.WeakSeparationProblems....................... 26 E. Screenable and Para-LindelöfProblems................ 28 F.ReflectionProblems........................... 28 G. Countable Chain Condition Problems . 30 H.RealLineProblems........................... 31 References................................... 32 Problems I wish I could solve by S. Watson ................................ 37 1.Introduction............................... 39 2. Normal not Collectionwise Hausdorff Spaces . 40 3. Non-metrizable Normal Moore Spaces . 43 4.LocallyCompactNormalSpaces.................... 44 5.CountablyParacompactSpaces.................... 47 6.CollectionwiseHausdorffSpaces.................... 50 7. Para-LindelöfSpaces.......................... 52 8.DowkerSpaces.............................. 54 9.ExtendingIdeals............................. 55 vii viii Contents 10.Homeomorphisms............................ 58 11.Absoluteness............................... 61 12.Complementation............................ 63 13.OtherProblems............................. 68 References................................... 69 Weiss’ Questions by W. Weiss ................................ 77 A.ProblemsaboutBasicSpaces...................... 79 B. Problems about Cardinal Invariants . 80 C.ProblemsaboutPartitions....................... 81 References................................... 83 Perfectly normal compacta, cosmic spaces, and some partition problems by G. Gruenhage .............................. 85 1. Some Strange Questions . 87 2.PerfectlyNormalCompacta...................... 89 3. Cosmic Spaces and Coloring Axioms . 91 References................................... 94 Open Problems on βω by K. P. Hart and J. van Mill ...................... 97 1.Introduction............................... 99 2.DefinitionsandNotation........................ 99 3.Answerstoolderproblems.......................100 4.Autohomeomorphisms..........................103 5. Subspaces . 105 6.IndividualUltrafilters..........................107 7.Dynamics,AlgebraandNumberTheory................109 8.Other...................................111 9.UncountableCardinals.........................118 References...................................120 On first countable, countably compact spaces III: The problem of obtain- ing separable noncompact examples by P. Nyikos ................................127 1. Topological background . 131 2. The γN construction...........................132 3. The Ostaszewski-van Douwen construction. 134 4. The “dominating reals” constructions. 140 5.Linearlyorderedremainders......................146 6.Difficultieswithmanifolds.......................152 7.IntheNoMan’sLand..........................157 References...................................159 Contents ix Set-theoretic problems in Moore spaces by G. M. Reed ...............................163 1.Introduction...............................165 2.Normality.................................165 3.ChainConditions............................169 4.ThecollectionwiseHausdorffproperty.................172 5.Embeddingsandsubspaces.......................172 6. The point-countable base problem for Moore spaces . 174 7.Metrization................................174 8. Recent solutions . 176 References...................................177 Some Conjectures by M. E. Rudin ..............................183 Small Uncountable Cardinals and Topology by J. E. Vaughan. With an Appendix by S. Shelah . 195 1.Definitionsandset-theoreticproblems.................197 2.Problemsintopology..........................206 3. Questions raised by van Douwen in his Handbook article . 209 References...................................212 II General Topology 219 A Survey of the Class MOBI by H. R. Bennett and J. Chaber .....................221 Problems on Perfect Ordered Spaces by H. R. Bennett and D. J. Lutzer ....................231 1.Introduction...............................233 2. Perfect subspaces vs. perfect superspaces . 233 3. Perfect ordered spaces and σ-discrete dense sets . 234 4. How to recognize perfect generalized ordered spaces . 235 5. A metrization problem for compact ordered spaces . 235 References...................................236 The Point-Countable Base Problem by P. J. Collins, G. M. Reed and A. W. Roscoe ............237 1.Origins..................................239 2.Thepoint-countablebaseproblem...................242 3.Postscript:ageneralstructuringmechanism.............247 References...................................249 xContents Some Open Problems in Densely Homogeneous Spaces by B. Fitzpatrick, Jr. and Zhou Hao-xuan ................251 1.Introduction...............................253 2.SeparationAxioms............................253 3. The Relationship between CDH and SLH . 254 4. Open Subsets of CDH Spaces. 255 5.LocalConnectedness...........................256 6.CartesianProducts...........................256 7.Completeness...............................257 8.ModificationsoftheDefinitions.....................257 References...................................257 Large Homogeneous Compact Spaces by K. Kunen ................................261 1.TheProblem...............................263

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