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 challenge 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 Problems become quickly outdated, and any list soon loses its value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 success in the fact that its preparation has inspired the solution to several longoutstanding 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 valuable advice on TEX and METAFONT. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steprāns’ Problems
by J. Steprans . . . . . . . . . . . . . . . .
1. The Toronto Problem . . . . . . . . . . .
2. Continuous colourings of closed graphs .
3. Autohomeomorphisms of the Čech-Stone
Integers . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . .
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Compactification
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Tall’s Problems
by F. D. Tall . . . . . . . . . . . . . . . . . . . . . . .
A. Normal Moore Space Problems . . . . . . . . . . . .
B. Locally Compact Normal Non-collectionwise Normal
C. Collectionwise Hausdorff Problems . . . . . . . . . .
D. Weak Separation Problems . . . . . . . . . . . . . .
E. Screenable and Para-Lindelöf Problems . . . . . . .
F. Reflection Problems . . . . . . . . . . . . . . . . . .
G. Countable Chain Condition Problems . . . . . . . .
H. Real Line Problems . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Problems
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21
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32
Problems I wish I could solve
by S. Watson . . . . . . . . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . .
2. Normal not Collectionwise Hausdorff Spaces
3. Non-metrizable Normal Moore Spaces . . . .
4. Locally Compact Normal Spaces . . . . . . .
5. Countably Paracompact Spaces . . . . . . .
6. Collectionwise Hausdorff Spaces . . . . . . .
7. Para-Lindelöf Spaces . . . . . . . . . . . . .
8. Dowker Spaces . . . . . . . . . . . . . . . . .
9. Extending Ideals . . . . . . . . . . . . . . . .
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viii
10. Homeomorphisms
11. Absoluteness . . .
12. Complementation
13. Other Problems .
References . . . . . . .
Contents
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58
61
63
68
69
Weiss’ Questions
by W. Weiss . . . . . . . . . . . . . .
A. Problems about Basic Spaces . . . .
B. Problems about Cardinal Invariants
C. Problems about Partitions . . . . .
References . . . . . . . . . . . . . . . . .
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77
79
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81
83
Perfectly normal compacta, cosmic spaces, and some partition
by G. Gruenhage . . . . . . . . . . . . . . . . . . . . . . .
1. Some Strange Questions . . . . . . . . . . . . . . . . . .
2. Perfectly Normal Compacta . . . . . . . . . . . . . . .
3. Cosmic Spaces and Coloring Axioms . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
problems
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85
87
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94
Open Problems on βω
by K. P. Hart and J. van Mill . . . . . . .
1. Introduction . . . . . . . . . . . . . . . .
2. Definitions and Notation . . . . . . . . .
3. Answers to older problems . . . . . . . .
4. Autohomeomorphisms . . . . . . . . . . .
5. Subspaces . . . . . . . . . . . . . . . . . .
6. Individual Ultrafilters . . . . . . . . . . .
7. Dynamics, Algebra and Number Theory .
8. Other . . . . . . . . . . . . . . . . . . . .
9. Uncountable Cardinals . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . .
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97
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120
On first countable, countably compact spaces III: The problem of obtaining separable noncompact examples
by P. Nyikos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Topological background . . . . . . . . . . . . . . . . . . . . . . . .
2. The γN construction. . . . . . . . . . . . . . . . . . . . . . . . . .
3. The Ostaszewski-van Douwen construction. . . . . . . . . . . . . .
4. The “dominating reals” constructions. . . . . . . . . . . . . . . . .
5. Linearly ordered remainders . . . . . . . . . . . . . . . . . . . . .
6. Difficulties with manifolds . . . . . . . . . . . . . . . . . . . . . .
7. In the No Man’s Land . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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127
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Contents
ix
Set-theoretic problems in Moore spaces
by G. M. Reed . . . . . . . . . . . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . . . .
2. Normality . . . . . . . . . . . . . . . . . . . . . . . .
3. Chain Conditions . . . . . . . . . . . . . . . . . . .
4. The collectionwise Hausdorff property . . . . . . . .
5. Embeddings and subspaces . . . . . . . . . . . . . .
6. The point-countable base problem for Moore spaces
7. Metrization . . . . . . . . . . . . . . . . . . . . . . .
8. Recent solutions . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
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163
165
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177
Some Conjectures
by M. E. Rudin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Small Uncountable Cardinals and Topology
by J. E. Vaughan. With an Appendix by S. Shelah . . . .
1. Definitions and set-theoretic problems . . . . . . . . . . .
2. Problems in topology . . . . . . . . . . . . . . . . . . . .
3. Questions raised by van Douwen in his Handbook article
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II
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General Topology
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209
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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 . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . . . .
2. Perfect subspaces vs. perfect superspaces . . . . . .
3. Perfect ordered spaces and σ-discrete dense sets . .
4. How to recognize perfect generalized ordered spaces
5. A metrization problem for compact ordered spaces .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
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231
233
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234
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235
236
The Point-Countable Base Problem
by P. J. Collins, G. M. Reed and A. W. Roscoe
1. Origins . . . . . . . . . . . . . . . . . . . . . .
2. The point-countable base problem . . . . . . .
3. Postscript: a general structuring mechanism .
References . . . . . . . . . . . . . . . . . . . . . . .
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237
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249
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x
Contents
Some Open Problems in Densely Homogeneous Spaces
by B. Fitzpatrick, Jr. and Zhou Hao-xuan . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . .
2. Separation Axioms . . . . . . . . . . . . . . . . .
3. The Relationship between CDH and SLH . . . .
4. Open Subsets of CDH Spaces. . . . . . . . . . .
5. Local Connectedness . . . . . . . . . . . . . . . .
6. Cartesian Products . . . . . . . . . . . . . . . .
7. Completeness . . . . . . . . . . . . . . . . . . . .
8. Modifications of the Definitions. . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
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251
253
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256
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257
257
257
Large Homogeneous Compact Spaces
by K. Kunen . . . . . . . . . . .
1. The Problem . . . . . . . . . .
2. Products . . . . . . . . . . . .
References . . . . . . . . . . . . . .
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261
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270
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Some Problems
by E. Michael . . . . . . . . . . . . . . . . . . . . .
0. Introduction . . . . . . . . . . . . . . . . . . . . .
1. Inductively perfect maps, compact-covering maps,
compact-covering maps . . . . . . . . . . . . . . .
2. Quotient s-maps and compact-covering maps . . .
3. Continuous selections . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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and countable. . . . . . . . .
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. 271
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277
Questions in Dimension Theory
by R. Pol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
III
Continua Theory
293
Eleven Annotated Problems About Continua
by H. Cook, W. T. Ingram and A. Lelek . . . . . . . . . . . . . . . . . 295
Tree-like Curves and Three Classical Problems
by J. T. Rogers, Jr. . . . . . . . . . . . . .
1. The Fixed-Point Property . . . . . . . . .
2. Hereditarily Equivalent Continua . . . . .
3. Homogeneous Continua . . . . . . . . . .
4. Miscellaneous Interesting Questions . . .
References . . . . . . . . . . . . . . . . . . . .
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303
305
307
308
310
310
Contents
IV
xi
Topology and Algebraic Structures
311
Problems on Topological Groups and other Homogeneous Spaces
by W. W. Comfort . . . . . . . . . . . . . . . . . . . . . . . .
0. Introduction and Notation . . . . . . . . . . . . . . . . . .
1. Embedding Problems . . . . . . . . . . . . . . . . . . . . .
2. Proper Dense Subgroups . . . . . . . . . . . . . . . . . . .
3. Miscellaneous Problems . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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313
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338
Problems in Domain Theory and Topology
by J. D. Lawson and M. Mislove . . . . . . . . . . .
1. Locally compact spaces and spectral theory . . . .
2. The Scott Topology . . . . . . . . . . . . . . . . .
3. Fixed Points . . . . . . . . . . . . . . . . . . . . .
4. Function Spaces . . . . . . . . . . . . . . . . . . .
5. Cartesian Closedness . . . . . . . . . . . . . . . .
6. Strongly algebraic and finitely continuous DCPO’s
7. Dual and patch topologies . . . . . . . . . . . . .
8. Supersober and Compact Ordered Spaces . . . . .
9. Adjunctions . . . . . . . . . . . . . . . . . . . . .
10. Powerdomains . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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349
352
354
357
358
360
362
364
367
368
369
370
V
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Topology and Computer Science
Problems in the Topology of Binary Digital Images
by T. Y. Kong, R. Litherland and A. Rosenfeld
1. Background . . . . . . . . . . . . . . . . . . . .
2. Two-Dimensional Thinning . . . . . . . . . . .
3. Three-Dimensional Thinning . . . . . . . . . .
4. Open Problems . . . . . . . . . . . . . . . . . .
Acknowledgement . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .
373
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On Relating Denotational and Operational Semantics
Languages with Recursion and Concurrency
by J.-J. Ch. Meyer and E. P. de Vink . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . .
Mathematical Preliminaries . . . . . . . . . . . . . .
Operational Semantics . . . . . . . . . . . . . . . . .
Denotational Semantics . . . . . . . . . . . . . . . .
Equivalence of O and D . . . . . . . . . . . . . . . .
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375
377
377
381
383
384
384
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387
389
390
394
396
398
for Programming
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xii
Contents
Conclusion and Open Problems . . . . . . . . . . . . . . . . . . . . . . . 402
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
VI
Algebraic and Geometric Topology
407
Problems on Topological Classification of Incomplete Metric
by T. Dobrowolski and J. Mogilski . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . . . . .
2. Absorbing sets: A Survey of Results . . . . . . . . . .
3. General Problems about Absorbing Sets . . . . . . . .
4. Problems about λ-convex Absorbing Sets . . . . . . .
5. Problems about σ-Compact Spaces . . . . . . . . . .
6. Problems about Absolute Borel Sets . . . . . . . . . .
7. Problems about Finite-Dimensional Spaces . . . . . .
8. Final Remarks . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spaces
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409
411
411
415
416
419
422
424
425
426
Problems about Finite-Dimensional Manifolds
by R. J. Daverman . . . . . . . . . . . . . . . . . . . . .
1. Venerable Conjectures . . . . . . . . . . . . . . . . . . .
2. Manifold and Generalized Manifold Structure Problems
3. Decomposition Problems . . . . . . . . . . . . . . . . .
4. Embedding Questions . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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431
434
437
440
447
450
A List of Open Problems in Shape Theory
by J. Dydak and J. Segal . . . . . . . . . .
1. Cohomological and shape dimensions . .
2. Movability and polyhedral shape . . . . .
3. Shape and strong shape equivalences . .
4. P -like continua and shape classifications .
References . . . . . . . . . . . . . . . . . . . .
Algebraic Topology
by G. E. Carlsson . . . . . . . .
1. Introduction . . . . . . . . . .
2. Problem Session for Homotopy
3. H-spaces . . . . . . . . . . . .
4. K and L-theory . . . . . . . .
5. Manifolds & Bordism . . . . .
6. Transformation Groups . . . .
7. K. Pawalowski . . . . . . . . .
References . . . . . . . . . . . . . .
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Theory:
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457
459
460
462
464
465
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Adams
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469
471
471
476
478
479
481
484
485