Open problems in topology.pdf

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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
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Introduction
This volume grew from a discussion by the editors on the di culty 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, gener-
alized metric spaces, geometric topology, homogeneity, infinite-dimensional
topology, knot theory, ordered spaces, set-theoretic topology, topological dy-
namics, 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 Problems become quickly outdated, and any list soon loses its value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 sucient 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 T E Xand
METAFONT
Jan van Mill
George M. Reed
. They also express their gratitude to
Eva Coplakova for composing the indexes, and to Eva Coplakova and Geertje
van Mill for typing the manuscript.
Table of Contents
Introduction................................... v
Contents..................................... vii
I Set Theoretic Topology
1
Dow’s Questions
by A. Dow ................................. 5
Steprans’ Problems
by J. Steprans ............................... 13
1.TheTorontoProblem.......................... 15
2.Continuouscolouringsofclosedgraphs ................ 16
3. Autohomeomorphisms of the Cech-Stone 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-Lindel¨ofProblems ................ 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-Lindel¨ofSpaces .......................... 52
8.DowkerSpaces.............................. 54
9.ExtendingIdeals............................. 55
vii

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