1978 Book CounterexamplesInTopo - Unknown.pdf

(19174 KB) Pobierz
Lynn Arthur Steen
J. Arthur Seebach, Jr.
Counterexamples
in Topology
Second Edition
Springer-Verlag
New York
Heidelberg
Berlin
Lynn Arthur Steen
Saint Olaf College
Northfield, Minn.
55057
USA
J. Arthur Seebach, Jr.
Saint Olaf College
Northfield, Minn.
55057
USA
AMS Subject Classification: 54·01
Library of Congress Cataloging in Publication Data
Steen, Lynn A 1941·
Counterexamples in topology.
Bibliography: p.
Includes index.
1. Topological spaces.
I.
Seebach,
J.
Arthur, joint author.
II.
Title
QA611.3.S74 1978
514'.3
78·1623
All rights reserved.
No part of this book may be translated or reproduced in
any form without written permission from Springer.Verlag
Copyright
©
1970, 1978 by Springer·Verlag New York Inc.
The first edition was published in 1970 by Holt, Rinehart, and Winston Inc.
9 8 7 6 5 4 321
ISBN-13: 978-0-387-90312-5
DOl: 1007/978-1-4612-6290-9
e-ISBN-13:978-1-4612-6290-9
Preface
The creative process of mathematics, both historically and individually,
may be described as a counterpoint between theorems and examples. Al-
though it would be hazardous to claim that the creation of significant
examples is less demanding than the development of theory, we have dis-
covered that focusing on examples is a particularly expeditious means of
involving undergraduate mathematics students in actual research. Not only
are examples more concrete than theorems-and thus more accessible-but
they cut across individual theories and make it both appropriate and neces-
sary for the student to explore the entire literature in journals as well as
texts. Indeed, much of the content of this book was first outlined by under-
graduate research teams working with the authors at Saint Olaf College
during the summers of 1967 and 1968.
In compiling and editing material for this book, both the authors and
their undergraduate assistants realized a substantial increment in topologi-
cal insight as a direct result of chasing through details of each example. We
hope our readers will have a similar experience. Each of the 143 examples in
this book provides innumerable concrete illustrations of definitions, theo-
rems, and general methods of proof. There is no better way, for instance, to
learn what the definition of metacompactness really means than to try to
prove that Niemytzki's tangent disc topology is not metacompact.
The search for counterexamples is as lively and creative an activity as
can be found in mathematics research. Topology particularly is replete
with unreported or unsolved problems (do you know an example of a
Hausdorff topological space which is separable and locally compact, but
not CT-compact?), and the process of modifying old examples or creating
new ones requires a wild and uninhibited geometric imagination. Far from
providing all relevant examples, this book provides a context in which to
iii
iv Preface
ask new questions and seek new answers. We hope that each reader will
share (and not just vicariously) in the excitement of the hunt.
Counterexamples in Topology
was originally designed, not as a text, but
as a course supplement and reference work for undergraduate and graduate
students of general topology, as well as for their teachers. For such use, the
reader should scan the book and stop occasionally for a guided tour of the
various examples. The authors have used it in this manner as a supplement
to a standard textbook and found it to be a valuable aid.
There are, however, two rather different circumstances under which this
monograph could most appropriately be used as the exclusive reference in
a topology course. An instructor who wishes to develop his own theory in
class lecture may well find the succinct exposition which precedes the
examples an appropriate minimal source of definitions and structure. On
the other hand,
Counterexamples in Topology
may provide sufficiently few
proofs to serve as a basis for an inductive, Moore-type topology course. In
either case, the book gives the instructor the flexibility to design his own
course, and the students a wealth of historically and mathematically sig-
nificant examples.
A counterexample, in its most restricted sense, is an example which dis-
proves a famous conjecture. We choose to interpret the word more broadly,
particularly since all examples of general topology, especially as viewed by
beginning students, stand in contrast to the canon of the real line. So in
this sense any example which in some respect stands opposite to the reals
is truly a
Gegenbeispiel.
Having said that, we should offer some rationale
for our inclusions and omissions. In general we opted for examples which
Wf're necessary to distinguish definitions, and for famous, well known, or
simply unusual examples even if they exhibited no new properties. Of course,
what is well known to others may be unknown to us, so we acknowledge
with regret the probable omission of certain deserving examples.
In choosing among competing definitions we generally adopted the
strategy of making no unnecessary assumptions. With rare exception
therefore, we define all properties for all topological spaces, and not just
for, for instance, Hausdorff spaces.
Often we give only a brief outline or hint of a proof; this is intentional, but
we caution readers against inferring that we believe the result trivial.
Rather, in most cases, we believe the result to be a worthwhile exercise
which could be done, using the hint, in a reasonable period of time. Some
of the more difficult steps are discussed in the Notes at the end of the book.
The examples are ordered very roughly by their appropriateness to the
definitions as set forth in the first section. This is a very crude guide whose

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin