Lecture Notes on General Topology

153 Pages·2021·6.096 MB·English

by  Guoliang Wang

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b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Higher
Education World Scientifi c
Press
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Published by
Higher Education Press Limited Company
4 Dewai Dajie, Beijing 100120, P. R. China
and
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
LECTURE NOTES ON GENERAL TOPOLOGY
Copyright © 2021 by Higher Education Press Limited Company and
World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 978-981-122-741-7 (hardcover)
ISBN 978-981-122-742-4 (ebook for institutions)
ISBN 978-981-122-743-1 (ebook for individuals)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/12020#t=suppl
Printed in P. R. China
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To Alice and Charles.
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Preface
This book is intended as a one-semester course in general topology for un-
dergraduate students as well as first-year graduate students.
One of the ways in which general topology has been influencing science
is by introducing a language consisting of terminologies and using it every-
where. The study of general topology therefore resembles a language study
that one has to learn many words as studying abstract algebra.
Here are some highlights of this book. This is a book with a concentra-
tion on basic concepts in general topology. Though it is hard to determine
precisely which concepts therein are basic and which are not, I make efforts
in selecting basic ones according to personal experience on the occurrence
frequency of notions in advanced mathematics. The selections are listed as
chapter titles: topological spaces, continuous maps and homeomorphisms,
connectedness, separation and countability axioms, compactness, and prod-
uct and quotient spaces. Interrelationships among concepts are also taken
into account carefully. Twelve particular topological spaces are labeled and
one may use them to learn concepts as examples. This book contains exer-
cises for each chapter, some solutions, and well-organized indices.
Much material of this book comes from Viro, Ivanov, Netsvetaev and
Kharlamov’s problem textbook [42, Part 1]. There are books on gen-
eral topology with an emphasis on intuitions like Prasolov [32], on geo-
metric insights like Armstrong [4] and Weeks [43], and on aspects that
are most useful in analysis like Pervin [31]. In comparison, this book fo-
cuses on general topology itself. Besides Kelley’s classical book [22] and
Munkres’ [29,30], textbooks on general topology with more extensive con-
tent include Dugundji [13], Hocking and Young [18], Willard [44], and
You [45].
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iv Preface
IamindebtedtoProfessorsFuguiShi,HuafeiSun,andChengyeYoufor
their valuable suggestions. I would like to show appreciation to Professors
Peng Cao, Lining Jiang, Bingzhao Li, and Yubin Tian, who encouraged me
to write this book in a high quality. Thank my students Dandan Jin and
Jiarui Zhang, who helped me draw some of the figures. Special thanks to
the editor Xiaoli Wu for her constant administrative assistance throughout
the preparation of this book. I am grateful to the typist Ming Yang for
his technical support in using LaTeX. For financial support of publishing
this book, I should thank School of Mathematics and Statistics at Beijing
Institute of Technology.
I gratefully acknowledge Robert Andre who kindly pointed out an error
at the beginning of Chapter 3.
Guoliang Wang
Beijing
July 16, 2020
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Notation
[a,b] the interval connecting the points a,b Rn, 65
∈
A the closure of a set A, 18
Ac the complement of a set A, 18
A the interior of a set A, 18
◦
B(x,r) the open ball of radius r centered at x, 25
Bn the unit open ball in Rn, 26
Bd the boundary operator, 18
β the topology generated by a collection β, 12
h i
C the set of complex numbers, 22
c the complement operator, 19
C(f,g) the coincidence set of maps f and g, 77
Cl the closure operator, 18
D(x,r) the disk of radius r centered at x, 25
Dn the unit disk in Rn, 26
Ext the exterior operator, 18
Γ the partition of a set X with respect to a map f from X, 107
f
the upper half plane, 83
H
H the set x R: x 1 + , 24
{ ∈ ≥ }∪{ ∞}
i the interior operator (in operator theory), 19
Int the interior operator, 18
k the closure operator (in operator theory), 19
K the set 1/n: n Z of unit fractions, 12
{ ∈ }
N the set n Z: n 0 of natural numbers, 22
{ ∈ ≥ }