Topology
Stefan Waldmann Topology An Introduction 123
Stefan Waldmann Julius Maximilian University of Würzburg Würzburg Germany ISBN 978-3-319-09679-7 ISBN 978-3-319-09680-3 (ebook) DOI 10.1007/978-3-319-09680-3 Library of Congress Control Number: 2014945348 Mathematical Classification Code: 54-XX, 54-01 Springer Cham Heidelberg New York Dordrecht London Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Robert, Sonja, Richard, Silvia, and Viola
Preface These lecture notes grew out of notes I prepared for my lecture Topology held in Erlangen in Summer Term 2012. The course is aimed at bachelor students in their second year being familiar with the basic notions from calculus. The purpose of this small course is to give some first introduction to the notions of general (or point set) topology as they are needed in many other areas of mathematics. Of course, there are many excellent textbooks on topology available. However, the aim of these notes as well as of the lecture itself is to give bachelor students after their first year a minimal amount of topology needed to continue with more advanced topics in a mathematics (or physics) programme but still providing detailed proofs. Moreover, the idea is to require only very basic preliminary knowledge as offered by the introductory calculus and linear algebra courses. The text is self-contained and provides many exercises which will enable the student to work through these notes on her own. Alternatively, the text may serve as a companion for a small lecture on topology. The first four chapters can be seen as the core of the theory which every mathematics student and hopefully also some physics students should be exposed to. The remaining two chapters give a certain personal preference: I have chosen to put some focus on possible applications in functional analysis. This explains why the last two chapters are on continuous functions as well as on Baire s Theorem in different formulations. On the other hand, I have omitted other important concepts like the fundamental groupoid or topological groups and their continuous actions due to the lack of time and space. The participants of the original lecture in Erlangen showed great patience with the first versions, not only of these notes but also with the lecture itself. I would like to thank all of them for their comments, remarks and suggestions, which all found their way into these notes in one form or the other. In particular, I would like to mention here Alexander Spies for numerous corrections and a careful proofreading of the entire manuscript. Moreover, I am much obliged to Florian Unger for taking care of the LATEX-files and all the typing of the first version of the draft. Without his help, the manuscript would never have been finished. It is a pleasure to thank Karl-Hermann Neeb for various discussions on the pedagogical aspects of teaching vii
viii Preface of topology. The anonymous referees pointed out many weak points in the original manuscript helping to improve it in many places. Their comments and remarks are much appreciated. Last but not least, I would like to thank my family for the patience and support throughout: without this the book would never have been possible. Würzburg, June 2014 Stefan Waldmann
Contents 1 Introduction.... 1 2 Topological Spaces and Continuity... 5 2.1 Metric Spaces.................................... 5 2.2 Topological Spaces................................ 12 2.3 Neighbourhoods, Interiors, and Closures.................. 16 2.4 Continuous Maps.................................. 20 2.5 Connectedness.................................... 25 2.6 Separation Properties............................... 29 2.7 Exercises....................................... 33 3 Construction of Topological Spaces... 41 3.1 The Product Topology and Initial Topologies.............. 41 3.2 Final Topologies and Quotients........................ 45 3.3 Topological Manifolds.............................. 48 3.4 Exercises....................................... 53 4 Convergence in Topological Spaces... 59 4.1 Convergence of Nets............................... 59 4.2 Nets and Filters................................... 63 4.3 Ultrafilters...................................... 67 4.4 Exercises....................................... 68 5 Compactness... 73 5.1 Compact Spaces.................................. 73 5.2 Continuous Maps and Compactness..................... 77 5.3 Tikhonov s Theorem............................... 79 5.4 Further Notions of Compactness....................... 80 5.5 Exercises....................................... 84 ix
x Contents 6 Continuous Functions.... 87 6.1 Urysohn s Lemma and Tietze s Theorem................. 87 6.2 The Stone-Weierstraß Theorem........................ 92 6.3 The Arzelà-Ascoli Theorem.......................... 99 6.4 Exercises....................................... 106 7 Baire s Theorem... 111 7.1 Meager Subsets and Baire Spaces...................... 111 7.2 Baire s Theorem.................................. 115 7.3 Discontinuous Functions............................. 118 7.4 Exercises....................................... 122 Appendix A: Not an Introduction to Set Theory... 125 References.... 131 Index... 133
Symbols M, N,... Sets 2 M Power set of M (M, d) Metric space with metric d S n n-sphere in R n T n ¼ S 1 S 1 n-torus B r ðpþ Open ball centred at p with radius r UðpÞ Set of all neighbourhoods of p ðp n Þ n2n Sequence of points ðm, MÞ Topological space with topology M2 M OM Open subset of topological space A M Closed subset of topological space S; BM Subbasis or basis of a topology Mj N Subspace topology for N M A Open interior of a subset A M A Boundary of a subset A M A cl Closure of a subset A M, also A CðM; NÞ Continuous maps from M to N CðMÞ ¼CðM; CÞ Continuous complex-valued functions on M C, D,... Categories ObjðCÞ Objects of C Morph(a, b) Morphisms from a to b for a, b ObjðCÞ top Category of topological spaces Top Category of Hausdorff spaces CðpÞ Connected component of p M Π(p) Path-connected component of p M T i Separation property i C b ðmþ Bounded continuous complex-valued functions on M kf k 1 Supremum norm of a function f C The Cantor set M ¼ P i2i M i Cartesian product of sets or topological spaces xi
xii Symbols pr i : M! M i Projection onto i-th factor Equivalence relation on M M= Set of equivalence classes p : M! M= Quotient map D M M M Diagonal in M M with diagonal map D U : G M! M Group action of G on M G p Orbit of p M M=G Set of orbits, orbit space (U, x) Topological chart with local coordinates RP n ; CP n Real and complex projective space ði; 4Þ Directed or partially ordered set ðp i Þ i2i Net of points p i M lim p i i2i Limit point(s) of a net A scl Sequential closure of a subset A M F Filter on M f F Push-forward filter F A Trace filter on a subset A M fo i g i2i Open cover of M K M Compact subset of M ðm ; M Þ Alexandroff compactification of ðm; MÞ A C -Algebra or subalgebra of CðMÞ fk n g n2n Exhausting sequence of compact subsets OðXÞ Holomorphic functions of open subset X C jj K Supremum norm over compact K M p Seminorm on a vector space B p,r (x) Open ball with respect to seminorm p B locðmþ Locally bounded functions on M dist(p, A) Distance of point p to closed subset A supp f Support of a continuous function C K ðmþ Continuous functions with support in K C 0 ðmþ Continuous functions with compact support