123

Marco Manetti

UN

ITEX

TU

NIT

EXT

Topology

UNITEXT - La Matematica per il 3+2

Volume 91

Editor-in-chief

A. Quarteroni

Series editors

L. AmbrosioP. BiscariC. CilibertoM. LedouxW.J. Runggaldier

More information about this series at http://www.springer.com/series/5418

Marco Manetti

Topology

123

Marco ManettiDepartment of MathematicsSapienza - Università di RomaRomeItaly

Translation from the Italian language edition: Topologia, Marco Manetti, © Springer-VerlagItalia, Milano 2014. All rights reserved.

ISSN 2038-5722 ISSN 2038-5757 (electronic)UNITEXT - La Matematica per il 3+2ISBN 978-3-319-16957-6 ISBN 978-3-319-16958-3 (eBook)DOI 10.1007/978-3-319-16958-3

Library of Congress Control Number: 2014945348

Springer Cham Heidelberg New York Dordrecht London© Springer International Publishing Switzerland 2015This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof 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 transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

Cover design: Simona Colombo, Giochi di Grafica, Milano, Italy

Printed on acid-free paper

Springer International Publishing AG Switzerland is part of Springer Science+Business Media(www.springer.com)

Translated by Simon G. Chiossi, UFBA—Universidade Federal da Bahia, Salvador (Brazil).

Preface

To the Student

This textbook offers a primer in general topology (point-set topology), together withan introduction to algebraic topology. It is meant primarily for students with amathematical background that is usually taught in the first year of undergraduatedegrees in Mathematics and Physics.

Point-set topology is the language in which a considerable part of mathematics iswritten. It is not an accident that the original name ‘analytic topology’ was replacedby ‘general topology’, a more apt term for that part of topology that is used by the vastmajority of mathematicians and is fundamental in many areas of mathematics. Overtime its unabated employment has had a constant polishing effect on its theorems anddefinitions, thus rendering it an extraordinarily elegant subject. There is no doubt thatpoint-set topology has a significant formative value, in that it forces the brain—andtrains it at the same time—to handle extremely abstract objects, defined solely byaxioms. In studying on this book, you will experience hands-on that the point-settopology resembles a language more than a theory. There are endless terms anddefinitions to be learnt, a myriad of theorems whose proof is often rather easy, onlyoccasionally exceeding 20 lines. There are, obviously, also deep and far-from-trivialresults, such as the theorems of Baire, Alexander and Tychonov.

The part on algebraic topology, details of which we will give in Chap. 9 togetherwith the mandatory motivations, is devoted to the study of homotopy, fundamentalgroups and covering spaces.

I included around 500 exercises in the text: trying to solve them with dedicationis the best way to attain a firm hold on the matter, adapt it to your own way ofthinking and also learn to develop original ideas. Some exercises are solved directlyin the text, either in full or almost. They are called ‘Examples’, and their importanceshould not be underestimated: understanding them is the correct way tomake abstract notions concrete. Exercises marked with ~, instead, are solved inChap. 16.

v

It is a matter of fact that the best way to learn a new subject is by attendinglectures, or studying on books, and trying to understand definitions, theorems andthe interrelationships properly. At the same time you should solve the exercises,without the fear of making mistakes, and then compare the solutions with the onesin the text, or those provided by teacher, classmates or the internet.

This book also proposes a number of exercises marked with , which I per-sonally believe to be harder than the typical exam question. These exercises shouldtherefore be taken as endeavours to intelligence, and incentives to be creative: theyrequire that we abandon ourselves to new synergies of ideas and accept to be guidedby subtler analogies, rather than trail patiently along a path paved by routine ideasand standard suggestions.

To the Lecturer

In the academic years 2004–2005 and 2005–2006, I taught a lecture course called‘Topology’ for the Bachelor’s degree in Mathematics at University of Rome‘La Sapienza’. The aim was to fit the newly introduced programme specificationsfor mathematical teaching in that part of the syllabus traditionally covered in‘Geometry 2’ course of the earlier 4-year degrees. The themes were carefullychosen so to keep into account on one side the formative and cultural features of thesingle topics, on the other their usefulness in the study of mathematics and researchalike. Some choices certainly break with a long-standing and established traditionof topology teaching in Italy, and with hindsight I suspect they might have beenelicited by my own research work in algebra and algebraic geometry. I decided itwould be best to get straight to the point and state key results and definitions asearly as possible, thus fending off the terato(po)logical aspects.

From the initial project to the final layout of my notes, I tried to tackle theconceptual obstacles gradually, and make both theory and exercises as interestingand entertaining as possible for students. Whether I achieved these goals the readerwill tell.

The background necessary to benefit from the book is standard, as taught in first-year Maths and Physics undergraduate courses. Solid knowledge of the language ofsets, of linear algebra, basic group theory, the properties of real functions, series andsequences from ‘Calculus’ are needed. The second chapter is dedicated to thearithmetic of cardinal numbers and Zorn’s lemma, two pivotal prerequisites that arenot always addressed during the first year: it will be up to the lecturer to decide—after assessing the students’ proficiency—whether to discuss these topics or not.

The material present here is more than sufficient for 90 hours of lectures andexercise classes, even if, nowadays, mathematics syllabi tend to allocate far lesstime to topology. In order to help teachers decide what to skip I indicated with thesymbol y ancillary topics, which may be left out at first reading. It has to be said,though, that Chaps. 3–6 (with the exception of the sections displaying y), form thebackbone of point-set topology and, as such, should not be excluded.

vi Preface

The bibliography is clearly incomplete and lists manuals that I found mostuseful, plus a selection of research articles and books where the willing student canfind further information about the topics treated, or mentioned in passing, in thisvolume.

Acknowledgments

I am grateful to Ciro Ciliberto and Domenico Fiorenza for reading earlier versionsand for the tips they gave me. I would like to thank Francesca Bonadei of Springer-Verlag Italia for helping with the final layout, and all the students (‘victims’) of mylectures on topology for the almost-always-useful and relevant observations thatallowed me to amend and improve the book.

This volume is based on the second Italian edition. Simon G. Chiossi has donean excellent work of translation; he also pointed out a few inaccuracies and pro-posed minor improvements. To him and the staff at Springer, I express here myheartfelt gratitude.

Future updates can be found athttp://www.mat.uniroma1.it/people/manetti/librotopology.html.

Rome, February 2015 Marco Manetti

Preface vii

Contents

1 Geometrical Introduction to Topology . . . . . . . . . . . . . . . . . . . . . 11.1 A Bicycle Ride Through the Streets of Rome . . . . . . . . . . . . 11.2 Topological Sewing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The Notion of Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Facts Without Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Notations and Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . 212.2 Induction and Completeness . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6 The Cardinality of the Product . . . . . . . . . . . . . . . . . . . . . . 36References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Topological Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Interior of a Set, Closure and Neighbourhoods . . . . . . . . . . . 433.3 Continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4 Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Subspaces and Immersions . . . . . . . . . . . . . . . . . . . . . . . . . 553.6 Topological Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.7 Hausdorff Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Connectedness and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Connected Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Covers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

ix

4.4 Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5 Wallace’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.6 Topological Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.7 Exhaustions by Compact Sets . . . . . . . . . . . . . . . . . . . . . . . 83Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Topological Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.1 Identifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Quotient Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Quotients by Groups of Homeomorphisms . . . . . . . . . . . . . . 925.4 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.5 Locally Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6 The Fundamental Theorem of Algebra y . . . . . . . . . . . . . . 101

6 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1 Countability Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.3 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.4 Compact Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.5 Baire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.6 Completions y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.7 Function Spaces and Ascoli-Arzelà Theorem y . . . . . . . . . . 1226.8 Directed Sets and Nets (Generalised Sequences) y . . . . . . . . 125

7 Manifolds, Infinite Products and Paracompactness . . . . . . . . . . . . 1297.1 Sub-bases and Alexander’s Theorem . . . . . . . . . . . . . . . . . . 1297.2 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.3 Refinements and Paracompactness y . . . . . . . . . . . . . . . . . 1337.4 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.5 Normal Spaces y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.6 Separation Axioms y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8 More Topics in General Topology y. . . . . . . . . . . . . . . . . . . . . . 1438.1 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.2 The Axiom of Choice Implies Zorn’s Lemma. . . . . . . . . . . . 1448.3 Zermelo’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.4 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.5 The Compact-Open Topology. . . . . . . . . . . . . . . . . . . . . . . 1518.6 Noetherian Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.7 A Long Exercise: Tietze’s Extension Theorem . . . . . . . . . . . 157References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

x Contents

9 Intermezzo y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.2 Polybricks and Betti Numbers . . . . . . . . . . . . . . . . . . . . . . 1629.3 What Algebraic Topology Is. . . . . . . . . . . . . . . . . . . . . . . . 163Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

10 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16510.1 Locally Connected Spaces and the Functor π0 . . . . . . . . . . . 16510.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16910.3 Retractions and Deformations . . . . . . . . . . . . . . . . . . . . . . . 17310.4 Categories and Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . 17510.5 A Detour y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

11 The Fundamental Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18111.1 Path Homotopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18111.2 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . 18611.3 The Functor π1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18911.4 The Sphere Sn Is Simply Connected (n� 2) . . . . . . . . . . . . . 19211.5 Topological Monoids y . . . . . . . . . . . . . . . . . . . . . . . . . . 196References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

12 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19912.1 Local Homeomorphisms and Sections . . . . . . . . . . . . . . . . . 19912.2 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20112.3 Quotients by Properly Discontinuous Actions . . . . . . . . . . . . 20412.4 Lifting Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20712.5 Brouwer’s Theorem and Borsuk’s Theorem . . . . . . . . . . . . . 21312.6 A Non-abelian Fundamental Group . . . . . . . . . . . . . . . . . . . 216Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

13 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21913.1 Monodromy of Covering Spaces . . . . . . . . . . . . . . . . . . . . . 21913.2 Group Actions on Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22213.3 An Isomorphism Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 22413.4 Lifting Arbitrary Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22713.5 Regular Coverings y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23013.6 Universal Coverings y . . . . . . . . . . . . . . . . . . . . . . . . . . . 23313.7 Coverings with Given Monodromy y . . . . . . . . . . . . . . . . . 236

14 van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23914.1 van Kampen’s Theorem, Universal Version . . . . . . . . . . . . . 23914.2 Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24414.3 Free Products of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 248

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14.4 Free Products and van Kampen’s Theorem. . . . . . . . . . . . . . 24914.5 Attaching Spaces and Topological Graphs . . . . . . . . . . . . . . 25314.6 Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

14.6.1 Attaching 1-cells. . . . . . . . . . . . . . . . . . . . . . . . . . 25714.6.2 Attaching 2-cells. . . . . . . . . . . . . . . . . . . . . . . . . . 25814.6.3 Attaching n-cells, n� 3 . . . . . . . . . . . . . . . . . . . . . 258

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

15 Selected Topics in Algebraic Topology y . . . . . . . . . . . . . . . . . . 26115.1 Natural Transformations and Equivalence of Categories . . . . . 26115.2 Inner and Outer Automorphisms . . . . . . . . . . . . . . . . . . . . . 26415.3 The Cantor Set and Peano Curves . . . . . . . . . . . . . . . . . . . . 26615.4 The Topology of SOð3;RÞ . . . . . . . . . . . . . . . . . . . . . . . . . 26815.5 The Hairy Ball Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 27215.6 Complex Polynomial Functions . . . . . . . . . . . . . . . . . . . . . 27415.7 Grothendieck’s Proof of van Kampen’s Theorem . . . . . . . . . 27515.8 A Long Exercise: The Poincaré-Volterra Theorem. . . . . . . . . 277References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

16 Hints and Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27916.1 Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27916.2 Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28116.3 Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28216.4 Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28616.5 Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28916.6 Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29016.7 Chapter 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29216.8 Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29416.9 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29616.10 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29716.11 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29816.12 Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30016.13 Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30116.14 Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

xii Contents

Chapter 1Geometrical Introduction to Topology

Let us start off with an excerpt from the introduction to Chapter V in What ismathematics? by R. Courant and H. Robbins.

‘In the middle of the nineteenth century there began a completely new development ingeometry that was soon to become one of the great forces in modern mathematics. Thenew subject, called analysis situs or topology, has as its object the study of the propertiesof geometrical figures that persist even when the figures are subjected to deformations sodrastic that all their metric and projective properties are lost.’ (...) ‘When Bernhard Riemann(1826–1866) came to Göttingen as a student, he found the mathematical atmosphere of thatuniversity town filled with keen interest in these strange new geometrical ideas. Soon herealized that here was the key to the understanding of the deepest properties of analyticfunctions of a complex variable.’

The expression ‘deformations so drastic’ is rather vague, and as a matter of factmodern topology studies several classes of transformations of geometrical figures, themost important among which are homeomorphisms and homotopy equivalences.

While precise definitions for these will be given later, in this chapter we shallencounter a preliminary, and only partial notion of homeomorphism, and discuss afew examples. We will stay away from excessively rigorous definitions and proofsat this early stage, and try instead to rely on the geometrical intuition of the reader,with the hope that in this way we might help newcomers acquire the basic ideas.

1.1 A Bicycle Ride Through the Streets of Rome

Problem 1.1 On a bright Sunday morning Mr. B. decides to go on a bike ride thatcrosses every bridge in Rome once, and only once. Knowing that he can decide whereto start from and where to end, will Mr. B. be able to accomplish his wish?

We remind those who aren’t familiar with the topography of Rome that the cityis divided by the river Tiber and its affluent Aniene in four regions, one being theTiberine island, lying in the middle of the river and joined to either riverbank bybridges.

© Springer International Publishing Switzerland 2015M. Manetti, Topology, UNITEXT - La Matematica per il 3+2 91,DOI 10.1007/978-3-319-16958-3_1

1

2 1 Geometrical Introduction to Topology

Fig. 1.1 The topology of thebridges of Rome

Vatican City Roma-Urbeairport

TheColosseum

Tiberineisland

p bridges

q bridgesr bridges

In order to answer the question we needn’t go around the streets of Rome on twowheels testing possibilities, nor mess up a street map with a marker.

The problem can be visualised by drawing four circles on a sheet of paper, one foreach region, and indicating how many bridges connect any two regions (Fig. 1.1).

A configuration of this kind contains all the information we need to solve theproblem. If you agree with this statement, then you are thinking topologically. Inother words you have understood that whether the proposed bicycle ride is possi-ble doesn’t depend on metric or projective properties like the bridges’ length, theirarchitectural structure, the extension of mainland areas and so on. But if you insiston solving the problem using a street map, you should imagine the map is drawn ona thin rubber sheet. Think of stretching, twisting and rumpling it, as much as youlike, without tearing it nor making different points touch one another by folding: youwill agree that the answer stays the same.

Problem 1.2 In the old town of Königsberg, on the river Pregel, there are two isletsand seven bridges as in Fig. 1.2. The story goes that Mr. C. wanted to cross on footevery bridge in Königsberg only once, and could start and finish his walk at any point.Would have Mr. C. been able to do so?

Fig. 1.2 The bridges of oldKönigsberg

1.1 A Bicycle Ride Through the Streets of Rome 3

Here’s another problem of topological flavour:

Problem 1.3 Can one drive from Rome to Venice by never leaving the motorway?

As the motorway goes through Rome and Venice, the problem is truly a questionabout the connectedness of the Italian road system: a network of roads is connectedif one can drive from any one point to any other. Here, too, the answer is clearlyindependent of how long the single stretches of road are, or how many turns orslopes there are etc.

A useful mathematical concept for solving the previous problems is that of agraph. In Euclidean space a graph is a non-empty set of V points, called nodesor vertices, certain pairs of which are joined by S segments called edges. Edgesare not necessarily straight, but can be parts of circles, parabolas, ellipses, or moregenerally ‘regular’ arcs that do not pass through the same point in space more thanonce. Another assumption is that distinct edges meet only at nodes. A walk of lengthp in a graph is a sequence v0, v1, . . . , vp of nodes and a sequence l1, l2, . . . , l p ofedges, with li joining vi−1 to vi for any i . The nodes v0 and vp are the endpoints ofthe walk. A graph is said to be connected if there is a walk beginning and ending atany two given nodes u, w.

If u is a node in a graph Γ , we call degree of u in Γ the number of edgescontaining u, counting twice edges with both endpoints in u. It’s straightforward thatthe sum of the degrees of all nodes equals twice the number of edges; in particular,any graph must have an even number of odd-degree nodes.

A graph is called Eulerian if there’s a walk that visits each edge exactly once: inthis case the length of the walk equals the number of edges in the graph.

Going back to the bridges’ problem, no matter whether in Rome or Königsberg,let’s construct a graph Γ having mainland areas as nodes and bridges as edges. Thecrossing problem has a positive answer if and only ifΓ is an Eulerian graph (Fig. 1.3).

Theorem 1.1 In any Eulerian graph there are at most two nodes of odd degree.More specifically, the answer to the problem of the Königsberg’s bridges is ‘no’.

Proof Choose a sequence of nodes v0, . . . , vS and a sequence of edges l1, . . . , lS

that give a walk visiting every edge precisely once. Every node u other than v0 andvS has degree equal twice the number of indices i such that u = vi . �

Fig. 1.3 The Königsbergbridges’ graph

BA

D

C