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From Geometry to Topology

H. Graham Flegg
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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The book under review, an unaltered reprint of a text first published in 1974, begins with a couple of chapters on Euclidean and projective geometry, segues into nine chapters on low-dimensional “surface-type” topology, and then ends with five chapters introducing metric and topological spaces. The exposition throughout is (by deliberate intent) intuitive and picture-based rather than rigorous, so the book is probably not suitable for upper-level mathematics courses in topology or geometry, but might find some use as supplemental reading in such courses or as a text for lower-level students in less rigorous courses. Students wanting to get a taste of what these subjects are about without getting bogged down in the details might also find this book useful.

In more detail: the first two chapters discuss geometry (Euclidean and then projective) from the standpoint of congruence, using this as a springboard for discussing transformations and their invariants. So, this is really an introduction to the ideas behind Klein’s Erlangen Programme, although that term is not used in these chapters (but is mentioned in a three-page Historical Note at the end of the book). Unfortunately, not much is actually said about the underlying geometry; it wasn’t even made clear, for example, that in projective geometry there are no parallel lines. It seems that the primary purpose of these chapters was to get the reader thinking in terms of transformations, so as to motivate the introduction to topology that begins in the next chapter, where the class of transformations now includes things like “stretching, bending and twisting” but not “cutting”.

Chapter 3 introduces these ideas, and then chapters 4 through 12 give a very informal introduction, with lots of pictures, to some of the ideas associated with low-dimensional topology, including classification of surfaces, the Euler characteristic, connections with graphs, planarity and map coloring and the Jordan curve theorem. The Brouwer fixed point theorem is also mentioned.

The chapter on coloring is, of course, now out of date, because the Four Color Theorem was proved several years after the publication of the original text and the Dover edition has not been updated to account for that. I was a little surprised, in fact, that the original text did not make more out of this then-unsolved question; there is a brief statement that “it has never been proved that the existence of a planar map requiring five colours is an impossibility”, but that’s about it; there is no really extended discussion of the history of this very important problem or the excitement it had engendered in the mathematical community.

This quibble aside, I found these chapters to be informative and nicely written; a student working his way through them would be rewarded with at least an intuitive understanding of the rudiments of basic surface theory. Since this material is not often covered in many undergraduate courses on topology that are based primarily on metric and topological spaces, these chapters could, as I indicated earlier, perhaps serve as supplemental reading for such a course.

The remaining chapters in the book take a first look at metric and topological spaces, after some prefatory work on basic set theory and functions. These chapters, I thought, were the least successful in the book. Take, for example, the final chapter, on topological spaces. It consists of a series of definitions, some examples and pictures, and the statements of some theorems, but does not provide much indication of just why these ideas are important. So, for example, after reading this chapter, the reader will know the technical definition of a compact topological space, but will not really know why anybody cares about them; surely, for example, the author could at least have referred to the familiar theorem from calculus that a continuous function defined on a closed and bounded interval is bounded, given a simple example of how this fails when the interval fails to be closed or bounded, and then explained that compactness was the real reason why this phenomenon occurred.

Another concern: The author mentions that the compact subsets of the real numbers are the closed and bounded subsets, but he does so in a single sentence, in the middle of a paragraph, without using an introductory phrase like “It can be shown that…”; students who have been told by their instructors to justify all steps in a math book as they read it might well think that they are supposed to be able to see why this is true, and that’s obviously unreasonable. Other examples like this pop up during the text as well.

There are no exercises in the main body of the text, but at the end of the book there is a collection of 40 of them, very few of which call for proofs. No solutions are provided.

Summary and conclusion: the best part of the book is the middle section on surfaces and graphs; that section might provide a beginning student with a good overview of these ideas in preparation for a more rigorous look at them.

Mark Hunacek ( teaches mathematics at Iowa State University.


Author's Preface
1 Congruence Classes
    What geometry is about
    "The rigid transformations: translation, reflection, rotation"
    Invariant properties
    Congruence as an equivalence relation
    Congruence classes as the concern of Euclidean geometry
2 Non-Euclidean Geometries
    Orientation as a property
    Orientation geometry divides congruence classes
    Magnification (and contraction) combine congruence classes
    Invariants of similarity geometry
    Affine and projective transformations and invariants
    Continuing process of combining equivalence classes
3 From Geometry to Topology
    Elastic deformations
    Intuitive idea of preservation of neighbourhoods
    Topological equivalence classes
    Derivation of 'topology'
    Close connection with study of continuity
4 Surfaces
    Surface of sphere
    "Properties of regions, paths and curves on a sphere"
    Similar considerations for torus and n-fold torus
    Separation of surface by curves
    Genus as a topological property
    Closed and open surfaces
    Two-sided and one-sided surfaces
    Special surfaces: Moebius band and Klein bottle
    Intuitive idea of orientability
    Important properties remain under one-one bicontinuous transformations
5 Connectivity
    Further topological properties of surfaces
    Connected and disconnected surfaces
    Contraction of simple closed curves to a point
    Homotopy classes
    Relation between homotopy classes and connectivity
    Cuts reducing surfaces to a disc
    Rank of open and closed surfaces
    Rank of connectivity
6 Euler Characteristic
    "Interrelation between vertices, arcs and regions"
    Euler characteristic as a topological property
    Relation with genus
    Flow on a surface
    "Singular points: sinks, sources, vortices, etc."
    Index of a singular point
    Singular points and Euler characteristic
7 Networks
    Odd and even vertices
    Planar and non-planar networks
    Paths through networks
    Connected and disconnected networks
    Trees and co-trees
    Specifying a network: cutsets and tiesets
    Traversing a network
    The Koenigsberg Bridge problem and extensions
8 The Colouring of Maps
    Colouring maps
    Chromatic number
    Regular maps
    Six colour theorem
    General relation to Euler characteristic
    Five colour theorem for maps on a sphere
9 The Jordan Curve Theorem
    Separating properties of simple closed curves
    Difficulty of general proof
    Definition of inside and outside
    Polygonal paths in a plane
    Proof of Jordan curve theorem for polygonal paths
10 Fixed Point Theorems
    Rotating a disc: fixed point at centre
    Contrast with annulus
    Continuous transformation of disc to itself
    Fixed point principle
    Simple one-dimensional case
    Proof based on labelling line segments
    Two-dimensional case with triangles
    Three-dimensional case with tetrahedra
11 Plane Diagrams
    Definition of manifold
    Constructions of manifolds from rectangle
    "Plane diagram represenations of sphere, torus, Moebius band, etc. "
    The real projective plane
    Euler characteristic from plane diagrams
    Seven colour theorem on a torus
    Symbolic representation of surfaces
    Indication of open and closed surfaces
12 The Standard Model
    Removal of disc from a sphere
    Addition of handles
    Standard model of two-sided surfaces
    Addition of cross-caps
    General standard model
    Relation to Euler characteristic
    Decomposition of surfaces
    "General classification as open or closed, two-sided or one-sided"
    Homeomorphic classes
13 Continuity
    Preservation of neighbourhood
    Continuous an discontinuous curves
    Formal definition of distance
    Triangle in-equality
    Distance in n-dimensional Euclidean space
    Formal definition of neighbourhood
    e-d definition of continuity at a point
    Definition of continuous transformation
14 The Language of Sets
    Sets and subsets defined
    Set equality
    Null set
    Power set
    Union and Intersection
    Laws of set theory
    Venn diagrams
    Index sets
    Cartesian product
    n-dimensional Euclidean space
15 Functions
    Definition of function
    Domain and codomain
    Image and image set
    "Injection, bijection, surjection"
    Examples of functions as transformations
    Complex functions
    Point at infinity
    Bilinear functions
    Inverse functions
    Identity function
    "Open, closed, and half-open subsets of R "
    Tearing by discontinuous functions
16 Metric Spaces
    Distance in Rn
    Definition of metric
    Continuity in terms of neighbourhoods
    Complete system of neighbourhoods
    Requirement for proof of non-continuity
    Functional relationships between d and e
    Limitations of metric
17 Topological Spaces
    Concept of open set
    Definition of a topology on a set
    Topological space
    Examples of topological spaces
    Open and closed sets
    Redefining neighbourhood
    Metrizable topological spaces
    "Interior, exterior, boundary"
    Continuity in terms of open sets
    Homeomorphic topological spaces
    Connected and disconnected spaces
    Completeness: not a topological property
    Completeness of the real numbers
    "Topology, the starting point of real analysis"
  Historical Note
  Exercises and Problems