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Publisher:

Dover Publications

Publication Date:

2001

Number of Pages:

208

Format:

Paperback

Price:

13.95

ISBN:

9780486419619

Category:

Textbook

[Reviewed by , on ]

Mark Hunacek

04/15/2015

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The chapter on coloring is, of course, now out of date, because the Four Color Theorem was proved several years after the publication of the original text and the Dover edition has not been updated to account for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 (mhunacek@iastate.edu) teaches mathematics at Iowa State University.

Author's Preface | ||||||||

Acknowledgements | ||||||||

1 | Congruence Classes | |||||||

What geometry is about | ||||||||

Congruence | ||||||||

"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 | ||||||||

Connectivity | ||||||||

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 | |||||||

Maps | ||||||||

"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 | |||||||

Netowrks | ||||||||

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 | ||||||||

Orientability | ||||||||

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 | ||||||||

Rank | ||||||||

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 | ||||||||

Distrance | ||||||||

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 | ||||||||

Complement | ||||||||

Laws of set theory | ||||||||

Venn diagrams | ||||||||

Index sets | ||||||||

Infinite | ||||||||

Intervals | ||||||||

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 | ||||||||

Inversion | ||||||||

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 | ||||||||

Neighbourhoods | ||||||||

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 | ||||||||

Closure | ||||||||

"Interior, exterior, boundary" | ||||||||

Continuity in terms of open sets | ||||||||

Homeomorphic topological spaces | ||||||||

Connected and disconnected spaces | ||||||||

Covering | ||||||||

Compactness | ||||||||

Completeness: not a topological property | ||||||||

Completeness of the real numbers | ||||||||

"Topology, the starting point of real analysis" | ||||||||

Historical Note | ||||||||

Exercises and Problems | ||||||||

Bibliography | ||||||||

Index | ||||||||

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