From Geometry to Topology

H. Graham Flegg

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486138497
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4–12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 edition.

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.

The traditional study of geometry is concerned with certain properties of figures in Euclidean space. For example, consider the triangle of

Fig. 1.1

Figure 1.1. This triangle has certain properties such as:

the values of its angles,
the lengths of its sides,
the number of sides,
its separation of a plane surface into a region inside and a region outside its perimeter,
the length of its perimeter,
the area enclosed by its perimeter,
its orientation with respect to some given axes in space,
its colour.

Not all these properties are geometric, and, in order to determine which are and which are not, it is necessary to introduce the concept of geometric equivalence, often termed congruence.

Intuitively, two plane figures are congruent if and only if one may be placed on top of the other so as to coincide perfectly. The properties which are shared by every figure congruent to a given figure are geometric properties. Clearly, all but the last two of the properties listed above are geometric.

The operation of placing one plane figure upon another needs more precise definition. The triangle of Figure 1.2, for example, is congruent to that of Figure 1.1. Superimposing this second triangle upon the first involves what is known as a rigid transformation (or isometry). There are three fundamental rigid transformations: translation, rotation and reflection. Every rigid transformation can be expressed in terms of these.

Fig. 1.2

Translation of a point P in a plane is shown in Figure 1.3. If P has co-ordinates (x, y) with respect to the given axes, then the point P′ to which it is translated has co-ordinates (x′, y′) where

x′ = x+a, y′ = y+b,

a being the distance moved in the positive x-direction and b the distance moved in the positive y-direction. (In fact, the figure shows that the transformation of P to P′ can be naturally decomposed into two translations, one in the positive x-direction and one in the positive y-direction.)

A plane figure, however, consists not of a single point but of an infinite number of points, though in the case of a triangle three points (the vertices) are sufficient to specify it uniquely. Figure 1.4 shows the translation of a triangle under the same transformation as that of Figure 1.3. Every point belonging to the original triangle is translated by the same amount a in the positive x-direction and by the same amount b in the positive y-direction. Thus the translation, T say, is given by

T:(x, y)

(x+a, y+b)

(which is read as “points (x, y) map to points (x+a, y+b)”), where the set of all points {(x, y)} is the subset of the plane consisting of the perimeter and interior of the original triangle. In a similar way, we can think of any plane figure, or the entire plane itself, being translated under T. In the latter case, x and y would be any real number pair, and the set of all points {(x, y)} would be the whole plane, R × R (the Cartesian product of the set of real numbers with itself).

Fig. 1.3

Fig. 1.4

Certain properties, such as the number of sides, the number of vertices, and the separation of the plane into an area inside and an area outside the perimeter of the triangle, are obviously preserved under translations such as T. To show that lengths are preserved, consider any two points P₁, P₂ with co-ordinates (x₁, y₁), (x₂, y₂) respectively. The length of the line P₁P₂ is defined as

√[(x₂–x₁)² + (y₂–y₁)²].

Under T, the line P₁P₂ is translated to P₁′P₂′, say, with co-ordinates (x₁+a, y₁+b), (x₂+a, y₂+b) respectively. The length of P₁′P₂′ is thus

√[((x₂+a)–(x₁ + a))² + (y₂+b)–(y₁ + b))²]
= √[(x₂–x₁)² + (y₂–y₁)²],

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Table of contents

• Title Page
• Copyright Page
• Author’s Preface
• Acknowledgements
• Table of Contents
• 1 - Congruence Classes
• 2 - Non-Euclidean Geometries
• 3 - From Geometry to Topology
• 4 - Surfaces
• 5 - Connectivity
• 6 - Euler Characteristic
• 7 - Networks
• 8 - The Colouring of Maps
• 9 - The Jordan Curve Theorem
• 10 - Fixed Point Theorems
• 11 - Plane Diagrams
• 12 - The Standard Model
• 13 - Continuity
• 14 - The Language of Sets
• 15 - Functions
• 16 - Metric Spaces
• 17 - Topological Spaces
• Historical Note
• A Selection of Exercises and Problems
• Bibliography
• Index
• A CATALOG OF SELECTED