From Geometry to Topology

by H. Graham Flegg

Overview

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.

Product Details

ISBN-13:	9780486138497
Publisher:	Dover Publications
Publication date:	03/08/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	208
File size:	8 MB

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From Geometry to Topology

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By H. Graham Flegg

Dover Publications, Inc.

Copyright © 1974 H. Graham Flegg
All rights reserved.
ISBN: 978-0-486-13849-7

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CHAPTER 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

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.

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).

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

showing that length is preserved under T. Since T represents any translation in the plane, length is preserved under all such translations.

Rotation of a point P about the origin of a plane co-ordinate system is shown in Figure 1.5. If P has co-ordinates (x, y), then P' will have co-ordinates (x cos φf-y sin φ, x sin φ + y cos φ), where φ is the angle through which the line OP is rotated, as shown, to give OP'. Consider again any two points P1, P2 with co-ordinates (x1, y1), (x2, y2) respectively. The length of the line joining the two points P1', P2' to which P1, P2 are transformed under a rotation through angle φ about the origin is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

showing that length is again preserved under rotation about the origin. This can be extended to rotations about any point in the plane quite simply. Figure 1.6 shows the rotation of a square in the plane about a point 0' with co-ordinates (a, b). This transformation must preserve length, since the axes can be regarded as temporarily translated (as shown) for the purposes of the rotation. 0' is now the new origin, and rotations about the origin have already been shown to preserve length. The temporary translation of the axes does not affect the situation, since it has previously been shown that length is preserved under all translations of the plane.

Reflection of two points P1, P2 in a given line is shown in Figure 1.7. Rather than repeat a direct formula method for showing that the length of any line P1P2 is preserved under reflection, it is simpler first to rotate the whole system about the point of intersection of the given line with the x-axis (or translate the system if the given line and the x-axis are parallel) so that they coincide. The rotation (or translation) preserves length. It is now necessary only to consider the situation shown in Figure 1.8. If the co-ordinates of P1, P2 are (x1, y1), (x2, y2) respectively, then the co-ordinates of P1', P2' are (x1,-y1), (x2,-y2) respectively; and since, in determining length according to [square root of [(x2-x1)2 + (y2-y1)2], the formula is unaffected by the substitution of -y1, -y2 for y1, y2 respectively, because the term involving the y's is squared, reflection in the x-axis, and hence in any line, preserves length.

The three rigid transformations, translation, rotation and reflection, thus all have this important property of preserving length. Length is therefore said to be invariant under these transformations. Clearly, many other properties of figures are also preserved under the rigid transformations, for example, values of angles, area, the number of sides of a polygon, and so on. One of the most obvious properties not preserved is orientation. Properties which are preserved are said to be geometric.

The examples of transformations considered so far have been confined to transformations in a plane. It is not difficult, however, to extend the same principles to three dimensions and to consider solid three-dimensional objects. The length of a line P1P2 is then defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the rectangular Cartesian co-ordinates of P1, P2 are (x1, y1, z1), (x2, y2, z2) respectively. Indeed, there is no mathematical reason for stopping at three dimensions, and the formula for length clearly has its general counterpart in n-dimensional space. The same extension to three- and higher dimensional space applies to the consideration of invariance under the rigid transformations, though it becomes extremely difficult to visualise what is happening in any space of dimension greater than three.

A space consisting of all points (x1, x2, ..., xn) where the distance between x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is termed a n-dimensional Euclidean space. The set of all figures in any n -dimensional Euclidean space can be divided up into distinct subsets such that in any given subset all the figures are equivalent, in the sense that they can be transformed into each other under one or more of the three rigid transformations. Thus the triangles of Figures 1.1 and 1.2 would each belong to one subset, the two squares of Figure 1.6 would each belong to another subset, and so on. The two triangles shown in Figure 1.9 would, however, belong to different subsets, as would the square and circle of Figure 1.10. Such subsets are termed equivalence classes, and the relation

"is congruent to"

on the set of all figures in Euclidean space, which holds for all members of any one equivalence class, is an equivalence relation. (An equivalence relation, i.e. a relation having the reflexive, symmetric, and transitive properties, separates the set on which it is defined into disjoint equivalence classes in a unique manner. Thus, if the equivalence relation is changed, the equivalence classes are also necessarily changed.)

In the study of Euclidean geometry, no distinction is made between the members within any one equivalence class. They all share identically the same geometric properties, each is congruent to the other, and hence the equivalence classes of Euclidean geometry are often termed congruence classes. To determine that two figures belong to different congruence classes, it is sufficient to find one geometric property which they do not have in common. For example, the triangles of Figure 1.9 do have the same area, since the lengths of their bases are the same and they have the same perpendicular heights. However, they have different angles, and this on its own is sufficient to determine that they belong to different classes, notwithstanding the fact that there are a number of geometric properties which they do share.

Euclidean geometry is thus concerned with the study of classes of figures, and in this context the properties of interest are those which enable it to be determined that two figures belong to different congruence classes by virtue of not sharing any one of these properties.

CHAPTER 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.

The individual congruence classes discussed in Chapter 1 can be further divided by taking account, in some way, of orientation in space. For example, in the plane, it may be required that the sides of equivalent polygonal figures make the same angles with some given line. In Figure 2.1 the two triangles are congruent, but in addition they are identically orientated with respect to the line PQ. Triangles not so orientated now belong to different equivalence classes. Within any one equivalence class, the members still share all the same geometric properties, but they share also the non-geometric property of defined orientation. In this new orientated geometry, the only transformation permitted is the rigid transformation of translation.

Free vectors provide an example of a set of one-dimensional 'figures' for which identical orientation is a requirement for equivalence. Thus the study of free vectors involves equivalence classes, within any one of which all the members have the same length and direction (orientation). Members of one such equivalence class are depicted in Figure 2.2. Each individual vector can be thought of as tied to its starting point in space, but, for the purposes of developing a vector algebra this distinction is ignored, and only the properties common to all, namely length and direction, are considered.

Certain of the geometric congruence classes may, however, be combined by permitting a difference in one or more geometric properties within one equivalence class. For example, it is possible to drop the requirement that lengths should be the same within a class, and to permit transformations which involve proportional magnification (or contraction) in addition to the rigid transformations. In such a geometry, which may be called similarity geometry, the two triangles of Figure 2.3 belong to the same equivalence class, and no distinction is made between them.

All straight line segments are equivalent in similarity geometry. All squares are equivalent, and all circles are equivalent. Rectangles having the same ratio of side lengths are equivalent, but rectangles of different side-length ratio belong to different equivalence classes. Clearly, area is no longer an invariant under the permitted transformations, but a considerable number of geometric properties are nevertheless retained. In particular, values of angles are preserved, straight line segments remain straight line segments (though their lengths are proportionately changed), and overall 'shape' is preserved without distortion. In three dimensions, no distinction is now made between spheres of differing radii, nor between cubes of differing edge lengths. Certain of the congruence classes of ordinary geometry have now been combined. Congruent figures are indeed still equivalent, but so are all figures which in terms of geometric properties would merely be classed as similar.

The pattern which is beginning to emerge is that by increasing the number of permitted transformations, equivalence classes of figures are combined as certain properties cease to be invariant. At each particular stage, it is the study of the invariant properties which forms the basis of the appropriate 'geometry'. This process may now be continued by permitting more and more transformations. For example, in the plane the transformations given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where a, b, c, d, e, f are real numbers and ae [not equal] bd, preserves neither length, nor angle, nor 'shape'. The geometry which now results is known asaffine geometry, and its equivalence classes are combinations of equivalence classes of similarity geometry. In affine geometry the two triangles of Figure 2.4 are equivalent, as are also the two triangles of Figure 2.5.

In Figure 2.4, the particular transformation involved, in addition to a translation, is known as a shear. The two triangles have the same base length and the same perpendicular height, but the upper vertex has been moved along a line parallel to the translated base. In Figure 2.5, the particular transformation involved, in addition to a translation, is known as a strain. Again each triangle has one side which is merely a translation of a corresponding side in the other, but following a translation the remaining vertex has been moved along a line not parallel to the common side. This can be seen more clearly in Figure 2.6, which depicts the strain transformation alone.

In the case of shear, it can be seen from Figure 2.4 that it so happens that the areas of the two triangles are the same. It is not generally true, however, that area is preserved under affine transformations, as can be immediately seen from Figures 2.5 and 2.6. Indeed, since magnifications and contractions are permitted as in similarity geometry, area cannot be an affine invariant.

Figure 2.7 depicts a square transformed under shear and also under strain. The two resulting figures are each equivalent to the original square and to each other. Thus, no distinction is made between squares and parallelograms. Further, no distinction is made between circles and ellipses. There are, however, a number of very important properties which are preserved under affine transformations.

Reference to Figure 2.7 shows that under shear and strain, lines which were originally parallel remain parallel although angles between lines are not invariant. It is not difficult to show that this is generally true under all transformations of the plane defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

ae [not equal] bd. If PQ and RS are parallel, and P, Q, R, S have co-ordinates (x1, y1), (x2, y2), (x3, y3), (x4, y4) respectively, then the equality of their slopes is expressed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Under the transformation, P, Q, R, S map to P', Q', R', S' with co-ordinates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. The slope of P'Q' is thus given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is the slope of R'S'. This parallel-preserving property of affine transformations means that not all four-sided polygons are equivalent. A square or a parallelogram cannot be transformed into, for example, a trapezium since this would contravene the invariance of parallelism. All triangles are, however, equivalent; no parallel lines are involved, and successive transformations of shear and strain in addition to the rigid transformations will transform any given triangle into any other triangle.

Another important invariant under affine transformations is the ratio in which points divide straight line segments. (A proof of this on lines similar to that for the case of parallelism is not difficult to construct.) A further invariant is that finite configurations remain finite.

(Continues...)

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Table of Contents

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