CHAPTER 1

INFORMAL TOPOLOGY

In this chapter the notion of a topological space is introduced, and informal ad hoc methods for identifying equivalent topological spaces and distinguishing between nonequivalent ones are provided.

The last book of Euclid’s opus Elements is devoted to the construction of the five Platonic solids pictured in Figure 1.1. A fact that Euclid did not mention is that the counts of the vertices, edges, and faces of these solids satisfy a simple and elegant relation. If these counts are denote by v, e, and f, respectively, then

(1)

Specifically, for these solids we have:

Figure 1.1 The Platonic solids.

A Platonic solid is defined by the specifications that each of its faces is the same regular polygon and that the same number of faces meet at each vertex. An interesting feature of Equation (1) is that while the Platonic solids depend on the notions of length and straightness for their definition, these two aspects are absent from the equation itself. For example, if each of the edges of the cube is either shrunk or extended by some factor, whose value may vary from edge to edge, a lopsided cube is obtained (Fig. 1.2) for which the equation still holds by virtue of the fact that it holds for the (perfect) cube. This is also clearly true for any similar modification of the other four Platonic solids. The fact of the matter is that Equation (1) holds not only for distorted Platonic solids, but for all solids as well, provided these solids are carefully defined. Thus, for the three solids of Figure 1.3 we have respectively 5 – 8 + 5 = 2, 6 – 9 + 5 = 2, and 7 – 12 + 7 = 2. The applicability of Equation (1) to all such solids was first noted by Leonhard Euler (1707–1783) in 1758, although some historians contend that this equation was presaged by certain observations of René Descartes (1596–1650).

Figure 1.2 A lopsided cube.

Euler’s equation remains valid even after the solids are subjected to a wider class of distortions which result in the curving of their edges and faces (see Figure 1.4). One need simply relax the definition of edges and faces so as to allow for any nonself intersecting curves and surfaces. Soccer balls and volleyballs, together with the patterns formed by their seams, are examples of such curved solids to which Euler’s equation applies. Moreover, it is clear that the equation still holds after the balls are deflated.

Topology is the study of those properties of geometrical figures that remain valid even after the figures are subjected to distortions. This is commonly expressed by saying that topology is rubber-sheet geometry. Accordingly, our necessarily informal definition of a topological space identifies it as any subset of space from which the notions of straightness and length have been abstracted; only the aspect of contiguity remains. Points, arcs, loops, triangles, solids (both straight and curved), and the surfaces of the latter are all examples of topological spaces. They are, of course, also geometrical objects, but topology is only concerned with those aspects of their geometry that remain valid despite any translations, elongations, inflations, distortions, or twists.

Figure 1.3 Three solids.

Figure 1.4 A curved cube.

Another topological problem investigated by Euler, somewhat earlier, in 1736, is known as the bridges of Koenigsberg. At that time this Prussian city straddled the two banks of a river and also included two islands, all of which were connected by seven bridges in the pattern indicated in Figure 1.5. On Sunday afternoons the citizens of Koenigsberg entertained themselves by strolling around all of the city’s parts, and eventually the question arose as to whether an excursion could be planned which would cross each of the seven bridges exactly once. This is clearly a geometrical problem in that its terms are defined visually, and yet the exact distances traversed in such excursions are immaterial (so long as they are not excessive, of course). Nor are the precise contours of the banks and the islands of any consequence. Hence, this is a topological problem. Theorem 2.2.2 will provide us with a tool for easily resolving this and similar questions.

The notorious Four-Color Problem, which asks whether it is possible to color the countries of every geographical map with four colors so that adjacent countries sharing a border of nonzero length receive distinct colors, is also of a topological nature. Maps are clearly visual objects, and yet the specific shapes and sizes of the countries in such a map are completely irrelevant. Only the adjacency patterns matter.

Every mathematical discipline deals with objects or structures, and most will provide a criterion for determining when two of these are identical, or equivalent. The equality of real numbers can be recognized from their decimal expansions, and two vectors are equal when they have the same direction and magnitude. Topological equivalence is called homeomorphism. The surface of a sphere is homeomorphic to those of a cube, a hockey puck, a plate, a bowl, and a drinking glass. The reason for this is that each of these objects can be deformed into any of the others. Similarly, the surface of a doughnut is homeomorphic to those of an inner tube, a tire, and a coffee mug. On the other hand, the surfaces of the sphere and the doughnut are not homeomorphic. Our intuition rejects the possibility of deforming the sphere into a doughnut shape without either tearing a hole in it or else stretching it out and juxtaposing and pasting its two ends together. Tearing, however, destroys some contiguities, whereas juxtaposition introduces new contiguities where there were none before, and so neither of these transformations is topologically admissible. This intuition of the topological difference between the sphere and the doughnut will be confrmed by a more formal argument in Chapter 3.

Figure 1.5 The city of Koenigsberg.

Figure 1.6 Homeomorphic open arcs.

The easiest way to establish the homeomorphism of two spaces is to describe a deformation of one onto the other that involves no tearing or juxtapositions. Such a deformation is called an isotopy. Whenever isotopies are used in the sequel, their existence will be clear and will require no formal justification. Such is the case, for instance, for the isotopies that establish the homeomorphisms of all the open arcs in Figure 1.6, all the loops in Figure 1.7, and all the ankh-like configurations of Figure 1.8. Note that whereas the page on which all these curves are drawn is two-dimensional, the context is definitely three-dimensional. In other words, all our curves (and surfaces) reside in Euclidean 3-space , and the isotopies may make use of all three dimensions.

The concept of isotopy is insufficient to describe all homeomorphisms. There are spaces which are homeomorphic but not isotopic. Such is the case for the two loops in Figure 1.9. It is clear that loop b is isotopic to all the loops of Figure 1.7 above, and it is plausible that loop a is not, a claim that will be justified in Chapter 5. Hence, the two loops are not isotopic to each other. Nevertheless, they are homeomorphic in the sense that ants crawling along these loops would experience them in identical manners. To express this homeomorphism somewhat more formally it is necessary to resort to the language of functions. First, however, it should be pointed out that the word function is used here in the sense of an association, or an assignment, rather than the end result of an algebraic calculation. In other words, a function f : S → T is simply a rule that associates to every point of S a point of T. In this text most of the functions will be described visually rather than algebraically.

Figure 1.7 Homeomorphic loops.

Figure 1.8 Homeomorphic ankhs.

Figure 1.9 Two spaces that are homeomorphic but not isotopic.

Given two topological spaces S and T, a homeomorphism is a function f : S – T such that

It is the vagueness of the notion of contiguity that prevents this from being a formal definition. Since any two points on a line are separated by an infinitude of other points, this concept is not well defined. The homeomorphism of S and T is denoted by S ≈ T. The homeomorphism of the loops of Figure 1.9 can now be established by orienting them, labeling their lowest points A and B, and matching points that are at equal distances from A and B, where the distance is measured along the oriented loop (Fig. 1.10). Of course, the positions of A and B can be varied without affecting the existence of the homeomorphism.

A similar function can be defined so as to establish the homeomorphism of any two loops as long as both are devoid of self-intersections. Suppose two such loops c and d, of lengths γ and δ respectively, are given (Fig. 1.11). Again begin by specifying orientations and initial points C...