Catálogo de publicaciones - libros

Topology is one of the better-known areas of modern mathematics. Most people have heard the statement that a topologist is someone who cannot tell the difference between a tea cup and a doughnut. This is true, and we will see why in Chapter 5. Clearly, then, topology ignores some things and perceives similarities between apparently dissimilar objects. As we will discover, the key to what topology ignores, and to what it concentrates on, is the behaviour of continuous functions. Topology studies the ways in which the properties of the domain and range determine the behaviour of a continuous function.

As topology is essentially just the study of continuous functions, we should start by clarifying exactly what we mean by the word “continuous”. We will begin by considering the most familiar type of function, namely those which are defined for some (or all) real numbers, and which return a real number

So far, we have only considered functions on the real line. We have seen how to hide those annoying єs and δs in the definition of continuity, replacing them with open sets. This enables us to consider functions with domains and ranges different from R; all we need is some notion of “open set”.

A typical example of the type of statement about continuous maps that topologists try to prove is the following.

In this chapter we consider ways of relating a new space to other spaces which are more familiar. Sometimes it will be the case that the new space is actually identical, topologically, with a space that we are more familiar with, and we will study this notion in Section 5.1. When this is not the case, it is often possible to express the new space in terms of other, perhaps more familiar, spaces. In the last three sections of this chapter, we will look at some different topological constructions which will enable us to deconstruct some of the more exotic spaces that we have met.

We said at the beginning of this book that topology is about the study of continuous functions and so the ultimate goal of topology should be to describe all the continuous maps between any given pair of topological spaces. Of course, with almost any pair of spaces, there are lots of continuous functions between them – far more than we can ever hope to list or understand. For example, it is not remotely feasible to list even the continuous functions from the interval [0, 1] to itself.

We now begin our study of topological invariants, by considering the “Euler number” or “Euler characteristic.” This assigns an integer to each topological space in a way that tells us something about the topology of the space. In particular, it can sometimes tell if two spaces are not homotopy equivalent, since spaces which are homotopy equivalent have the same Euler number.

In Chapter 6 we calculated the set [S, S] of homotopy classes of maps S1 → S1 and found that [S, S] = Z, which is an Abelian group. We can describe the group operation topologically as follows.

In Chapter 8, we have seen how to take a topological space and assign to it an algebraic object carrying some information about the topology of the space. However, homotopy groups are very hard to calculate. In this chapter we introduce “homology” groups, which can be thought of as a rough approximation to the homotopy groups of a space. In defining the homotopy groups of a space X, we considered all maps

For simplicial homology we supposed that our space had already been expressed as a simplicial complex, i.e., decomposed into a union of simplices. Some spaces cannot be expressed in such a way, and even those that can, can usually be expressed as a simplicial complex in many different ways. Choosing one way can obscure some details of the space. For these reasons “singular” homology was developed, which gets around this problem by, in a very loose sense, considering all possible simplicial decompositions.