Sets and Topology
Introduction to part II : Topology
Topology is the study of those properties of geometric configurations which remain invariant when these configurations are subjected to one-to-one bicontinuous transformations, or homeomorphisms (see Chapter XII, § 3). We call such properties topological invariants. For example, the property of a circle to separate the plane into two regions is a topological invariant; if we transform the circle into an ellipse or into the perimeter of a triangle, this property is retained. On the other hand, the property of a curve to have a tangent line at every point is not a topological property; the circle has this property but the perimeter of a triangle does not, although it may be obtained from the circle by means of a homeomorphism.
As can already be seen from the above example topology operates with more general concepts than analysis; differential properties of a given transformation are nonessential for topology, but bicontinuity is essential. As a consequence, topology is often suitable for the solution of problems to which analysis cannot give the answer.
The generality of topological methods rests not only on the generality of the assumptions concerning the transformations considered but also on the generality of the sets considered to which these transformations are applied. These can be arbitrary point sets on the real line or in the plane, or in n-dimensional space, or still more general sets, provided only that they be sets for which - roughly speaking - it is possible to define the concept of closed set, i. e. provided that they are topological spaces. This generality has not only a methodological significance; in modern mathematics there is a characteristic tendency to confer upon the set of objects considered in a given investigation (be these functions, sequences or curves). a topology, and hence - to a geometrisation or rather to a topologisation - of the investigation. This gives rise to numerous applications. Thus, e. g. theorems on the existence of a solution of certain types of differential equations can be expressed as theorems on the existence of invariant points of a function space (the space of continuous functions) under continuous transformations; these theorems can be proved by topological methods in a more general form and in a simpler way than was formerly done without the aid of topology.
How much more general ought the spaces considered in topology be in order that they suffice for applications and yet, because of undue generality, they do not become too artificial? The answer to this question depends on the aims which a given topological work is to serve. Because of the limited scope and elementary character of this book it seemed appropriate to limit ourselves to the spaces called metric (whose definition is given in Chapter IX, Sect 6 1). Their generality is sufficient for the majority of important applications; in particular, subsets of n-dimensional Euclidean space, sequence, spaces (of Hilbert. and Fréchet), and the space of continuous functions are metric spaces; at the same time, the very concept of a metric space is especially simple and geometrically clear.
In Chapters IX-XII we give the fundamental concepts with which we must deal in all parts of topology. The reader knows many of these concepts from analysis, in relation to the space of real or complex numbers (such as accumulation point, neighbourhood, closed set, and so on); this refers especially to Chapter XII which contains theorems on continuous functions. Theorems known from analysis, e. g. on uniform continuity, uniform convergence, the Darboux property, are proved here (and in Chapters XV and XVI) under significantly more general hypotheses. This permits us to recognize the proper extent of these theorems (which also is of didactic significance).
In the further chapters (XIII-XVIII) we gradually confine ourselves to more specific spaces: we give the important properties of separable spaces (still embracing the majority of spaces arising in applications), complete spaces (with the Baire theorem and its consequences), compact spaces (which form the generalization of closed bounded subsets of Euclidean space), connected spaces (connectedness is the precise statement of the concept of the continuity of a set) and locally connected spaces (as it turns out, curves, surfaces, multi-dimensional varieties or manifolds, with which we have to deal in differential geometry are as a rule locally connected continua).
Chapter XIX contains results from dimension theory. The concept of dimension - even though it dates from antiquity (it appears already in Euclid's Elements) - was properly defined only in recent times and this thanks to the use of topological methods. The limitations imposed on the present volume have forced us to refrain from giving some of the proofs.
We shall concern ourselves in more detail with the properties of the n-dimensional simplex, which is the fundamental concept of classical multi-dimensional geometry, in Chapter XX. In particular, we give a proof of the renowned fixed point theorem, due to L E J Brouwer, which has such extensive applications in the theory of differential equations.
Chapter XXI contains, in a very general outline, an introduction to homology theory which forms a fundamental part of algebraic topology. The latter has various applications in differential and algebraic geometry, the calculus of variations, and in other branches of analysis. This chapter depends on Chapter XX (viz., on the concept of simplex), in contrast, however, to the other chapters of this book, use is made here of algebraic concepts, especially from the theory of groups. This is the origin of the name algebraic topology in contrast to set-theoretic topology, in which we make use of the concepts and theorems of set theory. Worthy of remark is the relation of the individual branches of mathematics which we observe here: topology, being a powerful tool for functional analysis and for various branches of classical analysis, which in its turn is connected, because of its applications, with technology and the natural sciences, itself makes use of the methods of algebra and the theory of sets.
Finally, the last chapter, XXII, conceptually closely related to geometry, concerns theorems on the separation of the plane. Here is given a detailed proof of the Jordan theorem which is a classical theorem of analysis.
In its initial stages, set-theoretic and algebraic topology developed entirely independently and possessed completely different thematic. Set-theoretical topology, formerly called the theory of point sets, and concerning arbitrary subsets of Euclidean space, was begun by G Cantor, the creator of the theory of sets (circa 1880). Algebraic topology was created by H Poincaré in the last years of the past century; its objects were n-dimensional polygons and polyhedra. Some reconciliation of these two theories came rather late, about 35 years ago; this was, to a large degree, the work of P S Aleksandrov. This period was preceded by the transition from the investigation of subsets of Euclidean space in set-theoretic topology to the investigation of arbitrary topological spaces. This extension of the thematics of topology appeared to a significant degree in connection with the new mathematical investigations concerning the concept of function space and infinite-dimensional spaces introduced by Hilbert.
In the last thirty years or so there has appeared an unusually rich flourishing of topology; many fundamental problems of topology have been solved and new methods developed. Topology, which until recently was a conglomeration of loosely related theorems, became a systematic science, and topological methods penetrated into many other domains of mathematics.