10. Topological spaces.
The notions of limit and continuity go back to antiquity; it would be impossible to make a complete history of them without studying systematically from this point of view, not only the mathematicians, but also the Greek philosophers and in particular Aristotle, nor also without following the evolution of these ideas across the mathematics of the Renaissance and the beginnings of the differential and integral Calculus. Such a study, which it would certainly be very interesting to undertake, would go far beyond the framework of this note.
It is Riemann who must be considered as the creator of topology, as of so many other branches of modern mathematics: it is in fact he who, first, sought to disengage the notion of topological space, conceived the idea of an autonomous theory of these spaces, defined the invariants (the "Betti numbers") which were to play the greatest role in the later development of topology, and gave its first applications to analysis (periods of abelian integrals). But the movement of ideas in the first half of the XIXth century had not happened without preparing the way for Riemann in more than one way. Indeed, the desire to settle mathematics on a firm base, which was the cause of so much important research during the whole of the XIXth century and up to our day, had led to the correct definition of the notion of a convergent series and of a sequence of numbers tending to a limit (Cauchy, Abel) and that of a continuous function (Bolzano, Cauchy). On the other hand, the geometric representation (by points in the plane) of complex numbers, or, as had been said until then, "imaginary numbers" (qualified sometimes also, in the XVIIIth century, as "impossible" numbers), a representation due to Argand and Gauss (see p. 161), had become familiar to the majority of mathematicians: it constituted progress of the same order as in our days the adoption of geometric language in the study of Hilbert space, and contained in embryo the possibility of a geometric representation of every object which was susceptible to continuous variation; Gauss, who in any case was naturally led to such conceptions by his research on the foundations of geometry, on non-Euclidean geometry, on curved surfaces, seems already to have had this possibility in view, for he uses the words "size twice extended" in defining (independently of Argand and the French mathematicians) the geometric representation of imaginary numbers ([124 a], v. II, pp. 101-103 and pp. 175-178).
It is on the one hand his research on algebraic functions and their integrals, on the other hand his thoughts (largely inspired by his study of the works of Gauss) on the foundations of geometry, that led Riemann to formulate a programme of study which is exactly that of modern topology, and to give this programme a start towards its realisation. Here for example is how he expresses himself in his Theory of abelian functions ([259 a], p. 91):
"In the study of functions that are obtained by the integration of exact differentials, some theorems of analysis situs are almost indispensable. Under this name, which was used by Leibniz, although perhaps with a somewhat different meaning, it is in order to include that part of the theory of continuous quantities which studies these quantities, not as independent of their position and measurable the ones by means of the others, but in abstracting every idea of measurement and studying solely their relationships of position and inclusion. I reserve the right to treat this object later, in a way that is completely independent of all measurement ...".
And in his famous inaugural Lecture "On the hypotheses that form the foundation of geometry" ([259 a], p. 272):
"...The general notion of quantity several times extended, which contains that of spatial quantity as a particular case, has remained completely unexplored ...(p. 272)".
"...The notion of quantity presupposes an element susceptible of different determinations. Following on whether or not one can pass from one determination to another by continuos transitions, these determinations make up a continuous multiplicity (of which they will be called the points) or a discrete multiplicity (p. 273)".
"...The measurement consists of a superposition of quantities to compare; to measure, there must therefore be a means of bringing one quantity onto another. In the absence of such a means, it is not possible to compare two quantities except when one is part of the other ... The studies that can then be made about this subject constitute part of the theory of quantities, independent of the measurement, and where the quantities are not considered as having an existence independent of their position, nor as being expressible by means of a unit of measure, but as parts of a multiplicity. Such studies have become a necessity in many parts of mathematics, and in particular for the theory of analytic multiform functions ... (p. 274)".
"...The determination of the position in a given multiplicity is thus reduced, each time that it is possible, to a finite number of numerical determi-
Riemann means by that, as is shown in what follows, a part of topological space with an arbitrary number of dimensions.
nations. There are, it is true, some multiplicities, in which the determination of the position requires, not a finite number, but an infinite sequence or a continuous multiplicity of determinations of quantities. Such multiplicities are made up for example by the possible determinations of a function in a given domain, the position of a figure in space, etc." (p. 276).
In this last sentence will be noticed the first idea of a study of functional spaces; besides, already in Riemann’s Dissertation the same idea is to be found expressed "The set of these functions", he says in connection with the minimum problem known under the name of Dirichlet’s principle, "makes up a connected domain, closed in itself" ([259 a], p. 30), which is, in an imperfect form, nonetheless the embryo of the proof that Hilbert would give later of Dirichlet’s principle, and even of the majority of the applications of functional spaces to the calculus of variations.
As we have said, Riemann gave a start to the carrying out of this grandiose programme, by defining the "Betti numbers", first of a surface ([259 a], pp. 92-93), then (ibid., pp. 479-482; cf. also [259 c]) of a multiplicity of an arbitrary number of dimensions, and in applying this definition to the theory of integrals; results which inaugurate algebraic Topology, a branch of mathematics whose development has not stopped accelerating since the beginning of the XXth century, and should not be traced here.
As for the general theory of topological spaces, such as was foreseen by Riemann, it was necessary in order for it to develop, that the theory of real numbers, of sets of numbers, of sets of points on the straight line, in the plane and in space, should first be studied more systematically than had been the case in Riemann’s time: this study was linked, on the other hand, to researches (semi philosophical with Bolzano, essentially mathematical with Dedekind) on the nature of irrational numbers, as well as progress in the theory of functions of a real variable (to which Riemann himself brought an important contribution by his definition of the integral and his theory of trigonometric series, and which was the object, among others, of the work of du Bois-Reymond, Dini, Weierstrass); it was the work of the second half of the XIXth century and most particularly Cantor, who was the first to define (first on a straight line, then in Euclidean space of dimensions) the notions of accumulation point, closed and perfect set, and who obtained the essential results on the structure of these sets on the straight line(cf. p. 155): one should consult on this, not only the Works of Cantor [47], but also his very interesting correspondence with Dedekind [48], where will be found clearly expressed also the idea of the number of dimensions considered as a topological invariant.
Further progress of the theory is stated for example, in a semi-historical, semi-systematic form in the book of Schoenflies [275 a]: by far the most important was the Borel-Lebesgue theorem (proved first by Borel for a closed interval on the straight line and a countable family of open intervals covering it).
The ideas of Cantor had first met a fairly lively opposition (cf. p. 28). At least his theory of sets of points on the straight line and in the plane was soon used and widely spread by the French and German schools of the theory of functions (Jordan, Poincaré, Klein, Mittag-Leffler, then Hadamard, Borel, Baire, Lebesgue, etc.): the first volumes of the Borel collection, in particular, each contain an elementary statement of this theory (see for example [32 a]). As these ideas were spread, people began to think in different ways about their possible application to sets, not of points anymore, but of curves or functions: an idea that saw the light of day, for example, already in 1884, in the title “On the limit curves of a family of curves” of a memoir of Ascoli [10], and which expresses itself in a communication of Hadamard to the congress of mathematicians at Zurich in 1896 [141]; it is tightly linked also to the introduction of “line functions” by Volterra in 1887, and to the creation of the “functional calculus”, or theory of functions where the argument is a function (on that, the work of Volterra on functional Analysis [322 b] can be consulted). On the other hand, in the famous memoir ([163 a], v. III, pp. 10-37) where Hilbert, taking up again on this point the ideas of Riemann, proved the existence of the minimum in the Dirichlet principle and inaugurated the “direct method” of the calculus of variations, one sees clearly appearing the interest that there is in considering sets of functions where the Bolzano-Weierstrass principle is valid, that is to say every sequence contains a convergent sub sequence; such sets would soon indeed play an important role, not only in the calculus of variations, but in the theory of functions of a real variable (Ascoli, Arzelà) and in that of functions of a complex variable(Vitali, Carathéodory, Montel). Finally the study of functional equations, and most particularly the solution by Fredhom of the type of equation that bears his name [116], made familiar the consideration of a function as an argument, and of a set of functions as the analogue of a set of points, in connection with which it was quite as natural to use geometric language as for the points of a Euclidean space of dimensions (a space which, also, escapes “intuition”, and for that reason, remained for a long time an object of mistrust for many mathematicians). In particular, the memorable works of Hilbert on integral equations [163 b] end with the definition and the geometric study of Hilbert space by Erhard Schmidt [274 b], in complete analogy with Euclidean geometry (see p. 212).
Meanwhile the notion of axiomatic theory had taken an increasingly greater importance, thanks above all to numerous works on the foundations of geometry, among which those of Hilbert [163 c] exercised a particularly decisive influence; during this very same work, Hilbert had been led to state exactly, from 1902 ([163 c], p. 180), a first axiomatic definition of the “multiplicity twice extended” in the sense of Riemann, a definition which constituted, he said, “the foundation of a rigorous axiomatic treatment of analysis situs”, and already used neighbourhoods (in a meaning restricted by the exigencies of the problem to which Hilbert was then limiting himself).
The first attempts to bring out that which was in common to the properties of sets of points and of functions (without the intervention of a notion of "distance"), were made by Fréchet [115 a] and F. Riesz [260 b]; but the first, starting from the notion of countable limit, does not succeed in constructing, for non metrizable spaces, an axiom system which is convenient and fruitful; at least he recognised the relationship between the Bolzano-Weierstrass principle and the Borel-Lebesgue theorem; it is in this connection that he introduced the word "compact", even though with a meaning somewhat different from that given to it today. As for F. Riesz, who started from the notion of point of accumulation (or rather, which comes to the same thing, of "derived" set), his theory was still incomplete, and remained besides as only a sketch.
With Hausdorff ([152 a], chap. 7-8-9) general topology as it is understood today starts. Taking up again the notion of neighbourhood, he knew how to choose, among the axioms of Hilbert on neighbourhoods in the plane, those that could give to his theory at the same time all the precision and the generality that were desirable. The chapter where he develops its consequences remains a model of axiomatic theory, abstract but adapted to applications from the beginning. That was, quite naturally, the point of departure for subsequent research on general topology, and principally the work of the school of Moscow, directed for the greater part towards the problem of metrisation (cf. p. 166): we must remember from this especially here the definition, by Alexandroff and Urysohn, of compact spaces (under the name of "bicompact spaces"), then the proof by Tychonoff [313] of the compactness of products of compact spaces. Finally the introduction of filters by H. Cartan [53], which while supplying a very valuable instrument in view of all kinds of applications (where it is an advantageous substitute for the notion of "Moore-Smith convergence" [227]), came, thanks to the theorem on ultrafilters, to achieve a clarification and simplification of the theory.