Historical note

(Numbers in brackets refer to the bibliography at the end of this note.)

The ideas of limit and continuity go back to antiquity, and a complete history of them could not be written without studying systematically from this point of view not only the Greek mathematicians but also the Greek philosophers and Aristotle in particular. It would also be necessary to trace the evolution of these ideas through Renaissance mathematics and the beginnings of the differential and integral calculus. Such a study, though it would undoubtedly be interesting to undertake, would go far beyond the framework of this note.

It is Riemann who should be considered as the creator of topology, as of so many other branches of modern mathematics. He was the first to attempt to formulate the notion of a topological space; he conceived the idea of an autonomous theory of such spaces; he defined invariants (the "Betti numbers") which were to play a pre-eminent part in the later development of topology; and he was the first to apply topology to analysis (periods of abelian integrals). But the current of ideas in the first half of the nineteenth century had prepared the path for Riemann in more ways than one. In the first place, the desire to put mathematics on a firm basis, which was the cause of so many important researches throughout the nineteenth century and up to the present day, had led to a correct understanding of the notions of a convergent series and a sequence of numbers tending to a limit (Cauchy, Abel) and to the notion of a continuous function (Bolzano, Cauchy). On the other hand, the geometrical representation (by points of a plane) of the complex numbers (or, as they had hitherto been called, "imaginary" or even "impossible" numbers) which was due to Gauss and Argand, had become familiar to the majority of mathematicians; it constituted an advance of the same order as the adoption, in our century, of the language of geometry in the study of Hilbert space, and contained the germ of the possibility of a geometrical representation of every object capable of continuous variation. Gauss, who was in any case naturally led to such concepts by his researches on the foundations of geometry, on non-Euclidean geometry and on curved surfaces, seems to have had this possibility already in mind, for he uses the words "magnitude twice extended" when defining (independently of Argand and the French mathematicians) the geometrical representation of complex numbers ([1], pp. 101-103 and 175-178).

Riemann's work on algebraic functions and their integrals and his reflections on the foundations of geometry (largely inspired by his study of Gauss's work) led him to formulate a program of study, which is precisely that of modern topology, and to begin to realize this program. Here, for example, is what he says in his theory of abelian functions ([2], p. 91):

"In the study of functions obtained by integrating exact differentials, some theorems of Analysis situs are almost indispensable. By this name, which was used by Leibnitz, although perhaps in a somewhat different sense, should be called that part of the theory of continuous magnitudes which studies these magnitudes, not independently of their position and by measuring them in terms of each other, but rather by abstracting all ideas of measurement and considering only their relations of position and inclusion. I reserve to a later occasion an investigation completely independent of all measurement..."

And in his famous inaugural lecture "On the hypotheses which underlie Geometry" ([2], p. 272):

"...the general concept of a magnitude many times extended (*) which contains as a particular case that of spatial magnitude, has remained completely unexplored..." (p. 272)

"...The notion of magnitude presupposes that an element is capable of different determinations. According as one can pass from one determination to another by a continuous process of transition or not, these determinations form a continuous or a discrete manifold: in the former case the determinations are called points of the manifold..." (p. 273)

"...Measurement consists of superposition of the magnitudes to be compared, hence in order to measure we need some means of using one magnitude as a yardstick for another. In the absence of this we can compare two magnitudes only if one is part of the other... The investigations which can be undertaken in this context form a part of the theory of magnitudes which is independent of the theory of measurement and in which the magnitudes are considered not as existing independently of their position nor as expressible in terms of a unit of measurement, but as regions in a manifold. Such investigations have become necessary in several parts of mathematics, in particular in the theory of many-valued analytic functions..." (p. 274)

(*) As the sequel shows, Riemann means by this phrase a subset of a topological space of arbitrary dimension.

"... The determination of position in a given manifold, whenever this is possible, can be reduced to a finite number of numerical determinations. There are however manifolds in which the determination of position requires not a finite number but an infinite sequence or even a continuous manifold of determinations of magnitudes. For example, the possible determinations of a function on a given domain, or the possible forms of a spatial figure, give manifolds of this type." (p. 276)

Note in this last phrase the first idea of a study of functional spaces; Riemann had already expressed the same idea in his dissertation: "the totality of these functions", he stated in connection with the minimal problem known as Dirichlet's principle, "forms a connected domain which is closed in itself" ([2], p. 30); this, though imperfectly expressed, is nevertheless the germ of the proof which Hilbert was later to give of Dirichlet's principle, and of most of the applications of function spaces to the calculus of variations.

As we have said, Riemann began the execution of this grandiose program by defining the "Betti numbers", first for a surface ([2], pp. 92-93) and later ([2], pp. 479-482; cf. also [3]) for a manifold of any dimension, and applied this definition to the theory of integrals; for this and for the considerable development of this theory since Riemann's time we refer the reader to the Historical Notes to the chapters on algebraic topology in this series of volumes.

Before a general theory of topological spaces, such as Riemann had envisaged, could be developed, it was necessary that the theory of real numbers, of sets of numbers, of sets of points on a line, in a plane and in space should be more systematically investigated than they had been in Riemann's time. Such investigations were related on the other hand to research into the nature of irrational numbers (semi-philosophical by Bolzano and essentially mathematical by Dedekind) and to progress in the theory of functions of a real variable in which Riemann himself made an important contribution by his definition of the integral and his theory of trigonometrical series, and to which du Bois-Reymond, Dini and Weierstrass, among others, contributed; they were the work of the second half of the nineteenth century, and especially the work of Cantor, who was the first to define (originally on the line, later in Euclidean space of $n$ dimensions) the notions of point of accumulation, closed set, open set, perfect set, and obtained the essential results on the structure of these sets on the line (cf. the Historical Note to Chapter IV). In this context, not only the works of Cantor [4] should be consulted, but also his extremely interesting correspondence with Dedekind [5], where the idea of dimensionality as a topological invariant can be found clearly expressed. The later progress of the theory is traced in a semi-historical, semi-systematic form in Schöenflies' book [6]; by far the most important acquisition was the theorem of Borel-Lebesgue, namely the fact that every bounded closed subset of Euclidean $n$-space $\mathbf{R}^n$ (cf. Chapter VI, § 1) satisfies axiom $(\mathrm{C''})$ of § 9 of this chapter (the theorem was first proved by Borel for a closed interval on the line and a countable family of open intervals covering it).

Cantor’s ideas had originally met with vigorous opposition (cf. the Historical Note to Book I, Chapters I-IV). At any rate his theory of point-sets on the line and in the plane was quickly made use of and disseminated by the French and German schools of function-theory (Jordan, Poincaré, Klein, Mittag-Leffler, and later Hadamard, Bore, Baire, Lebesgue, etc.); each of the early Borel treatises, in particular, contains an elementary exposition of this theory (see for example [7]). As these ideas spread, their possible application to sets, not of points but of curves or functions, began to be considered in various quarters, as witness the title “On the limit curves of a variety of curves” of a memoir by Ascoli in 1883 [8] and a communication by Hadamard to the congress of mathematicians at Zürich in 1896 [9]; all this is closely related to the introduction of “line functions” by Volterra in 1887 and to the creation of “functional calculus” or theory of functions in which the argument is a function (cf. Voltera’s book on functional analysis [10]). On the other hand, Hilbert’s famous memoir [11], in which, taking up Riemann’s ideas again, he proved the existence of the minimum in Dirichlet’s principle and inaugurated the “direct method” in the calculus of variations, showed clearly the importance of considering sets of functions in which the Bolzano-Weierstrass principle holds, that is to say in which every sequence has a convergent subsequence. Such sets were beginning in any case to play an important part, not only in the calculus of variations but also in the theory of functions of a real variable (Ascoli, Arzelà) and a little later in the theory of functions of a complex variable (Vitali, Carathéodory, Montel). Finally the study of functional equations, and especially the solution by Fredholm of the type of equation which bears his name, made it commonplace to consider a function as an argument and a set of functions as a set of points, and as natural to use the language of geometry in this context as in Euclidean space of $n$ dimensions (a space which equally eludes “intuition” and for this reason remained long an object of distrust to many mathematicians). In particular the memorable work of Hilbert on integral equations [12] led to the definition and geometrical study of Hilbert space by Erhard Schmidt [14], in complete analogy with Euclidean geometry.

Meanwhile the concept of an axiomatic theory had acquired more and more importance, thanks to much work on the foundations of geometry; here Hilbert’s contributions [13] had a particularly decisive influence. In the course of this work, Hilbert had been led to formulate in 1902 ([13], p. 180) the first axiomatic definition of the “manifold twice extended” in the sense of Riemann, a definition which constituted, said Hilbert, “the foundation of a rigorous axiomatic treatment of Analysis situs''. Hilbert also made use of neighbourhoods (in a sense restricted by the demands of the problem to which he limited himself).

The first attempts to abstract what is common to properties of sets of points and sets of functions are due to Fréchet [15] and F. Riesz [16]. The former started from the notion of countable limit and did not succeed in constructing a convenient and fruitful system of axioms, but at least he recognized the relationship between the principle of Bolzano-Weierstrass [which is just axiom (C) of § 9, restricted to countable sequences] and the Borel-Lebesgue theorem [axiom (C'') of § 9]; in this connection he introduced the word "compact", although in a sense somewhat different from that in which it is used in this series of volumes. As to F. Riesz, who took as his starting point the concept of point of accumulation (or rather of "derived set", which amounts to the same thing), his theory was again incomplete and appeared only in outline form.

General topology as it is understood today began with Hausdorff ([17], Chapters 7, 8, 9), who again took up the concept of neighbourhood (by which he meant what in the terminology of this series of volumes is called an "open neighbourhood") and chose from Hilbert's axioms for neighbourhoods in the plane those which gave his theory all the precision and generality desired. The axioms he took as a starting-point were essentially (taking into account the difference between his concept of neighbourhood and ours) axioms (V_I), (V_{II}), (V_{III}), (V_{IV}) of § 1 and (H) of § 8, and the chapter in which he develops the consequences of these axioms has remained a model of axiomatic theory, abstract but adapted in advance to applications. Hausdorff's work was naturally the point of departure for later research in general topology and especially for the work of the Moscow school, which was largely directed towards the problem of metrization (cf. the Historical Note to Chapter IX); here we recall especially the definition of compact spaces (under the name of "bicompact spaces") by Alexandroff and Urysohn, and Tychonoff's proof of the compactness of products of compact spaces [19]. Finally, the introduction of filters by H. Cartan [20] has brought to topology a valuable instrument, usable in all sorts of applications (in which it replaces to advantage the notion of "Moore-Smith convergence" [18]). Furthermore, the development of the theorem on ultrafilters (Theorem 1, § 6), has clarified and simplified the theory.

[1] C. F. Gauss, Werke, Vol. 2, Göttingen, 1863. [2] B. Riemann, Gesammelte mathematische Werke, 2nd edition, Leipzig (Teubner), 1892. [3] B. Riemann, in Lettere di E. Betti a P. Tardy, Rend. Accad. Lincei (5) 24¹ (1915) pp. 517-519. [4] G. Cantor, Gesammelte Abhandlungen, Berlin (Springer), 1932. [5] G. Cantor and R. Dedekind, Briefwechsel, Actualités scientifiques et industrielles, no. 518, Paris (Hermann), 1937. [6] A. Schoenflies, Entwicklungen der Mengenlehre und ihrer Anwendungen, Part I, 2nd edition, Leipzig-Berlin (Teubner), 1913. [7] E. Borel, Leçons sur la théorie des fonctions, 2nd edition, Paris (Gauthier-Villars), 1914. [8] G. Ascoli, Le curvi limiti de una varietà data di curve, Mem. Accad. Lincei (3) 18 (1883) p. 521-586. [9] J. Hadamard, Sur certaines applications possibles de la théorie des ensembles, Verhandl. Intern. Math.-Kongress, Zürich, 1898, pp. 201-202. [10] V. Volterra, Theory of Functionals, London-Glasgow (Blackie and Son), 1930. [11] D. Hilbert, Gesammelte Abhandlungen, vol. 3, Berlin (Springer), 1935, pp. 10-37 [= Jahresbericht des D.M.V., 8 (1900), p. 184, and Math. Ann., 59 (1904), p. 161]. [12] D. Hilbert, Grundzüge einer allgemeinen Theorie der Integralgleichungen, 2nd edition, Leipzig-Berlin (Teubner), 1924. [13] D. Hilbert, Grundlagen der Geometrie, 7th edition, Leipzig-Berlin (Teubner), 1930.

[14] E. Schmidt, Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten, Rend. Palermo, 25 (1908), pp. 53-77. [15] M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Palermo, 22 (1906), pp. 1-74. [16] F. Riesz, Stetigkeitsbegriff und abstrakte Mengenlehre, Atti del IV Congresso Intern. dei Matem., Bologna, 1908, vol. 2, pp. 18-24. [17] F. Hausdorff, Grundzüge der Mengenlehre, 1st edition, Leipzig (Veit), 1914. [18] E. H. Moore and H. L. Smith, A general theory of limits, Am. Journ. of Math., 44 (1922) pp. 102-121. [19] A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Ann., 102 (1930), pp. 514-561. [20] H. Cartan, Théorie des filtres; filtres et ultrafiltres, C. R. Acad. Sc. Paris, 205 (1937), pp. 595-598 and pp. 777-779.