CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE SÉMINAIRE HISTOIRES DE GÉOMÉTRIES

FONDATION MAISON DES SCIENCES DE L’HOMME Année 2007

ÉQUIPE F2DS & CENTRE CHARLES MORAZÉ

54 bd Raspail 75006 Paris

Frigyes Riesz's approach to Hilbert’s problem of continuity of space. A chapter in the history of general topology Laura RODRIGUEZ

(Johannes Gutenberg – Universität, Mainz)

Lundi 23 avril de 10h à 12h, Salle 214

The question we will be dealing with is: to which extent is the concept of topological space historically related with the concept of space in geometry? For some time it was believed that Felix Hausdorff came to his definition of the concept of topological space starting from Hilbert’s notion of the plane as a two-dimensional manifold 1. It was Hermann Weyl, Hilbert’s best student, who set on the germ of this legend in an obituary dedicated to his master 2. In recent years Erhard Scholz showed that this assertion does not hold 3 . According to Scholz, Hausdorff developed his system of axioms for neighbourhoods when he was preparing a lecture on Riemannian surfaces in the spring of the year 1912. As for Hilbert’s work on the foundations of geometry, Hausdorff had known it for sure since some years before 1912 and so he had probably knew also Hilbert’s definition of the plane, and yet Scholz argues convincingly against any direct influence.

In this way Scholz proved that Hilbert’s work on the foundations of geometry did not play any essential role in the development of Hausdorff’s concept of topological

1 “With Hausdorff 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.” Nicolas Bourbaki. Elements of the History of Mathematics . Berlin: Springer 1994 (Original French edition from 1984), p. 143. A similar statement can be found in a text by C. Chevalley and A. Weil called “Hermann Weyl (1885-1955)” that appeared in L’Enseignement Mathématique tome 3, fasc. 3 (1957) and was partially reprint in: Hermann Weyl Gesammelte Abhandlungen , K. Chandrasekharan (ed.), Heidelberg: Springer 1968, 2 Hermann Weyl. David Hilbert and his mathematical work , Bulletin of AMS 50 (1944), 612-654. Here p. 638. 3 Erhard Scholz. Logische Ordnungen im Chaos: Hausdorffs frühe Beiträge zur Mengenlehre. In: Egbert Brieskorn (Hrsg.) Felix Hausdorff zum Gedächtnis. Band I: Aspekte seines Werkes. Wiesbaden, Vieweg 1996 space. This is true for Hausdorff’s concept. But some years before Hausdorff’s discovery another mathematician was working on the development of an abstract point set theory based on a general concept named “mathematical continuum”, which happens to be very closely related to Hausdorff’s notion of topological space. I am talking of Frigyes Riesz, who 1906 was a young Hungarian mathematician that had spent some years as a student and a research fellow in Göttingen and in Paris. Contrary to Hausdorff, Riesz did indeed attempt to contribute with his theory to Hilbert’s problem of the continuity of space. This is the story I want to tell you: how Riesz’s abstract point set theory was connected with Hilbert’ work on the foundations of geometry.

The aims of this talk are:

To present Hilbert’s problem of continuity of space in the context of his work on the foundations of geometry

To show how Riesz came to develop his notion of continuous space

And to evaluate the role that Riesz’s ideas played in the further development of mathematics

Hilbert and the foundations of Geometry

By 1900 David Hilbert had emerged as the leading mathematician in Germany. In 1899 Hilbert published his book The foundations of geometry, with which he succeeded in building up the Euclidean geometry axiomatically and in a strict systematic way, namely starting from a group of axioms and adding on subsequently the other four sets of axioms. The last group he added was the group of axioms of continuity, showing in this way how far the geometry can be built up without any assumptions of continuity.

In 1902 Hilbert pursued an approach to the foundations of geometry entirely different from the one followed in his book. The background was given by the so-called Riemann-Helmholtz-Lie problem of space. From the standpoint of mechanics Hermann von Helmoltz had asked in 1868: in which of all those possible spaces embraced by Riemann’s notion of a three dimensional manifold can the task be fulfilled of describing the mobility of a solid. This is known as Helmholtz’s postulate of the free mobility of rigid bodies. Helmholtz succeeded in limiting the class of those geometries to the now so-called simply connected three-dimensional manifolds with constant curvature. These are the Euclidean, the non-Euclidean by Bolya- Lobatschewski, the spherical, and the elliptical geometry.

The question had been taken up by Sophus Lie in the light of his general theory of continuous groups. Lie’s approach depends on certain assumptions of differentiability; to get rid of them is the task asked by Hilbert in his fifth Paris problem from 1900.

In the paper ”Ueber die Grundlagen der Geometrie” from 1902 Hilbert does get rid of any assumptions of differentiability as far as Riemann-Helmholtz-Lie’s problem in the plane is concerned, that means for the two-dimensional space. The proof is difficult and laborious. Very important here is that the notion of continuity is now the foundation and has therefore to be assumed from the beginning. So continuity is no more the last keystone of the building as it had been in Hilbert’s Grundlagen book.

Hilbert’s fifth mathematical problem

I would like to say some words on Hilbert’s fifth mathematical problem. In 1900 David Hilbert gave in the second International Congress of Mathematicians in Paris an invited paper. He spoke on The Problems of Mathematics and presented a list of unsolved problems that had become since quite famous. The so-called Hilbert's problems came in four groups. The first group concerned foundational questions and consisted of six problems. From them, the fifth one was related with the foundations of geometry. Hilbert described it in the following terms:

On sait qu’en employant la notion des groupes continues de transformations, Lie a établi un système d’axiomes pour la Géométrie, et a démontré au moyen de sa théorie des groupes continus de transformations que ce système d’axiomes suffit pour édifier la Géométrie.

Or dans l’exposition de sa théorie, Lie suppose toujours que les fonctions définissant les groupes sont susceptibles de différentiation; alors rien dans ces développements ne nous dit si, dans la question des axiomes de la Géométrie, cette hypothèse relative à la différentiation est de tout nécessité, ou si elle ne serait pas plutôt une conséquence du concept de groupes ainsi que des autres axiomes géométriques employés. 4

4 David Hilbert. Sur les problèmes futures des mathématiques. Traduite par M. L. Laugel. Deuxième Congrès International des Mathématiciens, Paris 1900. From this quotation I would like to point out two things:

First, the terms used by Hilbert suggest an interpretation of Lie’s contribution into a question of providing a foundation of geometry axiomatically. This is easily understood considering Hilbert’s own achievements at that time, namely his book on the foundations of geometry from 1899, which was precisely a very accurate attempt of building up the geometry systematically on the base of a system of axioms. But I have to stress that this was Hilbert’s very personal interpretation. Lie himself didn’t claim to have attempted an axiomatic foundation of geometry –and most surely not in the sense meant by Hilbert.

Second, in the fifth problem Hilbert wondered whether the assumptions of differentiability imposed by Lie had really to be included within the axioms. He wondered whether the properties of differentiability might actually follow as a consequence of the continuity of the transformations defining the rigid motions, together with the group property and the other axioms of geometry. So, looked from the perspective of his axiomatic approach, the task Hilbert was setting is actually that of finding a system of axioms of geometry and a more appropriated definition of the continuity of group of transformations from which the properties of differentiability could be proved to be a consequence. It is the question of the mutually independence of the elementary assumptions. The significance Hilbert saw in the independence of the elementary assumptions, as he stressed in his Foundations book, was to establish which of them were really unavoidable for the proof of a geometric elementary truth 5.

This lecture of Hilbert’s fifth mathematical problem is by no means new but I find it important to recall you in which way this problem was meant to clear also a foundational question, namely the question: how far can the geometry be built up using a group theoretical approach and starting alone from assumptions of continuity?

Hilbert’s problem of continuity of space from 1899 to 1902

For Hilbert that question was a double one: How shall continuous space be defined? How shall continuous group be defined? This last question will not be of our further

5 „In der Tat sucht die vorstehende geometrische Untersuchung allgemein darüber Aufschluß zu geben, welche Axiome, Voraussetzungen oder Hilfsmittel zum Beweise einer elementargeometrischen Wahrheit nötig sind.“ D. Hilbert, Grundlagen der Geometrie, Stuttgart: Teubner 1956 (8th ed.), p. 125. interest. Suffice it to say that its solution is known as the solution of Hilbert’s fifth mathematical problem and its importance is associated with the emergence of a whole new branch of mathematics: the theory of topological groups.

The question we will deal with is: how shall continuous space be defined? That is what I call “Hilbert’s problem of continuity of space”. How shall continuity be characterised in order to provide a foundation on which geometry can be built up? So Hilbert’s fifth mathematical problem as well as his paper on the foundations of geometry from 1902 concerned the problem of continuity of space.

In his paper from 1902 Hilbert explained his new approach to the foundations of geometry as follows [my translation]:

There [in his book from 1899, LR] the axioms have been ordered in a way that the continuity is assumed after all the others axioms as the last one, so that the next question follows up naturally: to what extent are the known theorems and results of the elementary geometry dependent of any assumptions of continuity?

In the present paper the continuity is, on the contrary, assumed before any other axiom, from the very beginning, via the definition of the plane and that of the motion, so that the main task becomes to find out the least possible amount of assumptions from which (and making extensively use of the continuity) the elemental objects of geometry (circle and line) can be obtained as well as those properties of them that are necessary for building up the geometry. 6

In his paper from 1902 “On the foundations of geometry” the space he studied was the two-dimensional one. He set the property of continuity from the beginning in the definition of the concept of the plane and in the definition of the notion of the motion. He took a topological approach: he applied the methods of the point-set theory and those of the analysis situs, From the point-set topology he used elementary concepts like those of open and closed set, but first of all he made extensively use of the notion of convergence of a sequence in the plane and of the concept of accumulation

6 „Dort ist eine solche Anordnung der Axiome befolgt worden, bei der die Stetigkeit hinter allen übrigen Axiomen an letzter Stelle gefordert wird, so daß dann naturgemäß die Frage in der Vordergrund tritt, inwieweit die bekannten Sätze und Schlußweisen der elementaren Geometrie von der Forderung der Stetigkeit unabhängig sind. In der vorstehenden Untersuchung dagegen wird die Stetigkeit vor allen übrigen Axiomen an erster Stelle durch die Definition der Ebene und der Bewegung gefordert, so daß vielmehr die wichtigste Aufgabe darin bestand, das geringste Maß an Forderungen zu ermitteln, um aus demselben unter weitester Benutzung der Stetigkeit die elementaren Gebilde der Geometrie (Kreis und Gerade) und ihre zum Aufbau der Geometrie notwendigen Eigenschaften gewinnen zu können.“ D. Hilbert. “Über die Grundlagen der Geometrie“ Math. Ann . 56 (1902). Reprint in D. Hilbert. Grundlagen der Geometrie . Stuttgart: Teubner 1956 (8th ed.) 178-230. Quotation p. 230. point in the plane. From the analysis situs the notion of Jordan curve and related results like the theorem of Jordan and its inverse were going to play an essential role.

Hilbert defined the plane as a two-dimensional manifold and by doing so he provided at the same time the most accurate definition for this concept of the time:

According to Hilbert, the plane E is a point-set in which to each point p corresponds a family of so-called neighbourhoods, formed by sets U ⊂ E containing p. The system of neighbourhoods fulfils six axioms. One of them postulates the existence of a one- to-one map onto a Jordan region, that means onto the inside of a closed Jordan curve in IR 2. We call these maps coordinate functions. Furthermore, when a neighbourhood has two different images, then Hilbert demanded the corresponding coordinate change to be continuous. That means: let f: U V and g: U  W be two coordinate functions of the same neighbourhood U onto the Jordan regions V and W, then the coordinate change g ∗f-1: V W is a continuous map of V onto W in IR 2. The other four axioms state: 1. For each Jordan domain V’ ⊂ V and containing the point ϕ(p), its counterimage

ϕ-1(V’) is also a neighbourhood of p. 2. A neighbourhood U of p containing the point q is also a neighbourhood of q. 3. Each two neighbourhoods U and U’ of p contain another one U’’ ⊂U∩U’ 4. To any two points p, q of the plane there exists a common neighbourhood.

Jordan regions: a basis of neighbourhoods for the plane

These are the neighbourhoods axioms from which it was once thought Hausdorff developed his concept of topological space. True is that the system of Jordan regions Hilbert used determine what in modern terms is called a neighbourhood basis for the topology of the plane. So Hilbert provided the plane with a topology but not having this notion he could not take full advantage of it. So for instance, he could not postulate that the coordinate functions shall be continuous mappings from the plane into the Cartesian space IR 2. He did not have the means for defining this notion.

Transferring Cantor’s point-set theory of IR 2

By defining the coordinate functions in that special way Hilbert introduced a series of concepts on the plane that depended of the natural topology of the Cartesian plane IR 2. He transferred point-set theoretical concepts valid for IR 2 onto the plane using the inverse functions of the coordinate functions. In this way Hilbert would initially use arguments of convergence as well as other methods and concepts of the point set- theory in IR 2 and then , using the coordinate functions, he would transfer the obtained results onto the plane. That means, all the point-set theoretical criterions used by Hilbert were to be checked on IR 2. Since the basic concept of the point-set theory in IR 2 was that of accumulation point, its importance became essential. Therefore Riesz’s attention was called onto this notion when he tried to follow Hilbert’s steps.

Motion and Jordan’s curves

For the further construction of geometry Hilbert introduced a restriction in his definition of the plane, namely that there exists an injective mapping κκκ from the plane to the Cartesian space IR 2. Then he defined the concept of motion as a bijective mapping b: E  E such that the corresponding mapping κκκbκκκ-1 : IR 2  IR 2 is a continuous mapping from IR 2 to IR 2 that preserves the orientation of any Jordan curve.

Thus, in these terms the notion of motion works only in a two-dimensional space. An extension of Hilbert’s concept of motion for a three dimensional space would need a concept that replaces that of Jordan curve.

Before we turn to discuss the Riesz’ contributions let us summarize what we have seen so far. In Hilbert’s second approach to the foundations of geometry the properties of a Jordan curve helped to define the basic concepts of neighbourhood and motion, with which Hilbert succeeded in building up the two-dimensional geometry only on assumptions of continuity. The importance he attributed to the problem of continuity of space can be best understood by referring to his fifth mathematical problem. Now, the solution he gave to his fifth mathematical problem for the special case of the two dimensional geometry made extensively use of the point-set theory of the real numbers and of its basic concept of accumulation point.

Thus, after all Hilbert did resort to point-set theoretical properties of the set of real numbers for his characterisation of the continuity of space. And it was not the first time he tried it this way.

In 1891 Hilbert considered to define continuity in projective geometry using a projective order relation with which he would have been able to define continuity either by introducing the Archimedian axiom or Dedekind’s notion of cut. Three years later he suggested a characterisation using the property of the Weierstraß’ theorem that every infinite monotone increasing and bounded sequence of points has a supremum. In 1898 Hilbert came to the false conclusion that the Archimedian axiom and the Weierstraß’ property are essentially equivalent forms of defining continuity.

So Hilbert’s idea of continuity was essentially that of the completeness of the set of real numbers IR. Searching for axioms of continuity he resorted to those contributions to the foundations of analysis of the last two decades of the 19 th century that had succeeded to a certain extent in defining the completeness of the set of real numbers IR: Dedekind’s notion of cut, Weierstraß’ theorem.

Riesz’s approach to Hilbert’s problem of continuity of space

Let’s turn to Riesz’s contribution to Hilbert’s problem of continuity of space. Frigyes Riesz is as a mathematician best known because of his contributions to functional analysis, a branch of mathematics on which he started to work in 1906. Therefore I would like to warn you, for Riesz’ work I will discuss today is prior to his first publications in that field. I will discuss some papers he published between 1904 and 1906.

Riesz was born in 1880 in Györ, Hungary. After spending two years in Zürich he went to Budapest in 1899, where he began his studies of mathematics. Then he spent one year in Göttingen as a student. In 1902 he got his doctor title in Budapest. In the winter of 1903 he spent again one semester in Göttingen as guest of the Göttinger Mathematical Society . At the end of the year 1904 he was in a scientific correspondence with David Hilbert. In 1906 he published an article called “Die Genesis des Raumbegriffs” (in English: “The origin of the concept of space”). This is the main paper I want to drew your attention to.

Riesz’s work between 1903 and 1906

In the winter of 1903, while staying in Göttingen, Riesz showed a deep knowledge of Hilbert’s work on the foundations of geometry by working out a proof of the inverse of the theorem of Jordan’s curve by using a method developed by Hilbert in 1902 in the paper mentioned above. In November of 1904 Riesz wrote to Hilbert a letter and sent him a paper for publication. The paper has not been preserved but from the description Riesz gave in his letter he was sending an earlier version of his paper “Über mehrfache Ordnungstypen” (in English: “On multiply order types”). This article did indeed appear in 1905 in the Mathematischen Annalen , a periodical that was run at the time by Hilbert.

In his letter Riesz said that he had attempted to transfer geometric properties of point-sets to order types using the concept of fundamental sequence and he expressed his conviction that the lack of a notion of distance can be compensated with the Heine-Borel and Bolzano-Weierstraß theorems. He announced to have given a general definition of the geometric concept of “region” (Bereich) and to have investigated its connectedness using methods of the analysis situs. Further and most surprising he claimed to have found a generalization of the notion of a Jordan curve for which he could define its two senses and also to have proved that it divides a region. It sounds like a generalization of the Jordan curve theorem, but of course he had not found it. In the final print version of his paper he did not claim to have these results any more. As a matter of fact, he did not mention either the notion of Jordan curve nor the Jordan theorem at all. 7 Bearing in mind the importance of the Jordan theorem as a tool in the Hilbert’s paper, all these claims give the impression that Riesz had in mind a generalization of Hilbert’s approach and that he had already started to work on it by generalizing the tools used by Hilbert. There are more reasons to think this way. Let us turn back to Riesz’s letter [my translation].

Now my investigations come very close to those ones on the foundations of geometry. More precisely to those investigations that assuming the continuity as the first property of space build up the notion of multiply extended manifold on the concept of number and so put upper bounds to the power of space that appear to a certain extent artificial.

[Riesz to Hilbert, 18th November 1904]

The contributions to the foundations of geometry he was talking about included not just the Hilbert’s one but also that by Lie. We know this because in Riesz’s article

7 As for the Jordan theorem that asserts that a Jordan curve divides the plane in two regions one lying inside the other outside the curve, before 1908 this theorem had no valid proof and its generalization in dimension three was proved by James Alexander in 1924, after the topology had been sufficiently developed. “The origin of the concept of space” from 1906 he wrote in the introduction a similar passage but there he pointed directly to Lie’s and Hilbert’s notion of manifold.

Back to the quotation of his letter: Riesz was criticizing the fact that Lie as well as Hilbert had defined the notion of a n-dimensional manifold using the Cartesian space IR n and so they had defined it in terms of the real numbers. First of all he pointed to the consequences for the power of the set of points forming the manifold, namely that it could not be bigger that the power of the set of real numbers. This Idea of Riesz is quite impressive then it means that he was approaching the notion of continuity of space from a quite general point of view. He continued his letter as follows:

In my opinion the only essential features of IR n are the n-tuply ordering, the being dense everywhere and the continuity (the being perfect and connected) as these clearly appear in your definition of the plane although with the mentioned limitation. According to this, the [space, LR] IR n is to be understood as a set of n-tuply order type.

[Riesz to Hilbert, 18th November 1904]

Thus, Riesz was searching an abstract characterisation of the Cartesian space IR n in terms of the point-set theory and multiply order types. Following Cantor, he suggested that a set shall be said to be continuous if it is perfect and connected 8, but Riesz was thinking of course in applying the notions of perfect set and connected set he had developed in his own theory of multiply order types --Cantor studied only linear order types.

Why was Riesz suggesting an abstract characterisation of IR n in terms of point-set theoretical properties for order types? Because he thought it possible to give a more general definition of the concept of n-dimensional manifold using those properties but he did not want to have the limitations Hilbert did have as a consequence of the direct relationship he postulated between the plane and the Cartesian space IR 2. Apart from the already mentioned implications on the power, Riesz was concerned with the problem of defining continuity in an abstract general way and independently of IR n in order to avoid using implicitly the point-set theoretical properties of this space.

8 In Cantor’s theory a set is said to be perfect if it is closed and dense in itself; the set is said to be connected if for every two points of the set and for every ε>0 there exists a finite subset of T of points t1,…,t n, such that the distances d(t 1, t 2), d(t 2, t 3), etc. are all smaller than ε. As a first attempt he thought in applying his theory of multiply order types, as he mentioned in his letter. But at some point he decided to take a more general approach, an abstract point-set theoretical approach without assumptions of ordering. Such assumptions concern much more than just the possibility of defining continuity and Riesz wanted to contribute to those investigations on the foundations of geometry that start from assumptions of continuity alone.

Having studied in Göttingen by Hilbert, Riesz knew very well Hilbert’s work on the foundations of geometry. Riesz became aware that Hilbert’s definition of a two- dimensional manifold guaranteed the transfer of the concept of accumulation point in IR 2 to the manifold and that this transfer played an essential role in Hilbert’s procedure.

So he decided to develop the first elements of an abstract point-set theory based on the concept of accumulation point. His theory appeared in his article “The origin of the concept of space”. He handed in the Hungarian original version of this paper to the Hungarian Academy of Sciences in January of 1906.

Developing an abstract point set theory.

In 1907 he published a German version of the same article in the Hungarian periodical Mathematische und Naturwissenschaftliche Berichte aus Ungarn . It is an extensive article of 35 pages.

In this paper Riesz was concerned with the role of experience in the development of the concept of continuous space in geometry. Riesz’ approach was connected to the contemporary philosophical discussion about the relation of the axioms of geometry to real space, and especially with Poincaré’s contributions to this discussion. The attempt to define the role of experience led Riesz to two new specific questions. First: What does it mean that the geometrical space is continuous? Second: What does it mean that the real space is empirically continuous? For this lecture we will focus in the first question because of its relation with Hilbert’s problem of continuity of space. For the task of characterising the notion of continuity of space Riesz suggested a more general definition of the concept of mathematical continuum, namely as an abstract set provided with a concept of accumulation point fulfilling four axioms. I am giving you Riesz' definition in modern terms: M’ symbolizes the set of all accumulation points of the set M. X is a mathematical continuum , if for every element x of X and for every subset M of X holds: either x is not an accumulation point of M or if x is an accumulation point of M and the following conditions are satisfied:

1. Let M be a finite set, then M has no accumulation point;

2. Let x be an accumulation point of M and let M be a subset of N, then x is an accumulation point of N too;

3. Let M be a subset of X that can be divided into two disjoint subsets P and Q and let x be an accumulation point of M, then x is an accumulation point of P or Q;

4. Let x be an accumulation point of M and let y be a different element of x, then there exists a subset P of M, such that x is an accumulation point of P but y is not an accumulation point of P.

This is the abstract concept of space that Riesz introduced in his paper „The origin of the concept of space“ and which is closely related with the modern concept of topological space. Examples of mathematical continua are the spaces IR n with their standard concept of accumulation point, but also any set of functions when considered as a metric space.

Riesz used his concept of mathematical continuum as the basic condition that makes possible to investigate the properties of continuity of an abstract set. That means, the continuity of a set that is a mathematical continuum can be further characterized. But if the set is not a mathematical continuum to investigate its continuity is nonsense. With regard to mathematical continua the interesting question is not if they are continuous but how good their continuity structure is.

Riesz demanded as the basic property space to be a mathematical continuum. For the further characterization of its continuity he introduced the concept of similarly condensed (ähnlich verdichtet') . According to Riesz two sets are similarly condensed, if their structures of mathematical continua are isomorphic to each other. Besides he defined the concept of condensation type (Verdichtungstypus) as „the totality of all properties common to all mathematical continua, which are similarly condensed to each other”. Furthermore he defined a general concept of neighbourhood based on the notion of accumulation point. Applying these theoretical tools Riesz chose following characterization of continuous geometrical space: In this way , one achieves a characterisation of space as a connected mathematical continuum, of which every element has a neighbourhood, which, considered as an independent continuum, has the same condensation type as the three-dimensional space of numbers. So one arrives at Hilbert‘s more general conception of continuous space. (Riesz 1907, 349)

That means: for Riesz the space is a connected mathematical continuum locally similarly condensed as IR 3.

Historical meaning of Riesz’s contribution to Hilbert’s problem of continuity of space

Now the question I would like to discuss is: what happened with this contribution of Riesz?

With relation to the research work on the foundations of geometry: there were not any reception of these Riesz’ ideas, as far as I know. There are several factors that might have played a negative role. First: the task set by Hilbert drew more urgently the attention to the problem of getting rid of Lie’s assumptions of differentiability in the concept of group of continuous transformations. On the other hand, Riesz presented his ideas as a part of a more ambitious project that went beyond the frame of pure mathematics. A third factor might be the fact that the paper appeared in an Hungarian Journal and not in a Journal of the Rang of the Mathematischen Annalen .

The historical meaning of Riesz’ theory of mathematical continuum is related with the history of general topology and functional analysis. Riesz’ general concept of mathematical continuum was the basic concept with which he set out to develop an abstract point set-theory. This development preceded Felix Hausdorff’s theory of topological space. When Hausdorff’s theory appeared in 1914 Riesz’ ideas were not discarded. On the contrary, until the beginning of the 1920ies Riesz’ concepts competed with those from Hausdorff in the work of young set-theoretical mathematicians. Since 1911 there were different efforts to develop an abstract point set-theory based on Riesz’ concept of the mathematical continuum. They took place in the USA in the work of Ralph Root, Edward Chittenden and Robert Lee Moore. Once Hausdorff introduced his concept of topological space in 1914 other pioneers of the general topology like Maurice Fréchet and Leopold Vietoris considered both Riesz’ as well as Hausdorff’s ideas as a starting point. That means: Riesz’ ideas were considered as a serious alternative to Hausdorff’s theory of topological space at least until the beginning of the 1920ies, something, it seems to me, which has not been properly acknowledged in most of the studies on the history of general topology. In fact, it can be said that the history of Riesz’ concept of mathematical continuum constitutes a chapter in the history of both modern branches of mathematics: general topology and functional analysis.