Abstract

ARBOLEDA, Luis Carlos. Introduction of the topology structure in Fréchet and Hausdorff works. Rev. acad. colomb. cienc. exact. fis. nat. [online]. 2017, vol.41, n.161, pp.528-537. ISSN 0370-3908.  https://doi.org/10.18257/raccefyn.510.

In the first part, we remind Fréchet’s testimonies about the approach of his early work (1904-1906) in the emerging fields of Functional Analysis and General Analysis, in relation to his idea of introducing a topological structure in an abstract space. In the second part, we highlight the influence that had on this idea, the algebraic point of view of the time of extending the Cantorian notions to an abstract space with a finite group structure. Fréchet took advantage of techniques such as the “composition mode” between the elements of the space, to axiomatize operations and structures of the “class ” with sequential convergence, the “class ” with neighborhood system, the “class ” with “écart” (metric). Then, new historical data are used to reaffirm the proximity of the philosophical conceptions underlying these investigations with the ideas of Leibniz, specifically with regard to the method of “analysis of principles”. The third part examines the contribution of Hausdorff of 1912 and 1914 to the establishment of the neighborhood axiomatics for the topology of an abstract space. Taking into account the observations of Weyl and Bourbaki that Hausdorff drew on Hilbert, we examine the system of axioms for the neighborhoods of the plane introduced by Hilbert in two 1902 papers devoted to the problem of the space continuity. As regards Fréchet’s influence in Hausdorff, we explore the connections of Hausdorff’s “topological space” based on neighborhoods with the notions of metric, sequential convergence and neighborhoods, proposed years earlier by Fréchet. From the beginning, Hausdorff argued that the topology of the separable space had the characteristics of generality and formal rigor that allowed it to adapt to applications better than others. It is shown that all this was consistent with the ideals of simplicity, unity and economy of thought Hausdorff had acquired in his early philosophical works.

Keywords : History of topology; Philosophy of mathematical practice; systems of axioms for neighborhoods; Fréchet; Hausdorff.

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