The problem of the invariance of dimension in the growth of modern topology. II
type: Item
instance of: scholarly article
title: The problem of the invariance of dimension in the growth of modern topology. II
zbMATH Open document ID: 0532.55001
DOI: 10.1007/BF02116242
author: Dale M. Johnson
published in: Archive for History of Exact Sciences
publication date: 1981-01-01T00:00:00Z
review text: This book-size paper continues the study of the development of the topological dimension concept, initiated in the earlier papers ibid. 17, 261-295 (1977; Zbl 0372.01002) and in particular ibid. 20, 97-188 (1979; Zbl 0424.54020), to which the present paper is the second part. It starts with Chapter 5 (42 pages), mainly devoted to the ideas of Henri Poincaré. The author discusses two theories of dimension by Poincaré, the first and older one based on group theory, the second one of a topological nature. In order to explain Poincaré's approach, his philosophical ideas (described as a ''philosophy of conventionalism'') are very lucidly discussed. On the basis of this discussion the contributions of the great Frenchman (and their limitations) are made clear. The last part of Chapter 5 deals with the work of F. Riesz, R. Baire and M. Fréchet (insofar as it touches dimension). In particular, Baire showed (1907) how an n-dimensional Jordan separation theorem entails n- dimensional domain invariance, and how domain invariance entails invariance of dimension (these interrelations were later taken up by Brouwer); however, he never succeeded in proving the n-dimensional Jordan theorem. Chapters 6 and 7 (together with 3 appendices; they cover 95 pages) form the main part of the paper. They are devoted to the contributions to dimension theory of L. E. J. Brouwer, and his controversy with Henri Lebesgue. The historical development of Brouwer's ideas is carefully reconstructed, using not only the material in the edition of the ''Collected Works'' of \textit{L. E. J. Brouwer} (1976; Zbl 0328.01020) and the important contributions of Hans Freudenthal, the editor of Vol. II of these ''Collected Works'', to the understanding of Brouwer's work, but also additional material, which was unavailable at the time the ''Collected Works'' were prepared for publication (much of this additional material is reproduced in the appendices). The author presents a detailed account of the evolution of …
Mathematics Subject Classification ID: 55-03
Mathematics Subject Classification ID: 54-03
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 55M10
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 3844618
zbMATH Keywords: topological dimension
zbMATH Keywords: n-dimensional Jordan separation theorem
zbMATH Keywords: n- dimensional domain invariance
zbMATH Keywords: invariance of dimension
zbMATH Keywords: inductive definition of dimension
zbMATH Keywords: bibliography
reviewed by: Cor Baayen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02116242
OpenAlex ID: W2024835680
description: scientific article; zbMATH DE number 3844618