Archive for History of Exact Sciences

, Volume 25, Issue 2, pp 85–266

The problem of the invariance of dimension in the growth of modern topology, part II

  • Dale M. Johnson
Article

DOI: 10.1007/BF02116242

Cite this article as:
Johnson, D.M. Arch. Hist. Exact Sci. (1981) 25: 85. doi:10.1007/BF02116242
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Summary

This work examines the historical origins of topological dimension theory with special reference to the problem of the invariance of dimension. Part I, comprising chapters 1–4, concerns problems and ideas about dimension from ancient times to about 1900. Chapter 1 deals with ancient Greek ideas about dimension and the origins of theories of hyperspaces and higher-dimensional geometries relating to the subsequent development of dimension theory. Chapter 2 treatsCantor's surprising discovery that continua of different dimension numbers can be put into one-one correspondence and his discussion withDedekind concerning the discovery. The problem of the invariance of dimension originates with this discovery. Chapter 3 deals with the early efforts of 1878–1879 to prove the invariance of dimension. Chapter 4 sketches the rise of point set topology with reference to the problem of proving dimensional invariance and the development of dimension theory. Part II, comprising chapters 5–8, concerns the development of dimension theory during the early part of the twentieth century. Chapter 5 deals with new approaches to the concept of dimension and the problem of dimensional invariance. Chapter 6 analyses the origins ofBrouwer's interest in topology and his breakthrough to the first general proof of the invariance of dimension. Chapter 7 treatsLebesgue's ideas about dimension and the invariance problem and the dispute that arose betweenBrouwer andLebesgue which led toBrouwer's further work on topology and dimension. Chapter 8 offers glimpses of the development of dimension theory afterBrouwer, especially the development of the dimension theory ofUrysohn andMenger during the twenties. Chapter 8 ends with some concluding remarks about the entire history covered.

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Dale M. Johnson
    • 1
  1. 1.The Hatfield PolytechnicHatfieldEngland