Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T02:29:36.372Z Has data issue: false hasContentIssue false
  • English
  • Français

What Is Topology?

Published online by Cambridge University Press:  14 March 2022

Philip Franklin
Philip Franklin*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass.
Get access Rights & Permissions [Opens in a new window]

Extract

Introduction. Topology (sometimes called analysis situs) is the most general and most fundamental branch of geometry. Logically its study should precede that of other kinds of geometry. But mathematical knowledge, whether regarded as part of our cultural heritage or as the possession of an individual, does not come into being like a building, from a completed foundation to a limited superstructure, but rather grows like a tree with ever-deepening roots as well as ever-spreading branches. So, historically, systematic studies in topology lagged behind Euclid's elements some 2000 years and were unknown a century ago. There has, however, been considerable activity in this field by the present generation, and a number of modern treatments are available for the specialist. In spite of its fundamental nature, the subject is difficult, and many of its problems remain unsolved even today. This difficulty makes it pedagogically advisable to avoid questions of topology germane to other branches of mathematics and there is a distinct scarcity of material available even to the mathematician, whose main interest lies outside this field.

Type
Research Article
Information
Philosophy of Science , Volume 2 , Issue 1 , January 1935 , pp. 39 - 47
Copyright
Copyright © The Philosophy of Science Association 1935

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Alexandroff, P. Einfachste Grundbegriffe der Topologie (Berlin, 1932).

Kerékjártó, B. v. Vorlesungen über Topologie (Berlin, 1923).

Lefschetz, S. Topology (New York, 1930).

Seifert, H. and Threlfall, W. Lehrbuch der Topologie (Leipzig, 1934).

Veblen, O. Analysis situs (New York, 1931).

Waerden, B. L. van der. Kombinatorische Topologie. Deutsch. Math. Vereinig., 39 (1930). p. 121.

2 Hilbert, D. and Cohn-Vossen, S. Anschauliche Geometrie (Berlin, 1932), last chapter.

3 Without giving the complete technical meaning of group, we note the principal property is that a succession of admissible transformations yields an admissible transformation.

4 Birkhoff, G. D. Dynamical Systems (New York, 1927).

5 Ahrens, W. Mathematische Unterhalten und Spiele (Leipzig, 1918–21).

6 Reynolds, C. N., Jr. On the problem of coloring maps in Four Colors, II. Annals of Mathematics, 28 (1927), p. 477–492.