Skip to main content
SpringerLink
Log in

From Developments to Developable Spaces

The Evolution of a Topological Idea

  • Chapter
Handbook of the History of General Topology

Part of the book series: History of Topology ((HIPO,volume 2))

  • 929 Accesses

Abstract

This investigation highlights a thread of mathematical activity that begins with E.H. Moore’s introduction of the mathematical notion of a “development for an abstract class” in 1910, before general topology was established as a field of mathematics, and suggests an evolution of ideas that leads to the present-day topological concept of “developable space”, as it was defined by R.H. Bing in 1951.

In addition, we are guided by the following principle:

to reconstruct the viewpoint of a writer, one must consider the backdrop of a classical past that remains in view in contrast with the powerful present that informs and judges this past.

John McCleary [1989]

Thus, rather than paraphrase or “report” the evolution of concepts, as I interpret it, I want, first, to create the historical context in which an activity took place; then, to allow the researcher himself to communicate his contribution by quoting the work; and finally, to add my reflection and observations about the work. In this way I hope “to reconstruct the viewpoint of the writer” and present that work in “the backdrop of a classical past” and still not judge the work in the light of “the powerful present”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 229.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alexandroff, P.S. and Nemytskii, V.V. (1938) Metrizability conditions for topological spaces and the axiom of symmetry [in Russian], Mat. Sb. 3, 663–672.

    Google Scholar 

  • Alexandroff, P. and Urysohn, P. (1923) Une condition nécessaire et suffisante pour qu’une classe (G) soit une classe (D), Comptes Rendus Acad. Sci. Paris 177, 1274–1276.

    Google Scholar 

  • Aull, C.E. (1979) E.R. Hedrick and generalized metric spaces and metrization, Metric Spaces, Generalized Metric Spaces, and Continua, G.R. Gordh, Jr. and J.R. Boyd, editors, Mathematics Department, Guilford College, 51–68.

    Google Scholar 

  • Aull, C.E. (1981) E.W. Chittenden and the early history of general topology, Topology and its Applications 12, 115–125.

    Article  MathSciNet  MATH  Google Scholar 

  • Bernkopf, M. (1966/67) The development of function spaces with particular reference to their origins in integral equation theory, Archive for History of Exact Sciences 3, 1–96.

    Google Scholar 

  • Bing, R.H. (1951) Metrization of topological spaces, Canadian J. Math. 3, 175–186.

    Article  MathSciNet  MATH  Google Scholar 

  • Bing, R.H. (1981) Metrization problems, General Topology and Modern Analysis, L.F. McAuley and M.M. Roa, editors, Academic Press, 3–16.

    Google Scholar 

  • Birkhoff, G.D. (1911) Review of “Introduction to a Form of General Analysis”, Bulletin Amer. Math. Soc. 17, 414–428.

    Article  MathSciNet  Google Scholar 

  • Bliss, G.A. (1934) The scientific work of E.H. Moore, Bulletin Amer. Math. Soc. 40, 501–514.

    Article  MathSciNet  Google Scholar 

  • Chittenden, E.W. (1915) Infinite developments and the composition property (K12B1)* in general analysis, Rendiconti del Circolo Matematico di Palermo 39, 81–108.

    Article  MATH  Google Scholar 

  • Chittenden, E.W. (1917) On the equivalence of relations Kgig2m, American J. Math. 39, 263–271.

    Article  MathSciNet  MATH  Google Scholar 

  • Chittenden, E.W. (1917) On the equivalence of écart and voisinage, Transactions Amer. Math. Soc. 18, 161–166.

    MathSciNet  MATH  Google Scholar 

  • Chittenden, E.W. (1927) On the metrization problem and related problems in the theory of abstract sets, Bulletin Amer. Math. Soc. 33, 13–34.

    Article  MathSciNet  MATH  Google Scholar 

  • Chittenden, E.W. (1929) On general topology and the relation of the properties of the class of all continuous functions to the properties of space, Transactions Amer. Math. Soc. 31, 290–321.

    Article  MathSciNet  MATH  Google Scholar 

  • Chittenden, E.W. and Pitcher, A.D. (1919) On the theory of developments of an abstract class in relation to the calcul fonctionnel, Transactions Amer. Math. Soc. 20, 213–233.

    Article  MathSciNet  MATH  Google Scholar 

  • Fitzpatrick, B. (1997) Some aspects of the work and influence of R.L. Moore, Handbook of the History of General Topology, Volume 1, C.E. Aull and R. Lowens, editors, Kluwer Academic Publishers, The Netherlands, 41–61.

    Google Scholar 

  • Fréchet, M. (1906) Sur quelques points de Calcul Fontionnel, Rendiconti del Circolo Matematico di Palermo 22, 1–74.

    Article  MATH  Google Scholar 

  • Fréchet, M. (1910) Les ensembles abstraits et le Calcul Fontionnel, Rendiconti del Circolo Matematico di Palermo 30, 1–26.

    Article  MATH  Google Scholar 

  • Fréchet, M. (1913) Sur les classes V normales, Transactions Amer. Math. Soc. 14, 320–324.

    MATH  Google Scholar 

  • Fréchet, M. (1917) Sur la notion de voisinage dans les ensembles abstraits, Comptes Rendus Acad. Sci. 165, 359–360.

    MATH  Google Scholar 

  • Fréchet, M. (1918) Relations entre les notions de limite et de distance, Transactions Amer. Math. Soc. 19, 53–65.

    MATH  Google Scholar 

  • Fréchet, M. (1918) Sur la notion de voisinage dans les ensembles abstraits, Bulletin des Sciences Mathematiques 42, 138–156.

    Google Scholar 

  • Fréchet, M. (1921) Sur les ensembles abstraits, Annales Scientifiques de l’École Normale Supérieure 38, 314–388.

    Google Scholar 

  • Fréchet, M. (1922) Esquisse d’une théorie des ensembles abstraits, Sir Asutosh Mookerjee’s Commemorative volumes II, Baptist Mission Press, Calcutta, 1922, 333–394.

    Google Scholar 

  • Fréchet, M. (1928) Les espaces abstraits et leurs théorie considérée comme introduction à l’anslyse générale, Gauthier—Villars, Paris, 1928.

    Google Scholar 

  • Frink, A.H. (1937) Distance functions and the metrization problem, Bulletin Amer. Math. Soc. 43, 133–142.

    Article  MathSciNet  Google Scholar 

  • Hahn, H. (1908) Bemerkungen zu den Untersuchungen des Herrn M. Fréchet: Sur quelques points du calcul fonctionnel, Monatshefte für Mathematik und Physik 19, 247–257.

    Article  MATH  Google Scholar 

  • Hausdorff, F. (1914) Grundzüge der Mengenlehre, Verlag von Veit & Company, Leipzig.

    MATH  Google Scholar 

  • Hedrick, E.R. (1911) On properties of a domain for which any derived set is closed, Transactions Amer. Math. Soc. 12, 285–294.

    Article  MathSciNet  MATH  Google Scholar 

  • Hilbert, D. (1902a) Ueber die Grundlagen der Geometrie, Math. Ann. 56, 381–422.

    Article  MathSciNet  Google Scholar 

  • Hilbert, D. (1902b) Ueber die Grundlagen der Geometrie, Göttinger Nachrichten (Math. Phys. Klasse), 233–241.

    Google Scholar 

  • Hilbert, D. (1906) Grundzüge einer allgemeine Theorie der linearen Integralgleichen, Gottinger Nachrichten, 157–227 and 439–480.

    Google Scholar 

  • Hilbert, D. (1909) Wesen und Ziele einer Analysis der unendlichvielen unabhängigen Variabeln, Rendiconti del Circolo Matematico di Palermo 27, 59–74.

    Article  MATH  Google Scholar 

  • Hildebrandt, T.H. (1912) A contribution to the foundations of Fréchet’s calcul fonctionnel, American J. Math. 34, 237–290.

    Article  MathSciNet  MATH  Google Scholar 

  • Jones, F.B. (1937) Concerning normal and completely normal spaces, Bulletin Amer. Math. Soc. 43, 671–677.

    Article  Google Scholar 

  • Jones, F.B. (1979) Metrization, non—metrization and a bit of history, Metric Spaces, Generalized Metric Spaces, and Continua, G.R. Gordh, Jr. and J.R. Boyd, editors, Mathematics Department, Guilford College, 1–4.

    Google Scholar 

  • Jones, F.B. (1997) The beginning of topology in the United States and the Moore school, Handbook of the History of General Topology, Volume 1, C.E. Aull and R. Lowen, editors, Kluwer Academic Publishers, The Netherlands, 97–103.

    Google Scholar 

  • Kelley, J.L. (1955) General Topology, D. Van Nostrand Company, Inc., Princeton, NJ.

    MATH  Google Scholar 

  • McCleary, J. (1989) A Theory of Reception for the History of Mathematics, The History of Modern Mathematics, D.E. Rowe and J. McCleary, editors, Academic Press, 3–14.

    Google Scholar 

  • Moore, E.H. (1910) Introduction to a Form of General Analysis, The New Haven Mathematical Colloquium, Yale University Press, 1–150.

    Google Scholar 

  • Moore, R.L. (1915) On a set of postulates which suffice to define a number-plane, Transactions Amer. Math. Soc. 16, 27–32.

    Article  MATH  Google Scholar 

  • Moore, R.L. (1916) On the foundations of plane analysis situs, Transactions Amer. Math. Soc. 17, 131–164.

    Article  Google Scholar 

  • Moore, R.L. (1919) Concerning a set of postulates for plane analysis situs, Transactions Amer. Math. Soc. 17, 169–178.

    Article  Google Scholar 

  • Moore, R.L. (1927) Abstract sets and the foundation of analysis situs, Bulletin Amer. Math. Soc. 33, p. 141.

    MATH  Google Scholar 

  • Moore, R.L. (1932) Foundations of Point Set Theory, American Mathematical Society Colloquium Publications, Volume 13.

    Google Scholar 

  • Moore, R.L. (1935) A set of axioms for plane analysis situs, Fund. Math. 25, 13–28.

    Google Scholar 

  • Niemytzki, V.W. (1927) On the “third axiom of metric space”, Transactions Amer. Math. Soc. 29, 507–513.

    MathSciNet  MATH  Google Scholar 

  • Nyikos, P. (1979) F. Burton Jones’s contribution to the normal Moore space problem, Metric Spaces, Generalized Metric Spaces, and Continua, G.R. Gordh, Jr. and J.R. Boyd, editors, Mathematics Department, Guilford College, 27–38.

    Google Scholar 

  • Pitcher, A.D. and Chittenden, E.W. (1918) On the foundations of the calcul fonctionnel of Fréchet, Transactions Amer. Math. Soc. 19, 66–78.

    MathSciNet  MATH  Google Scholar 

  • Riesz, F. (1905) Über mehrfache Ordungstypen, Math. Ann. 61, 406–421.

    Article  MathSciNet  MATH  Google Scholar 

  • Riesz, F. (1907) Die Genesis des Raumgriffs, Math. u Naturwiss. Berichte aus Ungarn 24, 309–353;

    Google Scholar 

  • Riesz, F. (1907) Die Genesis des Raumgriffs, Oeuvres Completes I, 110–154.

    Google Scholar 

  • Riesz, F. (1909) Stetigkeitsbegriff und Abstrakte Mengenlehre, Attti del IV Congr. Internaz. dei Mat. (Roma 1908) 2, 18–24;

    Google Scholar 

  • Riesz, F. (1909) Stetigkeitsbegriff und Abstrakte Mengenlehre, Oeuvres Completes I, 155–161.

    Google Scholar 

  • Root, R.E. (1911) Iterated limits of functions on an abstract range, Bulletin Amer. Math. Soc. 17, 538–539.

    Article  MathSciNet  MATH  Google Scholar 

  • Root, R.E. (1914) Iterated limits in general analysis, American J. Math. 36, 79–133.

    Article  MathSciNet  MATH  Google Scholar 

  • Root, R.E. (1914) Limits in terms of order, with example of limiting element not approachable by a sequence, Transactions Amer. Math. Soc. 15, 51–71.

    Article  MathSciNet  MATH  Google Scholar 

  • Rudin, M.E. (1997) The early work of F.B. Jones, Handbook of the History of General Topology, Volume 1, C.E. Aull and R. Lowens, editors, Kluwer Academic Publishers, The Netherlands, 85–96.

    Google Scholar 

  • Stone, A.H. (1948) Paracompactness and product spaces, Bulletin Amer. Math. Soc. 54, 977–982.

    Article  MATH  Google Scholar 

  • Taylor, A.E. (1982) A Study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals, Archive for Hist. of Exact Sci. 27, 233–295.

    Article  MATH  Google Scholar 

  • Taylor, A.E. (1985) A Study of Maurice Fréchet: II. Mainly about his Work on General Topology, 1909–1928, Archive for Hist. of Exact Sci. 34, 279–380.

    Article  MATH  Google Scholar 

  • Taylor, A.E. (1987) A Study of Maurice Fréchet: III. Fréchet as Analyst, 1909–1930, Archive for Hist. of Exact Sci. 37, 25–76.

    Article  MATH  Google Scholar 

  • Tychonoff, A. (1926) Über einen Metrisationssatz von P. Urysohn, Math. Annalen 95, 139–142.

    Article  MathSciNet  Google Scholar 

  • Tychonoff, A. and Vedenissoff, N. (1926) Sur le développement moderne de la théorie des espaces abstraits, Bulletin des Sciences Mathematiques 50, 15–27.

    Google Scholar 

  • Urysohn, P. (1925) Zum Metrisationproblem, Math. Annalen 94, 262–295.

    Article  MathSciNet  MATH  Google Scholar 

  • Urysohn, P. (1925) Über die Mächtigkeit der zusammenhängenden Mengen, Math. Annalen 94, 309–315.

    Article  MathSciNet  MATH  Google Scholar 

  • Veblen, O. (1905) Definition in terms of order alone in the linear continuum and in well-ordered sets, Transactions Amer. Math. Soc. 6, 165–171.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of New Hampshire, Durham, NH, USA

    S. D. Shore

Authors
  1. S. D. Shore
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA

    C. E. Aull

  2. Department of Mathematics and Computer Science, University of Antwerp, RUCA, Antwerp, Belgium

    R. Lowen

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Shore, S.D. (1998). From Developments to Developable Spaces. In: Aull, C.E., Lowen, R. (eds) Handbook of the History of General Topology. History of Topology, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1756-4_5

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1756-4_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5023-6

  • Online ISBN: 978-94-017-1756-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation