I have been looking into the history of topology. One thing I am very curious about is the history of the open cover definition of compactness. According to Raman-Sundström, this goes back to a lemma in Borels 1894 thesis. However, I have not been able to find what problem Borel was trying to solve or what kind of insight mathematicians like Heine, Borel and Lebesgue were trying to gain in studying open sets.

What motivated the open cover definition of compactness?

  • $\begingroup$ You can see : E.Borel, Sur quelques points de la théorie des fonctions, Annales scientifiques de l'École Normale Supérieure, Sér. 3, 12 (1895), page 51 and page 26. $\endgroup$ – Mauro ALLEGRANZA Nov 25 '15 at 15:27
  • $\begingroup$ Well, I do not read French. I can read the mathematics, but it does not tell me enough. $\endgroup$ – Avatrin Nov 25 '15 at 21:44
  • $\begingroup$ A good survey article, written long enough ago to be fairly complete with the early history of this idea, is Hildebrandt's 1926 Bull. AMS survey article The Borel theorem and its generalizations. $\endgroup$ – Dave L Renfro Nov 30 '15 at 21:50

There is a famous paper by Alexandroff & Urysohn, I believe, where they analyze many compactness-type definitions for the recently-defined notion of "topological space". Since "compact" was already used to mean: every sequence has a convergent subsequence, they came up with other names. They used the term "bicompact" for: every open cover has a finite subcover. In subsequent years, mathematicians came to see that this was the more fundamental notion, and started to use "compact" for it. But you may still find the term "bicompact" used in some Russian papers.

  • $\begingroup$ And not only in Russian. See, for example the remarkable book Young, L. C. Lectures on the calculus of variations and optimal control theory. Philadelphia-London-Toronto, Ont. 1969, where the matter of terminology is also discussed:-) $\endgroup$ – Alexandre Eremenko Nov 26 '15 at 19:20

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