# What are Normal Sets (Fréchet)?

In 1913, LEJ Brouwer started a new approach to give a topologist's definition of the notion dimension ("Über den natürlichen Dimensionsbegriff", Journal für die reine und angewandte Mathematik, 142, 1913, pp. 146--152".) In this paper, Brouwer starts with a "Normalmenge" (Normal Set), referring to Maurice René Fréchet.

• Can anyone explain in modern terms by which properties Normal Sets are characterized?
• Where can Fréchets definition be found?
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Meanwhile, I think what Brouwer meant is called Normal spaces today. –  Andreas Loos Jan 25 '13 at 12:48

The "normal sets" are separable metric spaces with no isolated points, as introduced by Fréchet in Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo 22, 1-74 (1906). See in particular pages 23-24, where the "classes normales" are defined as being [1] "parfaites, séparables et admettant une généralisation du théorème de Cauchy".

For an extensive discussion of Brouwer's paper in the historical context see D.M. Johnson's 1981 article in the Archive for History of Exact Sciences. Johnson notes that Brouwer is largely following F. Hausdorff's Grundzüge der Mengenlehre in his classification of the normal sets.

[1] The reference to Cauchy's theorem is the requirement that the limit of every subsequence of a sequence converging to an element $A$ is also $A$.

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So "généralisation du théorème de Cauchy" perhaps is completeness. –  Gerald Edgar Jan 27 '13 at 23:27

The most complete study in English of Fréchet's work that I know of is a series of three long papers (total of 217 pages) by Angus Ellis Taylor that were published in the 1980s:

A study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals, Archive for History of Exact Sciences 27 #3 (1982), 233-295.

A study of Maurice Fréchet: II. Mainly about his work on general topology, 1909–1928, Archive for History of Exact Sciences 34 #4 (1985), 279-380.

A study of Maurice Fréchet: III. Fréchet as analyst, 1909–1930, Archive for History of Exact Sciences 37 #1 (1987), 25-76.

Near the top of p. 256 of the first paper Taylor writes:

In a number of theorems Fréchet deals with $V$-classes that are complete and separable. He calls them normal. This terminology has not survived; in later developments of abstract topology the word normal is given an entirely different meaning.

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Great answer, thank you! –  user30980 Jan 28 '13 at 12:11