The "normal sets" are separable metric spaces with no isolated points, as introduced by Fréchet in <A HREF="http://dx.doi.org/10.1007/BF03018603">*Sur quelques points du calcul fonctionnel*</A>, 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 <A HREF="http://dx.doi.org/10.1007/BF02116242">Archive for History of Exact Sciences</A>. Johnson notes that Brouwer is largely following F. Hausdorff's *Grundzüge der Mengenlehre* in his classification of the normal sets.

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[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 $A$.