What are Normal Sets (Fréchet)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:38:57Z http://mathoverflow.net/feeds/question/119829 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119829/what-are-normal-sets-frechet What are Normal Sets (Fréchet)? Andreas Loos 2013-01-25T12:24:34Z 2013-01-28T14:56:46Z <p>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.</p> <ul> <li>Can anyone explain in modern terms by which properties Normal Sets are characterized?</li> <li>Where can Fréchets definition be found?</li> </ul> http://mathoverflow.net/questions/119829/what-are-normal-sets-frechet/120060#120060 Answer by Carlo Beenakker for What are Normal Sets (Fréchet)? Carlo Beenakker 2013-01-27T22:11:35Z 2013-01-28T09:58:42Z <p>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" rel="nofollow"><em>Sur quelques points du calcul fonctionnel</em></A>, Rendiconti del Circolo Matematico di Palermo <strong>22</strong>, 1-74 (1906). See in particular pages 23-24, where the "<em>classes normales</em>" are defined as being [1] "<em>parfaites, séparables et admettant une généralisation du théorème de Cauchy</em>".</p> <p>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" rel="nofollow">Archive for History of Exact Sciences</A>. Johnson notes that Brouwer is largely following F. Hausdorff's <em>Grundzüge der Mengenlehre</em> in his classification of the normal sets.</p> <hr> <p>[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\$. </p> http://mathoverflow.net/questions/119829/what-are-normal-sets-frechet/120105#120105 Answer by Andreas Loos for What are Normal Sets (Fréchet)? Andreas Loos 2013-01-28T12:11:12Z 2013-01-28T12:11:12Z <p>Great answer, thank you!</p> http://mathoverflow.net/questions/119829/what-are-normal-sets-frechet/120114#120114 Answer by Dave L Renfro for What are Normal Sets (Fréchet)? Dave L Renfro 2013-01-28T14:50:29Z 2013-01-28T14:56:46Z <p>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:</p> <p><em>A study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals</em>, <strong>Archive for History of Exact Sciences</strong> 27 #3 (1982), 233-295.</p> <p><em>A study of Maurice Fréchet: II. Mainly about his work on general topology, 1909–1928</em>, <strong>Archive for History of Exact Sciences</strong> 34 #4 (1985), 279-380.</p> <p><em>A study of Maurice Fréchet: III. Fréchet as analyst, 1909–1930</em>, <strong>Archive for History of Exact Sciences</strong> 37 #1 (1987), 25-76.</p> <p>Near the top of p. 256 of the first paper Taylor writes:</p> <blockquote> <p>In a number of theorems Fréchet deals with \$V\$-classes that are complete and separable. He calls them <em>normal</em>. This terminology has not survived; in later developments of abstract topology the word normal is given an entirely different meaning.</p> </blockquote>