Why the name 'separable' space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:58:01Z http://mathoverflow.net/feeds/question/51494 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51494/why-the-name-separable-space Why the name 'separable' space? minimax 2011-01-08T20:36:56Z 2011-08-09T01:23:30Z <p>It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?</p> http://mathoverflow.net/questions/51494/why-the-name-separable-space/51498#51498 Answer by Daniel Pape for Why the name 'separable' space? Daniel Pape 2011-01-08T22:24:07Z 2011-01-08T22:24:07Z <p>According to <a href="http://books.google.de/books?id=zlmJIO02NlsC&amp;pg=PA169&amp;dq=%2522separable+menge%2522&amp;hl=de&amp;ei=BeIoTfzWEcX1sgaal9i6Ag&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CCcQ6AEwAA#v=onepage&amp;q=%2522separable%2520menge%2522&amp;f=false" rel="nofollow">this part</a> of Hausdorff's collected works, the name "separable" was coined by Fréchet. Hausdorff writes this denotation wouldn't be very suggestive but established. The earliest use in the German Zentralblatt, where the word occurs in a review by Hahn, dates back to 1918. I am sure one can find the first use of the word in the given context in Fréchet's works but I doubt that he will explain his motivation for the use of this particular word there. So we will (presumably) never know...</p> http://mathoverflow.net/questions/51494/why-the-name-separable-space/51501#51501 Answer by Theo Buehler for Why the name 'separable' space? Theo Buehler 2011-01-08T22:35:47Z 2011-08-09T01:23:30Z <p>As far as I know the word <em>separable</em> was introduced by M. Fréchet in <em>Sur quelques points du calcul fonctionnel</em>, Rend. Circ. Mat. Palermo <strong>22</strong> (1906), 1-74. The paper can be obtained via <a href="http://dx.doi.org/10.1007/BF03018603" rel="nofollow">this link</a> (Springer). It's the famous paper in which he introduced metric spaces. He considers first slightly more general objects which he calls <em>classes (V)</em>: where (V) stands for <em>voisinage</em> &mdash; neighborhood.</p> <p><strong>Remark:</strong> Metrics are introduced under the name <em>&eacute;cart</em> in n<sup>o</sup>&nbsp;49 on page&nbsp;30. It is peculiar that the symmetry condition is <em>not</em> explicitly mentioned but it seems to be understood as Fr&eacute;chet immediately mentions that metric spaces generalize <em>classes (V)</em> cf.&nbsp;n<sup>o</sup>&nbsp;27 on page&nbsp;17f. However, I couldn't find an instance where he actually uses it, he is always careful to respect the order&nbsp;&mdash;&nbsp;I may have missed something since I haven't read the paper in detail.</p> <p>I quote the relevant passage [from n<sup>o</sup>&nbsp;37 on page 23f]:</p> <blockquote> <p>Nous appellerons ensuite <em>classe s&eacute;parable</em> une classe qui puisse &ecirc;tre consid&eacute;r&eacute;e d'au moins une fa&ccedil;on comme l'ensemble d&eacute;riv&eacute; d'un ensemble d&eacute;nombrable de ses propres &eacute;l&eacute;ments.</p> <p>[...]</p> <p>Ceci &eacute;tant, nous nous bornerons maintenant &agrave; l'&eacute;tude des <em>classes (V)</em> NORMALES, <em>c'est-&agrave;-dire parfaites, s&eacute;parables et admettant une g&eacute;n&eacute;ralisation du th&eacute;or&egrave;me de</em> CAUCHY. Cette limitation n'a du reste rien d'artificiel, elle provient directement de la comparaison des classes (V) avec les ensembles lin&eacute;aires [...]</p> <p>[...]</p> <p>Passons maintenant aux classes s&eacute;parables. On peut qualifier ainsi les ensembles lin&eacute;aires en consid&eacute;rant la droite ind&eacute;finie comme l'ensemble d&eacute;riv&eacute; de l'ensemble des points d'abscisses rationnelles. Mais il n'en est pas de m&ecirc;me pour toute classe parfaite (V).</p> </blockquote> <p>I am unable to translate this in a reasonable way (but see Amit's comment below for a translation). Very roughly: Fr&eacute;chet defines separable spaces as we do it today and says that in the following he will restrict attention to complete, perfect and separable metric spaces. The last quoted paragraph indeed confirms Qiaochu's comment.</p> http://mathoverflow.net/questions/51494/why-the-name-separable-space/51516#51516 Answer by Patrick I-Z for Why the name 'separable' space? Patrick I-Z 2011-01-09T02:54:13Z 2011-01-09T09:59:59Z <p>Well "séparer" means just "disjoin", "split up" etc. There are few meanings for "séparation" in french. For exemple "séparer par des fonctions". If you have a space $X$ and a set of functions ${\cal F}$ from $X$ to ${\bf R}$ (or anything else), you say that "${\cal F}$ sépare les points de $X$" iff for two different point $x$ and $x'$ there exists a function $f \in {\cal F}$ such that $f(x) \neq f(x')$, this is a common use of the word "séparer" in french, nothing mysterious. And this vocabulary can be applied to any analogous situations, in topology or whatever else context. I heard the first time this wording (when I was a student) in the case I mention above, long before I have heard using this wording in a topology course.</p>