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?

36$\begingroup$ My understanding is it comes from the special case of R, where it means that any two real numbers can be separated by, say, a rational number. $\endgroup$ – Qiaochu Yuan Jan 8 '11 at 20:39

$\begingroup$ Ha, that's a nice observation! $\endgroup$ – minimax Jan 8 '11 at 20:47

$\begingroup$ Maybe, it refers to the separation of points by means of countable collection of small open sets (where separation means that for any two points we may find their disjoint neighborhoods in our collection). It is probably not formally equivalent to the existence of a countable dense subset, but something similar. $\endgroup$ – Fedor Petrov Jan 8 '11 at 22:23
As far as I know the word separable was introduced by M. Fréchet in Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 174. The paper can be obtained via this link (Springer). It's the famous paper in which he introduced metric spaces. He considers first slightly more general objects which he calls classes (V): where (V) stands for voisinage — neighborhood.
Remark: Metrics are introduced under the name écart in n^{o} 49 on page 30. It is peculiar that the symmetry condition is not explicitly mentioned but it seems to be understood as Fréchet immediately mentions that metric spaces generalize classes (V) cf. n^{o} 27 on page 17f. However, I couldn't find an instance where he actually uses it, he is always careful to respect the order — I may have missed something since I haven't read the paper in detail.
I quote the relevant passage [from n^{o} 37 on page 23f]:
Nous appellerons ensuite classe séparable une classe qui puisse être considérée d'au moins une façon comme l'ensemble dérivé d'un ensemble dénombrable de ses propres éléments.
[...]
Ceci étant, nous nous bornerons maintenant à l'étude des classes (V) NORMALES, c'estàdire parfaites, séparables et admettant une généralisation du théorème de CAUCHY. Cette limitation n'a du reste rien d'artificiel, elle provient directement de la comparaison des classes (V) avec les ensembles linéaires [...]
[...]
Passons maintenant aux classes séparables. On peut qualifier ainsi les ensembles linéaires en considérant la droite indéfinie comme l'ensemble dérivé de l'ensemble des points d'abscisses rationnelles. Mais il n'en est pas de même pour toute classe parfaite (V).
Below is a translation into English (made by several people here).
Very roughly: Fréchet 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.
We will henceforth define a separable class as a class that can be considered in at least one way as the derived set of a countable set of its own elements.
[...]
This being said, we shall restrict ourselves now to the study of (V) NORMAL classes, that is to say perfect, separable and admitting a generalization of Cauchy's theorem. This limitation has in fact nothing artificial; it comes directly from the comparison of the classes (V) with linear sets.
[...]
We now pass to separable classes. We can qualify in this way linear sets by viewing the indefinite line as the derived subset of the set of its points with rational abscissa. But it isn't so for all perfect classes (V).

7$\begingroup$ We will henceforth call a class separable if it can be considered in at least one way as the derived set of a denumerable set of its elements. [...] We restrict ourselves now to the study of (V) NORMAL classes, that is to say perfect, separable and admitting a generalization of Cauchy's theorem. This limitation has nothing artificial, it comes directly from the comparison of the classes (V) with the linear sets. [...] We pass now to separable classes. [Not sure about this part] as the derived set of the set of rational points. But it isn't so for all perfect (V) classes. $\endgroup$ – Amit Kumar Gupta Jan 9 '11 at 17:57

$\begingroup$ Thank you very much! I'd translate the second to last sentence by "One may describe the 'subsets of the real line' (ensembles linéaires) in this way by considering the 'real line' (droite indéfinie) as the derived set of the set of the points with rational abscissae." $\endgroup$ – Theo Buehler Jan 9 '11 at 19:02
According to this part 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...
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.

1$\begingroup$ This does not explain the terminological connection between "separable" and "existence of countable dense subset" $\endgroup$ – Martin Brandenburg Jan 9 '11 at 10:02

$\begingroup$ No, I just mean that separable has been used for many different situations, the common origin is separability of points by a mean or another. I guess there is no more to look for. $\endgroup$ – Patrick IZ Jan 9 '11 at 12:37

$\begingroup$ I have a question: do you use in english the verb "to separate" in the same broad meaning than in french? $\endgroup$ – Patrick IZ Jan 9 '11 at 12:43

2$\begingroup$ Concerning your specific example: I have seen the phrase "$\mathcal{F}$ separates the points of $X$", yes. I believe it's standard terminology. $\endgroup$ – Zhen Lin Jan 9 '11 at 13:05