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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), 1-74. 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'est-à-dire 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).

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), 1-74. 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'est-à-dire 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).

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), 1-74. 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'est-à-dire 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).

I am unable to translate this in a reasonable way (but see Amit's comment below for a translation). 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.

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), 1-74. 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'est-à-dire 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).

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), 1-74. The paper can be obtained via this link (Springer). It's the famous paper in which he introduced metric spaces (he calls them classes (V), where (V) stands for voisinage -- neighborhood).

I quote the relevant passage [p. 23f]:

Nous appellerons ensuite classe séparable une classe qui puisse êre 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'est-à-dire 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).

I am unable to translate this in a reasonable way. Very roughly he 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.

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), 1-74. 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'est-à-dire 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).

I am unable to translate this in a reasonable way (but see Amit's comment below for a translation). 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.