Motivation for the term “separable” in topology

A topological space is called separable if contains a countable dense subset. This is a standard terminology, but I find it hard to associate the term to its definition. What is the motivation for using this term? More vaguely, is it meant to capture any suggestive image or analogy that I am missing?

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On Srivatsan’s request I’m making my comment into an answer, even if I have little to add to what I said in the MO-thread.

As Qiaochu put it in a comment there:

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.

In my answer on MO I provided a link to Maurice Fréchet’s paper Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1-74 and quoted several passages from it in order to support that view:

Title page of Fréchet's paper

The historical importance of that paper is (among many other things) that it is the place where metric spaces were formally introduced.

Separability is defined as follows:

passage defining separability

Amit Kumar Gupta’s translation in a comment on MO:

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.

And here’s the excerpt from which I quoted on MO with some more context — while not exactly accurate, I think it is best to interpret classe $(V)$ as metric space in the following:

Excerpt where separable is formally introduced
Continuation of the excerpt
Third part of the excerpt

Felix Hausdorff, in his magnum opus Mengenlehre (1914, 1927, 1934) wrote (p.129 of the 1934 edition):

Excerpt from Hausdorff

My loose translation:

The simplest and most important case is that a countable set is dense in $E$ [a metric space]; $E = R_{\alpha}$ has at most the cardinality of the continuum $\aleph_{0}^{\aleph_0} = \aleph$. A set in which a countable set is dense is called separable, with a not exactly very suggestive but already established term by M. Fréchet. A finite or separable set is called at most separable.