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:
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:
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:
Amit Kumar Gupta's translation in a comment on MO:
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:
My loose translation: