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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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See this MO thread for a detailed discussion. As Qiaochu stated in a comment there "My understanding is it comes from the special case of $\mathbb R$, where it means that any two real numbers can be separated by, say, a rational number." –  t.b. Sep 12 '11 at 2:36
    
@Theo Oh I considered this question too basic, so I didn't check MO. :-) I'll perhaps keep the question. Someone can post the MO thread as the answer, if we get no other answers... –  Srivatsan Sep 12 '11 at 2:38
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Given the passage from Fréchet that Theo posted in the MO thread, it seems very likely that Qiaochu is right (and if he isn’t, we’ll probably never know!). –  Brian M. Scott Sep 12 '11 at 2:51
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Qiaochu's explanation sounds reasonable. Let me also say though that the word "separable" is a notorious example of an overloaded term in topology, such that I think no one in the last 50 years or more has drawn any real intuition from the term. Perhaps it would be better if we spoke of spaces of countable density instead... –  Pete L. Clark Sep 12 '11 at 16:29
    
@Theo I am not sure waiting longer would actually help for this question. So if you write an answer giving the link, possibly highlighting Qiaochu's comment, I will accept it. :-) (Feel free to add anything else you feel relevant.) –  Srivatsan Sep 19 '11 at 0:50
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1 Answer

up vote 8 down vote accepted

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.

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