# What is the significance of the term "separable" in the context of countability properties?

In the context of topological spaces, I see the following major countability properties:

A space is:

• "separable" iff it has a countable dense subset

• "second countable" iff if has a countable basis

• "first countable" iff the neighbourhood system of every point has a countable local basis.

(Definitions taken from Counterexamples in Topology by Steen and Seebach, 2nd ed. 1978 -- there may be differences in wording from other sources.)

The question I have is: "separable" into what, exactly? By which I mean to say: what is the thinking behind calling such a condition "separable"?

It appears not to be related to the concept of "separation axioms", which do immediately and obviously invoke an intuitive notation of separation, neither does it seem to have anything to do with "separated sets".

(Anyone using the spelling "seperable" or "seperated" will be immediately downvoted. :-) )

• Munkres (1st edition, p 192) calls separable "an unfortunate choice of terminology" with a footnote saying: "This is a good example how a word can be overused. We have already defined what we mean by a separation of a space; and we shall discuss separation axioms shortly." Commented May 21, 2020 at 7:50

It's not a very good term and you're right that it has nothing to do with separation axioms. The intuition comes from thinking about, for example, $$\mathbb{R}$$: a subset $$S \subseteq \mathbb{R}$$ is dense iff any two distinct real numbers $$a < b$$ can be separated by an element of $$S$$ in the sense that there exists $$s \in S$$ such that $$a < s < b$$.

• Brilliant! Do you know of anywhere in the literature where this explanation can be found? I may want to cite it. Commented May 21, 2020 at 7:31
• @PrimeMover maybe you could dig up Fréchet's article? Maybe he says something about the choice of words? Commented May 21, 2020 at 7:53
• @Henno Brandsma Yes, I'll take a look at the Fréchet article -- I see it is already on my radar, so to speak. Commented May 21, 2020 at 7:59

Second countable has an old-fashioned alternative name "completely separable" (or "perfectly separable"), and indeed second countable is a stronger property than separable.. (just as completely normal is stronger than normal, and completely regular stronger than regular..)

I've always found second/first countable somewhat nondescript names too. Nowadays we can just state the conditions as $$d(X)=\aleph_0$$ (separable), $$\chi(X)=\aleph_0$$ (first countable) and $$w(X)=\aleph_0$$ (second countable); the historical names can be a bit confusing (the same might be said of "normal" and "(completely) regular" spaces).

The handbook of history of general topology might have more info on its first use (and who came up with it). Engelking (General Topology in his historical notes) says (and I believe him) that Fréchet first defined the notion in (his doctoral thesis)

Fréchet, M.M. Sur quelques points du calcul fonctionnel. Rend. Circ. Matem. Palermo 22, 1–72 (1906). https://doi.org/10.1007/BF03018603

(but I don't have access to a digital library to view it). So it's an old name, probably one of the oldest still in use in topology (next to open/closed, limit point..)

• To get a taste of Fréchet's theory (which was one of the early idea systems to formalise "topology") can be gleaned from this 1918 paper (In English) that also shows how much terminology has moved on since then. It's hard to get through; it doesn't mention separable I think.. Commented May 21, 2020 at 8:02
• The symbology used is okay from the perspective of someone who is deep into the modern paradigm but the specific meanings of $d(X)$, $\chi{X}$ and $w(X)$ are obscure for the autodidact whose library is limited to texts over 30 years old. Commented May 21, 2020 at 9:29
• @PrimeMover engelking does use these notations too, and his text is over 30 years old Get it. Commented May 21, 2020 at 9:41
• Well yes, that is a possibility, but it's still better to communicate clearly than to use more-or-less esoteric conventions that require one to have read specific books in order to understand what is being said. Commented May 21, 2020 at 10:00
• @PrimeMover all recent papers use this kind of notation. So it’s better to know it; it’s not esoteric at all. Commented May 21, 2020 at 10:01