In the context of topological spaces, I see the following major countability properties:
A space is:
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"separable" iff it has a countable dense subset
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"second countable" iff if has a countable basis
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"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. 🙂 )
Best Answer
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 $a < s < b$.