Are there any good intuitions to understand why countable dense subset implies separability? Is the separability related to the opposite of connectedness? In Munkres, I am a bit confused after he mentioned that this is an unfortunate choice of terminology.

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    $\begingroup$ It's the definition of separability that there is a countable dense subset. It's unfortunate terminology because separability has nothing to do with separation (or connectedness). $\endgroup$ – Daniel Fischer Aug 18 '14 at 21:09
  • $\begingroup$ The reason it's unfortunate is that there are a large suite of so-callled separation axioms that talk about separating points/closed sets, and then there's suite of axioms that mandate various things be countable. separability is a countability axiom, not a separation axiom $\endgroup$ – JHance Aug 18 '14 at 21:10

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