I was browsing through Paul Halmos' classic book on measure theory, when I came by the following definition of separability on page $3$ in the chapter on prerequisites:

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Today a separable space is one, which contains a countable, dense subset, while a second countable space is defined as above (and a second countable space is thus separable).

As for the other direction, from this wiki page on separability, a separable space is second countable, iff it is Lindelöf.

What is the reason that Halmos used this definition of separability? Is it because the term has evolved over time? Or perhaps all relevant spaces in measure theory are Lindelöf, so it is not important?

I have cross-posted this question on History of science and mathematics stackexchange.

  • $\begingroup$ Does Halmos perhaps only consider metric/metrisable spaces? $\endgroup$ – Daniel Fischer Dec 29 '15 at 14:11
  • $\begingroup$ @DanielFischer he hasn't mentioned metrics at this point. He has only given a general definition of topological space. $\endgroup$ – Mankind Dec 29 '15 at 14:13
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    $\begingroup$ In that case, I'm scratching my head. I think that by the time Halmos was writing books, any historical uncertainties about what should be called separable were long settled. But I'm not a historian of topology, so I may be wrong. $\endgroup$ – Daniel Fischer Dec 29 '15 at 14:16
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    $\begingroup$ My professor in probability theory, who wrote one of the most popular books on the subject, also mixed these terms. In certain areas of math, people only work with very nice topological spaces. So you say separable or second countable and its all the same. This is probably Halmos's perspective. $\endgroup$ – Forever Mozart Jan 6 '16 at 23:34

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