I was browsing through Paul Halmos' classic book on measure theory from 1950, when I came by the following definition of separability on page 3 in the chapter on prerequisites:
Today a separable space is one, which contains a countable, dense subset, while a second countable space is defined as a space having a countable base (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?
He has only discussed general topological spaces up to this point, and has made no mention of metric spaces.
I also posted this question on math.stackexchange a week ago, but so far there are no answers.