Separability and second countability is the same thing to Halmos

Question

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.

Answer 1

J. L. Kelley, General Topology (1955) defines the second axiom of countability then notes:

(The terms separable and perfectly separable are also used in this connection but we shall use neither.)

Answer 2

I don't know exactly who introduced the terminology, but "separable" comes from functional analysis, probably Hilbert spaces, where "having a countable dense subset" and "having a countable basis of open sets" are equivalent (as for metric spaces in general).

I agree that now most, if not all, authors seem to use "separable" for topological spaces with the weaker property of the two, one should keep in mind that terminology in general topology fluctuates a lot (e.g., as to whether "compact" implies Hausdorff). So any reasonable paper will redefine every term for which there is even the slightest possibility of confusion.

Also, "second countable" is horrendous terminology (as anything that entails remembering what is "first" and what is "second", e.g., "first category" instead of "meager"); Steen and Seebach's 1970 book Counterexamples in Topology define the terms "perfectly separable" and "completely separable" to be synonymous with "second-countable" (i.e., "having a countable basis of open sets"), which is probably better terminology if the term has to be redefined in any case.

Similarly, I remember finding at least one author who used "locally separable" for "first-countable" (i.e., "having a countable basis of neighborhoods at each point"), which would be better if it were not also confusing (because it could mean something like every neighborhood having a separable subneighborhood).

An even better terminology is probably "of countable density" (for "separable") and "of countable weight" (for "perfectly separable", i.e. "second-countable"), because density and weight are very standard names for the smallest infinite cardinal of a dense subset, resp. a basis for the topology. Similarly, a space "of countable character" would be a first-countable one.