definition of countably generated Borel space

Question

I have one source

http://www.ams.org/journals/tran/1957-085-01/S0002-9947-1957-0089999-2/S0002-9947-1957-0089999-2.pdf

page 137, first paragraph of section 2

which says a countably generated Borel space is a Borel space with a countable generating family, which is also separated. I don't know if that is supposed to mean that the countable generating family is also a separating family. On the other hand I have another source

http://www.math.missouri.edu/~jan/thesis/3meas

page 39

which does not have the additional requirement "which is also separated", but then goes on to say that it is clear that a countably generated Borel space is also countably separated. I do not know why this would be clear if we are not making any assumption that the countable generating family is also a separating family. I was wondering if I could find out what the correct definition is?

Answer 1

A $\sigma$-algebra is said to be countably generated if it is generated by a countable subset (v.g., the Borel sets of Euclidean $n$-space are generated by the open $n$-parallelepipeds with rational vertices). By extension, we may say that a measurable space is countably generated if its $\sigma$-algebra is.

This is independent of the concept separated. Nowadays it seems that the more explanatory phrase “separates points” is in use.

For instance, you have the following statement:

Let $\mathbf{S}=\langle S,\Sigma\rangle$ be a measurable space. $\mathbf{S}$ is the Borel space of a separable metrizable space iff $\Sigma$ is countably generated and separates points (in $S$).

The trivial $\sigma$-algebra $\{\emptyset, S\}$ on any set $S$ with more than two elements is countably generated but (obviously) does not separates points.