So I'm reading Folland and on p. 22 he says
"If X is any metric space..., the sigma algebra generated by the family of open sets in X is called the Borel sigma algebra and is denoted Bx."
He just went over how a subset of P(X) generates a sigma algebra but I don't understand how a family of open sets in X can generate a sigma algebra.
If we have a subset E of P(X), the sigma algebra generated by E is the smallest sigma algebra containing E. I don't quite see how this extends to a family of subsets of P(X). Should I just assume that, when multiple sets generate a sigma algebra, this sigma algebra is the smallest one containing all of the generating subsets?
Thanks for the help!