# Borel Sigma Algebra? How is it generated?

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!

• Exactly.   – Did Dec 17 '13 at 22:27
Let $E:=\{$open sets in $X\}$. Then we have $E\subseteq P(X)$. All is fine.
• Yes. I don't really understand where your confusion comes from. $E$ is one subset of $P(X)$, and it generates the smallest sigma algebra which contains $E$, as you once understood within the body of the question. – Berci Dec 17 '13 at 20:37