Showing that the family of Borel sets is countably generated I am taking a course in Measure Theory this year, and as an exercise from the lectures we have been given the following: 
Show that B(R) is countably generated; that is, show the Borel sets are
generated by a countable class A.
We have defined the Borel sets as:
Let X be a metric space, the family of Borel sets in X is the sigma algebra B(X) generated by the family of open sets. A set A in B(X) is called a Borel set.
So far, my thoughts are showing that the Borel sets can be generated by the intervals [q, infinity) where q is rational. 
I was wondering if anyone can help me formulate a proof of this, hopefully using my idea, as we don't get support classes or solutions and I would really like to understand this. Thank you.
 A: Your idea is correct. Take the generating set $\mathcal{G} = \{[q, \infty): q \in \mathbb{Q}\}$ and show that any open interval in $\mathbb{R}$ can be expressed as a countable intersection/union of intervals in $\mathcal{G}$, or their complements.
It's also well-known that any open set in $\mathbb{R}$ is a countable union of open intervals, so now you've shown that $\mathcal{G}$ generates all the open sets.
So you conclude that the $\sigma$-algebra generated by $\mathcal{G}$ is at least as big as the $\sigma$-algebra generated by the open subsets of $\mathbb{R}$. Obviously, it cannot be bigger (to be sure, show that all intervals in $\mathcal{G}$ can be generated by unions and intersections of open sets and their complements).
A: I think you are correct.
Consider $x \in \mathbb{R}$, and a sequence $(q_n)$ such that:
1) $\forall n \in \mathbb{N}, q_n \in \mathbb{Q}$
2) $\forall n \in \mathbb{N}, q_n \leq x$
3) $\lim_{n \to \infty} q_n = x$
(such a sequence exists since $\mathbb{Q}$ is dense in $\mathbb{R}$)
Then $\bigcap_{n \in \mathbb{N}} [q_n, \infty[ = [x, \infty[$
Therefore the intervals $[q, \infty[, q \in \mathbb{Q}$ generate the intervals $[x, \infty[, x \in \mathbb{R}$
And these intervals then generate the Borel sets (you can show it easily if you don't know the result already)
