TITLE: Countable sets in $\mathbb R$ are Borel sets
QUESTION [7 upvotes]: I am aware that this is a very general question, but why is every countable set in the real numbers a Borel set?

REPLY [14 votes]: Every singleton is a Borel set, $\{x\}=\bigcap_{n\in\Bbb N}(x-\frac1n,x+\frac1n)$.
And the countable union of Borel sets is a Borel set.