# Countable sets in $\mathbb R$ are Borel sets

I am aware that this is a very general question, but why is every countable set in the real numbers a Borel set?

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Do you know what "Borel set" means? –  Chris Eagle Mar 13 '13 at 10:19
And no, this isn't a very general question. This is a very specific question. –  Chris Eagle Mar 13 '13 at 10:19

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.

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thank you! with more or less the same approach I could show now that countable sets in R have measure zero. –  user62487 Mar 13 '13 at 16:09