Can any Borel set be generated by a countable number of operations starting from the open sets?

Question

Can any Borel set be written as a countable number of operations (union,intersection and complementation) starting from the open sets?

Answer 1

Yes, but you should define the operations more clearly. For example, if you let $G_0$ be the family of open sets and let $G_{n+1}$ be the family of sets which are countable intersections (respectively unions) of elements of $G_{n}$ if $n$ is even (respectively, $n$ is odd) for $n\in\mathbb N$, then it is NOT true that every Borel set is an element of some $G_n$.

But this form is true: Let $\Omega$ be the smallest uncountable well-ordered set (ordinal). For $\alpha\in\Omega$, of $\alpha$ is the least element of $\Omega$, let $G_{\alpha}$ be the family of open sets. If not, if $\alpha$ is even (resp. odd), let $G_{\alpha}$ be the family of countable intersections (resp. unions) of the sets that belong to some $G_{\beta}$, where $\beta<\alpha$. Limit ordinals are considered even here. Then it is true that $\cup_{\alpha\in\Omega} G_{\alpha}$ is the family of all Borel sets.

See Section 30.II of Kuratowski's book.