What are the minimum $\sigma$-ring and $\sigma$-algebra on $\mathbb R$ which contain the open intervals with rational endpoints?
Is there a relation between this $\sigma$-algebra and Borels?
What are the minimum $\sigma$-ring and $\sigma$-algebra on $\mathbb R$ which contain the open intervals with rational endpoints?
Is there a relation between this $\sigma$-algebra and Borels?
Let $\mathcal C:=\{(a,b),a,b\in\Bbb Q\}$. Denote $\mathcal R$ the $\sigma$-ring generated by $\mathcal C$ and $\mathcal A$ the $\sigma$-algebra generated by $\mathcal C$.
Let $a
We have $\mathbb R=\bigcup_{n\geq 1}\underbrace{(-n,n)}_{\in\mathcal C}$ hence $\Bbb R\in\mathcal R$. Hence $\mathcal R$ is stable by countable union, complementation and contains the whole real line: it's a $\sigma$-algebra. By the same argument, $\mathcal R=\mathcal B(\Bbb R)$.