Let $B_2$ be the $\sigma$-algebra of Borel subsets of $\mathbb R^2$, i.e. $B_2=\sigma (\mathbb G_2)$, where $\mathbb G_2$ is the collection of all open subsets of $\mathbb R^2$. Prove $B_2= \sigma (R_2)$ where $$R_2=\{ (a,b) \times (c,d): a<b \, \, \text{and} \, \, c<d\, \, \text{are real numbers} \}$$
I have been looking at this for ages. Please can someone show me how to do this.
Many thanks in advance.
Restrict $a$, $b$, $c$ and $d$ to being rational in your definition of $R_2$. Every open subset of the plane is a union of elements of this "slimmed down" $R_2$. Therefore every open subset of the plane is in $\sigma($"smaller $R_2$"$)$.