Definition of Borel on $\mathbb R$ ($\mathcal B(\mathbb R)$) is that it's the $\sigma$-algebra generated by all open sets in $\mathbb R$.
OK, if I take some open set like $C = (0,1)$, by definition of $\sigma$-algebra the complement of $C$ must also be in $\mathcal B(\mathbb R)$, so something like $(-\infty, 0] \cup [1, \infty)$ must be in $\mathcal B(\mathbb R)$. Is that right?