Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the topology generated by open intervals of $\mathbb R$ on $\mathbb R$?
I'd just like to know if I am correct.
Proof : Let $S$ be the sub-basis formed with the rays $(-\infty, a)$ and $(b, \infty)$. We have to check that the topology $T_{S}$ generated by the sub-basis $S$ is equal to the topology on $\mathbb{R}$. As $(-\infty, a) \cup ((-\infty, a)\cap(b, \infty)) \cup (b, \infty) = \mathbb{R} $ Then $S$ is a sub-basis for the topology on $\mathbb{R}$.
Am I right?