Describe the topology of Spec$(\mathbb{R}[x])$

Question

I am supposed to describe the points and topology of Spec$(\mathbb{R}[x])$, I managed to describe the points but I dont understand the "topology" of the set, what does this mean? Are they asking for the Zariski topology and how can I describe this?

Answer 1

The irreducible closed subsets of $\mathbb A_\mathbb R^1=Spec(\mathbb R[x])$ with its Zariski topology are:
$\bullet$ $\mathbb A_\mathbb R^1$
$\bullet \bullet$ The prime ideals $\langle x-r\rangle$ with $r\in \mathbb R$
$\bullet \bullet \bullet$ The prime ideals $\langle x^2+px+q\rangle$ with $p,q\in \mathbb R$ satisfying $p^2-4q\lt0$

A possibly reducible closed subset is then a finite union of the above (including $\emptyset$).