$\operatorname{Spec}(\mathbb{Z}[x])$ as a ''classical'' topological space?

Question

I've read recently the definition of the prime spectrum associated to a ring and how it can be made into a topological space.

I've read that $\operatorname{Spec}(\mathbb{Z}[x])$ is really important and that it can be represented by the Mumford's treasure map.

But, can be this topological space seen as a usual set ($\mathbb{R}$, $\mathbb{C}$ for example) with a particular topology?