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

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?

• Recall: $\mathrm{Spec} \Bbb Z [x]$ is not Hausdorff, thus it is not a subspace of any Hausdorff space. – Crostul Feb 20 at 18:00
• Perhaps you should tell us what you mean by "represented". But to write in more words what @Crostul is saying: $\text{Spec}\mathbb Z[x]$ is not homeomorphic to $\mathbb R$, nor to $\mathbb C$, nor to any other Hausdorff space, nor to any subspace of a Hausdorff space, because a subspace of a Hausdorff space must be Hausdorff, and a space homeomorphic to a Hausdorff space must be Hausdorff. – Lee Mosher Feb 20 at 18:10
• What are you asking? $\text{Spec}(\mathbb{Z}[x])$ is a topological space: this means it's a set equipped with topology. I don't know what you mean by a "usual set"... The set of prime ideals in the ring $\mathbb{Z}[x]$ is a perfectly good set. Of course, $\text{Spec}(\mathbb{Z}[x])$ is countably infinite, so it's in bijection with $\mathbb{N}$, and hence it's homeomorphic to $\mathbb{N}$ equipped with some topology. But this topology won't be very natural, and you won't gain anything by thinking about this homeomorphism. – Alex Kruckman Feb 20 at 18:15
• General topological spaces, and especially the non-Hausdorff ones ubiquitous in algebraic geometry, are hard to visualize in a "real-world way". The sketch in Mumford's Red Book that you refer to in the question is a serious attempt to help you visualize this space and come up with some kind of geometric intuition. You're not going to get a much more helpful representation than that. – Alex Kruckman Feb 20 at 18:21
• I do not think it should be deleted, because through your question I learned about Mumford's treasure map, and others may also. I do feel there is perhaps a better question to be asked, though. One might ask a detailed question about how to understand Mumford's treasure map, assuming that one has put some time into trying to understand the map. – Lee Mosher Feb 20 at 18:55