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?
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Tell us what you know about the Zariski topology, and that will help guide the answers. – rschwieb Dec 08 '14 at 16:33
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Yes, the Zariski topology. Unlike many other topological spaces, you can explicitly say what every single closed set, and hence open set, is. I would start with describing $\mathrm{Spec}(\mathbb{C}[x])$ first, this is easier. – RghtHndSd Dec 08 '14 at 16:33
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@rschwieb One way to describe this in the book was $V(I)={ P \in \text{Spec} A : I \subset P }$, but $I \subset P$ means that every $f \in I$ maps to $0$ in the map $A \to A/P$, so one writes $V(I)= { P \in \text{Spec}A : f(P)=0 $ for all $f \in I }$. Does $f(P)$ for all $f \in I$ mean that (for fix $f$) that $fp$ maps to $0$ for all $p \in P$ under the map $A \to A/P$? – Dec 08 '14 at 20:51
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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 $\emptyset$).
Georges Elencwajg
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