Editing Spectrum of a ring (section)

== Zariski topology ==

For any [[ideal (ring theory)|ideal]] <math>I</math> of <math>R</math>, define <math>V_I</math> to be the set of [[Prime ideal|prime ideals]] containing <math>I</math>. We can put a [[topology]] on <math>\operatorname{Spec}(R)</math> by defining the [[Characterizations of the category of topological spaces#Definition via closed sets|collection of closed sets]] to be
:<math>\big\{ V_I \colon I \text{ is an ideal of } R \big\}.</math>
This topology is called the [[Zariski topology]].

A [[Base (topology)|basis]] for the Zariski topology can be constructed as follows: For <math>f\in R</math>, define <math>D_f</math> to be the set of prime ideals of <math>R</math> not containing <math>f</math>.  Then each <math>D_f</math> is an open subset of <math>\operatorname{Spec}(R)</math>, and <math>\big\{D_f:f\in R\big\}</math> is a basis for the Zariski topology.

<math>\operatorname{Spec}(R)</math> is a [[compact space]], but almost never [[Hausdorff space|Hausdorff]]: In fact, the [[maximal ideal]]s in <math>R</math> are precisely the closed points in this topology. By the same reasoning, <math>\operatorname{Spec}(R)</math> is not, in general, a [[T1 space|T<sub>1</sub> space]].{{sfnp|Arkhangel'skii|Pontryagin|1990|loc=example 21, section 2.6|ps=}} However, <math>\operatorname{Spec}(R)</math> is always a [[Kolmogorov space]] (satisfies the T<sub>0</sub> axiom); it is also a [[spectral space]].