---
tag: EXJAXVXW51
kind: definition
name: Zariski topology
msc: [14A05, 13A15]
defines: 8SQK7RTB63
uses: [0310REJJY8, 00MHHZ9SNE, 0CQHFPGJ45, 0ZCK8DQYN1]
sources:
  - url: https://en.wikipedia.org/wiki/Zariski_topology
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $X$ be the set of [[prime ideals|00MHHZ9SNE]] of $R$.
For a subset $E \subseteq R$, let
$$V(E) = \set{\mathfrak p \in X : E \subseteq \mathfrak p}.$$
The *Zariski topology* on $X$ is the [[topology|0CQHFPGJ45]] whose [[closed sets|0ZCK8DQYN1]] are exactly the sets $V(E)$ for $E \subseteq R$.

## Remarks

These sets are the closed sets of a topology:

1. $V(\set{0}) = X$ and $V(\set{1}) = \emptyset$;
2. $\bigcap_i V(E_i) = V\left(\bigcup_i E_i\right)$ for any family of subsets $E_i$;
3. $V(I) \cup V(J) = V(IJ)$ for ideals $I, J$, because a prime ideal contains $IJ$ only if it contains $I$ or $J$.

Also $V(E) = V(I)$ for the ideal $I$ generated by $E$, and $V(I) = V(\sqrt I)$.
The closure of a point $\mathfrak p$ is $V(\mathfrak p)$, so $\set{\mathfrak p}$ is closed exactly when $\mathfrak p$ is a [[maximal ideal|0PV74TR5FP]].

Example: in $\operatorname{Spec} \Z$, the closed sets are $\operatorname{Spec} \Z$ itself and the finite sets of nonzero prime ideals.
Indeed $V(n\Z)$ for $n \ne 0$ is the finite set of $p\Z$ with $p \mid n$.

Non-example: $\operatorname{Spec} \Z$ with this topology is not [[Hausdorff|0S1FJ8VEX9]].
The point $(0)$ is not closed: its closure is $V(0)$, the whole space.
Any nonempty open set contains $(0)$, so no two nonempty open sets are disjoint.

The open sets $D(f) = X \setminus V(\set{f})$ are the [[basic open sets|N6JX9BCKQP]] and form a basis of the topology.