Definition:Zariski Topology

On an Affine Space
Let $k$ be a field.

Let $\mathbb A^n\left({k}\right) = k^n$ denote the standard affine space of dimension $n$ over $k$.

The Zariski topology on $\mathbb A^n\left({k}\right)$ is the topology on the direct product $k^n$ whose closed sets are the affine algebraic sets in $\mathbb A^n\left({k}\right)$.

On a Commutative Ring
Let $A$ be a commutative ring with unity.

Let $\operatorname{Spec}(A)$ be the prime spectrum of $A$.

The Zariski topology is that with closed sets $V \subseteq \operatorname{Spec}(A)$ such that for some $S \subseteq A$:


 * $V = \{ \mathfrak p \in \operatorname{Spec}(A) : \mathfrak p \supseteq S \}$

We usually write $V = V(S)$, and specify $S$ to determine the set $V$.

Also see

 * Zariski Topology is Topology