Definition:Zariski Topology

Geometric Definition
Let $k$ be a field.

The Zariski topology is the topology on the direct product $k^n$ which sets:
 * $X \subset k^n$ is open $\iff k^n \setminus X$ is an affine algebraic variety

where $k^n \setminus X$ denotes set difference.

Algebraic Definition
Let $A$ be a commutative ring with unity.

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

The Zariski topology says that a set $V \subseteq \operatorname{Spec}(A)$ is closed if 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 $V$.