# Definition:Zariski Topology

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## Definition

### On an Affine Space

Let $k$ be a field.

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

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

### On the spectrum of a ring

Let $A$ be a commutative ring with unity.

Let $\Spec A$ be the prime spectrum of $A$.

The **Zariski topology** on $\Spec A$ is the topology with closed sets the vanishing sets $\map V S$ for $S \subseteq A$.

## Also defined as

Some sources use the term **Zariski topology** for the **finite complement topology**.

This usage is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Results about
**Zariski topology**can be found**here**.

## Source of Name

This entry was named for Oscar Zariski.

## Source

- 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**Zariski topology**