Definition:Zariski Topology

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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.