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