---
tag: 8SQK7RTB63
kind: concept
name: Zariski topology
msc: [14A05, 13A15]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Zariski_topology
same_as:
  wikidata: Q147978
  wikipedia: Zariski topology
---

## Summary

The Zariski topology is a topology on the [[spectrum|FB45BHHA2J]] of a [[commutative ring|0310REJJY8]].
Its closed sets are the sets of [[prime ideals|00MHHZ9SNE]] containing a given set of elements.
On an affine variety it restricts to the topology whose closed sets are the zero sets of polynomials.
It has few open sets: it is [[quasi-compact|J4YW23MD6N]] and usually not [[Hausdorff|0S1FJ8VEX9]].
It is the topology used throughout algebraic geometry.

## Explanations

### L1 human

In the Zariski topology a closed set is a set of common zeros of some functions.
On the line $\C$, the zero set of a nonzero polynomial is finite, so the closed subsets of the line are the finite sets and the whole line.
Any two nonempty open sets then meet, which is very different from the usual topology.
On $\operatorname{Spec} R$ one says that $f$ "vanishes" at a prime $\mathfrak p$ when $f \in \mathfrak p$.
The [[definition|EXJAXVXW51]] uses this to define the closed sets.