---
tag: G3W2WXY2VN
kind: concept
name: Spectral space
msc: [54F65, 06D50, 14A05]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_space
same_as:
  wikidata: Q7575210
  wikipedia: Spectral space
---

## Summary

A spectral space is a compact [[sober|4N4DNDR22Z]] topological space whose compact open sets are closed under finite intersections and form a basis of the topology.
The prime spectrum of any [[commutative ring|0310REJJY8]] with the [[Zariski topology|8SQK7RTB63]] is spectral, and by [[Hochster's theorem|C82AXZK2BS]] every spectral space arises in this way.
The compact open sets of a spectral space form a bounded [[distributive lattice|0HM0HQCK6N]], and every bounded distributive lattice arises in this way.
So spectral spaces connect commutative algebra, lattice theory and topology.

## Explanations

### L1 human

The points of a spectral space behave like the prime ideals of a ring: some points are "general" and have other points in their closure.
The space is compact, and it has many open sets that are compact as well, enough to describe the whole topology.
Each closed set that cannot be split into two smaller closed sets has exactly one "generic" point whose closure is that set.
For example, the prime ideals of the integers form a spectral space whose generic point $(0)$ has every point in its closure.