---
tag: 4N4DNDR22Z
kind: concept
name: Sober space
msc: [54D10, 06D22]
specializes: [07SG12TQH3]
sources:
  - url: https://en.wikipedia.org/wiki/Sober_space
same_as:
  wikidata: Q8778429
  wikipedia: Sober space
---

## Summary

A sober space is a topological space in which every irreducible closed set is the closure of exactly one point.
That point is called the generic point of the set.
Every sober space is $T_0$, and every Hausdorff space is sober.
Spectra of commutative rings with the Zariski topology are sober.
A sober space can be recovered, up to homeomorphism, from its lattice of open sets.

## Explanations

### L1 human

Some closed sets cannot be split into two smaller closed pieces; they are called irreducible.
A sober space has, for each such set, one special point whose closure is the whole set.
In algebraic geometry, the irreducible curve $y = x^2$ in the plane has such a point in the spectrum of the polynomial ring: the prime ideal $(y - x^2)$.
In a Hausdorff space, the only irreducible closed sets are single points, so each is its own generic point.