# Sober space

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

## Contents

## Definition

### Symbol-free definition

A topological space is said to be **sober** if the only irreducible closed subsets are the closures of one-point sets.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Hausdorff space | any two distinct points can be separated by disjoint open subsets | Hausdorff implies sober | Sober T0 space, Sober T1 space|FULL LIST, MORE INFO | |

sober T1 space | the irreducible closed subsets are precisely the singleton subsets | Sober T0 space|FULL LIST, MORE INFO | ||

compact sober T1 space | compact and sober T1 | Sober T1 space|FULL LIST, MORE INFO | ||

sober T0 space | sober and a Kolmogorov space (T0 space): any two points are topologically distinguishable | |FULL LIST, MORE INFO | ||

compact sober T0 space | compact, sober, and T0. Spaces that arise via Zariski topology on the prime spectrum of a commutative unital ring are of this kind, but very rarely Hausdorff | Sober T1 space|FULL LIST, MORE INFO |