Sober space
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
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. Applying the Zariski topology often gives this kind of a space | Sober T0 space|FULL LIST, MORE INFO | ||
| compact sober T1 space | compact, sober, and T1. Many spaces that arise via Zariski topology are of this kind, but very rarely Hausdorff | Sober T1 space|FULL LIST, MORE INFO |