Sober space

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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 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