Difference between revisions of "Sober space"

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(Stronger properties)
(Stronger properties)
 
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| [[Weaker than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || [[Hausdorff implies sober]] || || {{intermediate notions short|sober space|Hausdorff space}}
 
| [[Weaker than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || [[Hausdorff implies sober]] || || {{intermediate notions short|sober space|Hausdorff space}}
 
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| [[Weaker than::sober T1 space]] || the irreducible closed subsets are precisely the singleton subsets. Applying the [[Zariski topology]] often gives this kind of a space || || || {{intermediate notions short|sober space|sober T1 space}}
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| [[Weaker than::sober T1 space]] || the irreducible closed subsets are precisely the singleton subsets || || || {{intermediate notions short|sober space|sober T1 space}}
 
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| [[Weaker than::compact sober T1 space]] || compact, sober, and T1. Many spaces that arise via [[Zariski topology]] are of this kind, but very rarely Hausdorff || || || {{intermediate notions short|sober space|compact sober T1 space}}
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| [[Weaker than::compact sober T1 space]] || compact and sober T1 || || || {{intermediate notions short|sober space|compact sober T1 space}}
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| [[Weaker than::sober T0 space]] || sober and a [[Kolmogorov space]] (T0 space): any two points are topologically distinguishable || || || {{intermediate notions short|sober space|sober T0 space}}
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| [[Weaker than::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 || || || {{intermediate notions short|sober space|compact sober T1 space}}
 
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Latest revision as of 01:33, 5 January 2017

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