Difference between revisions of "Sober space"

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(Stronger properties)
(Stronger properties)
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
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| [[Weaker than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || [[Hausdorff implies sober]] || || {{intermediate notions|sober space|Hausdorff space}}
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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}}
 
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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}}
 
| [[Weaker than::sober T1 space]] || the irreducible closed subsets are precisely the singleton subsets || || || {{intermediate notions short|sober space|sober T1 space}}

Revision as of 00:51, 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, sober, and T1. Spaces that arise via Zariski topology are of this kind, but very rarely Hausdorff Sober T1 space|FULL LIST, MORE INFO