Sober space: Difference between revisions
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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}} | | [[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 |