# Difference between revisions of "Sober 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}} | | [[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. 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::compact sober T1 space]] || compact, sober, and T1. | + | | [[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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## Revision as of 01:18, 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

## Contents

## 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 |