Difference between revisions of "Sober T1 space"
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Revision as of 01:24, 5 January 2017
This article describes a property of topological spaces obtained as a conjunction of the following two properties: sober space and T1 space
Contents
Definition
A topological space is termed a sober T1 space if, for any nonempty subset of , the following are equivalent:
 is a singleton subset, i.e., it has precisely one element.
 is an irreducible closed subset of , i.e., it is a closed subset and cannot be expressed as a union of two proper closed subsets of it (note that it does not matter for this definition whether we consider "closed" inside via the subspace topology, or closed inside )
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  FULL LIST, MORE INFO  
compact sober T1 space  compact space as well as a sober T1 space  FULL LIST, MORE INFO 
Weaker properties
Property  Meaning  Proof of implication  Proof of strictness (reverse implication failure)  Intermediate notions 

sober space  the only irreducible closed subsets are closures of onepoint subsets  Sober T0 spaceFULL LIST, MORE INFO  
T1 space  every singleton subset is closed  FULL LIST, MORE INFO 