Sober T1 space: Difference between revisions

Latest revision as of 01:27, 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

A topological space $X$ is termed a sober T1 space if, for any nonempty subset $A$ of $X$ , the following are equivalent:

$A$ is a singleton subset, i.e., it has precisely one element.
$A$ is an irreducible closed subset of $X$ , 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 $A$ via the subspace topology, or closed inside $X$ )

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

Property	Meaning	Intermediate notions
sober space	the only irreducible closed subsets are closures of one-point subsets	Sober T0 space\|FULL LIST, MORE INFO
sober T0 space	sober and a Kolmogorov space: any two points are topologically distinguishable	\|FULL LIST, MORE INFO
T1 space	every singleton subset is closed	\|FULL LIST, MORE INFO

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 | [[Stronger than::sober space]] || the only irreducible closed subsets are closures of one-point subsets || || || {{intermediate notions short|sober space|sober T1 space}}
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+| [[Stronger than::sober T0 space]] || sober and a [[Kolmogorov space]]: any two points are topologically distinguishable || || || {{intermediate notions short|sober T0 space|sober T1 space}}
 |-
 | [[Stronger than::T1 space]] || every singleton subset is closed || || || {{intermediate notions short|T1 space|sober T1 space}}
 |}