Sober T0 space

This article describes a property of topological spaces obtained as a conjunction of the following two properties: sober space and T0 space

Statement

A topological space $X$ is termed a sober T0 space if it is both a sober space and a Kolmogorov space (T0 space). Explicitly:

• It is sober: the only irreducible closed subsets of $X$ are closures of singleton subsets.
• It is T0: for any two distinct points $x,y\in X$, we can find either an open subset containing $x$ and not $y$, or an open subset containing $y$ and not $x$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
sober T1 space	sober and every singleton subset is closed			|FULL LIST, MORE INFO
Hausdorff space	any two distinct points can be separated by disjoint open subsets			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 singleton subsets			|FULL LIST, MORE INFO
Kolmogorov space	any two points are topologically distinguishable			|FULL LIST, MORE INFO