Sober T1 space

This article describes a property of topological spaces obtained as a conjunction of the following two properties: sober space and T1 space

Definition

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 $X$)

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 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