Sober T0 space

From Topospaces
Jump to: navigation, search

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