# Sober T0 space

From Topospaces

*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 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 are closures of singleton subsets.
- It is T0: for any two distinct points , we can find either an open subset containing and not , or an open subset containing and not .

## 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 |