# Sober T1 space

From Topospaces

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

## Contents

## Definition

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

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

## 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 |