Sober 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 $$X$$)