The Stacks project

Definition 5.8.6. Let $X$ be a topological space.

1. Let $Z \subset X$ be an irreducible closed subset. A generic point of $Z$ is a point $\xi \in Z$ such that $Z = \overline{\{ \xi \} }$.

2. The space $X$ is called Kolmogorov, if for every $x, x' \in X$, $x \not= x'$ there exists a closed subset of $X$ which contains exactly one of the two points.

3. The space $X$ is called quasi-sober if every irreducible closed subset has a generic point.

4. The space $X$ is called sober if every irreducible closed subset has a unique generic point.