Definition 5.8.4. Let $X$ be a topological space.

Let $Z \subset X$ be an irreducible closed subset. A

*generic point*of $Z$ is a point $\xi \in Z$ such that $Z = \overline{\{ \xi \} }$.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.The space $X$ is called

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

*sober*if every irreducible closed subset has a unique generic point.

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