# The Stacks Project

## Tag: 004X

This tag has label topology-definition-generic-point and it points to

The corresponding content:

Definition 5.5.4. 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 sober if every irreducible closed subset has a unique generic point.

\begin{definition}
\label{definition-generic-point}
Let $X$ be a topological space.
\begin{enumerate}
\item Let $Z \subset X$ be an irreducible closed subset.
A {\it generic point} of $Z$ is a point $\xi \in Z$ such
that $Z = \overline{\{\xi\}}$.
\item The space $X$ is called {\it 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.
\item The space $X$ is called {\it sober} if every
irreducible closed subset has a unique generic point.
\end{enumerate}
\end{definition}


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