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.

## Comments (0)

There are also:

• 8 comment(s) on Section 5.8: Irreducible components

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 004X. Beware of the difference between the letter 'O' and the digit '0'.