The Stacks project

Lemma 5.19.2. Let $X$ be a topological space.

  1. Any closed subset of $X$ is stable under specialization.

  2. Any open subset of $X$ is stable under generalization.

  3. A subset $T \subset X$ is stable under specialization if and only if the complement $T^ c$ is stable under generalization.

Proof. Let $F$ be a closed subset of $X$, if $y\in F$ then $\{ y\} \subset F$, so $\overline{\{ y\} } \subset \overline{F} = F$ as $F$ is closed. Thus for all specialization $x$ of $y$, we have $x\in F$.

Let $x, y\in X$ such that $x\in \overline{\{ y\} }$ and let $T$ be a subset of $X$. Saying that $T$ is stable under specialization means that $y\in T$ implies $x\in T$ and reciprocally saying that $T$ is stable under generalization means that $x\in T$ implies $y\in T$. Therefore (3) is proven using contraposition.

The second property follows from (1) and (3) by considering the complement. $\square$

Comments (0)

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Your email address will not be published.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 0062. Beware of the difference between the letter 'O' and the digit '0'.