The Stacks project

Lemma 5.15.2. The collection of constructible sets is closed under finite intersections, finite unions and complements.

Proof. Note that if $U_1$, $U_2$ are open and retrocompact in $X$ then so is $U_1 \cup U_2$ because the union of two quasi-compact subsets of $X$ is quasi-compact. It is also true that $U_1 \cap U_2$ is retrocompact. Namely, suppose $U \subset X$ is quasi-compact open, then $U_2 \cap U$ is quasi-compact because $U_2$ is retrocompact in $X$, and then we conclude $U_1 \cap (U_2 \cap U)$ is quasi-compact because $U_1$ is retrocompact in $X$. From this it is formal to show that the complement of a constructible set is constructible, that finite unions of constructibles are constructible, and that finite intersections of constructibles are constructible. $\square$

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Your email address will not be published.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 005H. Beware of the difference between the letter 'O' and the digit '0'.