The Stacks project

Lemma 5.15.12. Let $X$ be a topological space. Assume $X$ has a basis consisting of quasi-compact opens. Let $E$ be constructible in $X$ and $F \subset E$ constructible in $E$. Then $F$ is constructible in $X$.

Proof. Observe that any retrocompact subset $T$ of $X$ has a basis for the induced topology consisting of quasi-compact opens. In particular this holds for any constructible subset (Lemma 5.15.10). Write $E = E_1 \cup \ldots \cup E_ n$ with $E_ i = U_ i \cap V_ i^ c$ where $U_ i, V_ i \subset X$ are retrocompact open. Note that $E_ i = E \cap E_ i$ is constructible in $E$ by Lemma 5.15.11. Hence $F \cap E_ i$ is constructible in $E_ i$ by Lemma 5.15.11. Thus it suffices to prove the lemma in case $E = U \cap V^ c$ where $U, V \subset X$ are retrocompact open. In this case the inclusion $E \subset X$ is a composition

\[ E = U \cap V^ c \to U \to X \]

Then we can apply Lemma 5.15.9 to the first inclusion and Lemma 5.15.5 to the second. $\square$

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Your email address will not be published.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 09YJ. Beware of the difference between the letter 'O' and the digit '0'.