Lemma 5.15.9. Let $Z \subset X$ be a closed subset whose complement is retrocompact open. Let $E \subset Z$. If $E$ is constructible in $Z$, then $E$ is constructible in $X$.
Proof. Suppose that $V \subset Z$ is retrocompact open in $Z$. Consider the open subset $\tilde V = V \cup (X \setminus Z)$ of $X$. Let $W \subset X$ be quasi-compact open. Then
\[ W \cap \tilde V = \left(V \cap W\right) \cup \left((X \setminus Z) \cap W\right). \]
The first part is quasi-compact as $V \cap W = V \cap (Z \cap W)$ and $(Z \cap W)$ is quasi-compact open in $Z$ (Lemma 5.12.3) and $V$ is retrocompact in $Z$. The second part is quasi-compact as $(X \setminus Z)$ is retrocompact in $X$. In this way we see that $\tilde V$ is retrocompact in $X$. Thus if $V_1, V_2 \subset Z$ are retrocompact open, then
\[ V_1 \cap (Z \setminus V_2) = \tilde V_1 \cap (X \setminus \tilde V_2) \]
is constructible in $X$. We conclude since every constructible subset of $Z$ is a finite union of subsets of the form $V_1 \cap (Z \setminus V_2)$. $\square$
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