Lemma 5.15.5. Let $U \subset X$ be a retrocompact open. Let $E \subset U$. If $E$ is constructible in $U$, then $E$ is constructible in $X$.

**Proof.**
Suppose that $V, W \subset U$ are retrocompact open in $U$. Then $V, W$ are retrocompact open in $X$ (Lemma 5.12.2). Hence $V \cap (U \setminus W) = V \cap (X \setminus W)$ is constructible in $X$. We conclude since every constructible subset of $U$ is a finite union of subsets of the form $V \cap (U \setminus W)$.
$\square$

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