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. $\square$

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