Lemma 5.15.13. Let $X$ be a quasi-compact topological space having a basis consisting of quasi-compact opens such that the intersection of any two quasi-compact opens is quasi-compact. Let $T \subset X$ be a locally closed subset such that $T$ is quasi-compact and $T^ c$ is retrocompact in $X$. Then $T$ is constructible in $X$.

**Proof.**
Note that $T$ is quasi-compact and open in $\overline{T}$. Using our basis of quasi-compact opens we can write $T = U \cap \overline{T}$ where $U$ is quasi-compact open in $X$. Then $V = U \setminus T = U \cap T^ c$ is retrocompact in $U$ as $T^ c$ is retrocompact in $X$. Hence $V$ is quasi-compact. Since the intersection of any two quasi-compact opens is quasi-compact any quasi-compact open of $X$ is retrocompact. Thus $T = U \cap V^ c$ with $U$ and $V = U \setminus T$ retrocompact opens of $X$. A fortiori, $T$ is constructible in $X$.
$\square$

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