The Stacks project

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 $X$. $\square$