Grothendieck defined retrocompact subsets of a topological space in EGA III-1. p.12. The notion of retrocompact open subsets is used in the definition of constructible subsets of a scheme which is important in algebraic geometry.
Let $X$ be a topological space. We say a subset $Z$ of $X$ is retrocompact if $Z \cap U$ is quasi-compact for every quasi-compact open subset $U$ of $X$.
If $X$ is a separated scheme, every quasi-compact open subset is retrocompact.
Suppose $X$ is a locally Noetherian scheme. Let $U$ be an open subset of $X$. Let $V$ be a quasi-compact open subset of $X$. Since $V$ is a Noetherian topological space, $U \cap V$ is quasi-compact. Hence $U$ is retrocompact.
So let us suppose $X$ is a separated scheme which is not locally Noetherian. I would like to know an example of an open subset $U$ of $X$ satisfiying the following conditions.
(1) $U$ is not quasi-compact.
(2) $U$ is retrocompact.