Let $X$ be a [topological space](/page/Topological%20Space) such that every descending chain of closed subsets of $X$ stabilizes, and let $Y \subset X$ be closed. Equip $Y$ with the [subspace topology](/page/Subspace%20Topology) inherited from $X$. Then every descending chain of closed subsets of $Y$ stabilizes; equivalently, $Y$ is Noetherian.