Let $X$ be a nonempty Noetherian [topological space](/page/Topological%20Space). Then there exist finitely many nonempty irreducible closed subsets $X_1,\dots,X_r\subset X$ such that
are two such decompositions into nonempty irreducible closed subsets, and each decomposition is irredundant in the sense that no listed subset is contained in another listed subset in the same decomposition, then $r=s$ and, after reordering, $X_i=Y_i$ for every $i\in\{1,\dots,r\}$.