Closed Sets in Noetherian Topological Space

Theorem
Let $\struct {X, \tau}$ be a Noetherian topological space.

Let $Y \subseteq X$ be a non-empty closed subset.

Then there exist closed irreducible subsets $Y_1,\ldots,Y_r$ such that:
 * $Y = Y_1 \cup \cdots \cup Y_r$

Furthermore, if we require:
 * $\forall i,j : i \ne j \implies Y_i \not \subseteq Y_j$

then $Y_1,\ldots,Y_r$ are uniquely determined.