Closed Sets in Noetherian Topological Space

Theorem

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

Let $Y \subseteq X$ be a non-empty closed set of $T$.

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.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1977: Robin Hartshorne: Algebraic Geometry Proposition $1.5$