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

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Sources

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