(Irreducible)
A nonempty topological space is irreducible if it is not the union of two proper closed subsets. Equivalently, any two nonempty open subsets meet.
(Noetherian)
A topological space is Noetherian if its closed subsets satisfy the descending chain condition.
\(\AA^n\) is Noetherian: the map \(Z\mapsto I(Z)\) is injective and inclusion-reversing on closed subsets, so it turns a descending chain of closed sets into an ascending chain of radical ideals in \(k[x_1,\ldots,x_n]\), which stabilises by the Hilbert basis theorem.
What does it mean for a topological space to be irreducible?
What does it mean for a topological space to be Noetherian?
Why is $\AA^n$ Noetherian?
What is Noetherian induction?
Show that a closed subset of a Noetherian space has a unique decomposition into irreducible components.
Show that a nonempty open subset of an irreducible space is dense.
Show that the nodal cubic $V(y^2 - x^3 - x^2)$ and the cuspidal cubic $V(y^2 - x^3)$ are irreducible.
Backlinks
D-5LJUXDimension of a varietyusesP-AGHNOETHSPACEFour characterisations of a Noetherian space, and what they forceusesPR-TNVSIIrreducible decomposition in a Noetherian spaceusesP-AGHOPENDENSEOpens are dense in an irreducible space, and closures stay irreducibleuses