def ProperClosedFinite (X : Type u_1) [TopologicalSpace X] : Prop

A topological space has the proper closed finite property if every proper closed subset is finite (characteristic of a one-dimensional space such as an irreducible curve).

theorem continuous_of_injective_of_T1_properClosedFinite {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space X] (hY : ProperClosedFinite Y) {f : X → Y} (hf_inj : Function.Injective f) : Continuous f

Any injective map from a T1 space to a space whose proper closed subsets are finite is automatically continuous.

def Homeomorph.ofEquivProperClosedFinite {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space X] [T1Space Y] (hX : ProperClosedFinite X) (hY : ProperClosedFinite Y) (e : X ≃ Y) : X ≃ₜ Y

Promote an equivalence between two T1 spaces with the proper-closed-finite property to a homeomorphism.

theorem irreducible_curves_homeomorphic {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] [T1Space X] [T1Space Y] (hX : ProperClosedFinite X) (hY : ProperClosedFinite Y) (hcard : Cardinal.mk X = Cardinal.mk Y) : Nonempty (X ≃ₜ Y)

Lecture 5, Proposition 6 (irreducible curves are homeomorphic): any two T1 spaces with the proper-closed-finite property and equal cardinality are homeomorphic.

theorem WithBot.le_zero_of_add_one_le_one (x : WithBot ℕ∞) (h : x + 1 ≤ 1) : x ≤ 0

In WithBot ℕ∞, x + 1 ≤ 1 implies x ≤ 0.

theorem isArtinianRing_quotient_of_dim_le_one {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] (hdim : ringKrullDim A ≤ 1) (I : Ideal A) (hI : I ≠ ⊥) : IsArtinianRing (A ⧸ I)

For a Noetherian domain of Krull dimension at most 1, every quotient by a nonzero ideal is Artinian.

theorem zeroLocus_finite_of_dim_le_one {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] (hdim : ringKrullDim A ≤ 1) (I : Ideal A) (hI : I ≠ ⊥) : (PrimeSpectrum.zeroLocus ↑I).Finite

In a Noetherian domain of Krull dimension at most 1, the zero locus of any nonzero ideal is finite.

theorem properClosedFinite_primeSpectrum_of_dim_le_one {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] (hdim : ringKrullDim A ≤ 1) : ProperClosedFinite (PrimeSpectrum A)

The prime spectrum of a Noetherian domain of Krull dimension at most 1 satisfies the proper-closed-finite property: every proper closed subset is finite.