feat(algebraic_geometry): The underlying topological space of a Schem…

…e is T0 (#10869)




Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

src/algebraic_geometry/prime_spectrum/basic.lean

@@ -657,6 +657,10 @@ lemma le_iff_mem_closure (x y : prime_spectrum R) :
 by rw [← as_ideal_le_as_ideal, ← zero_locus_vanishing_ideal_eq_closure,
     mem_zero_locus, vanishing_ideal_singleton, set_like.coe_subset_coe]
 
+instance : t0_space (prime_spectrum R) :=
+by { simp [t0_space_iff_or_not_mem_closure, ← le_iff_mem_closure,
+  ← not_and_distrib, ← le_antisymm_iff, eq_comm] }
+
 end order
 
 end prime_spectrum

src/algebraic_geometry/properties.lean

@@ -28,6 +28,21 @@ namespace algebraic_geometry
 
 variable (X : Scheme)
 
+-- TODO: add sober spaces, and show that schemes are sober
+instance : t0_space X.carrier :=
+begin
+  rw t0_space_iff_distinguishable,
+  intros x y h h',
+  obtain ⟨U, R, ⟨e⟩⟩ := X.local_affine x,
+  have hy := (h' _ U.1.2).mp U.2,
+  erw ← subtype_indistinguishable_iff (⟨x, U.2⟩ : U.1.1) (⟨y, hy⟩ : U.1.1) at h',
+  let e' : U.1 ≃ₜ prime_spectrum R :=
+    homeo_of_iso ((LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _).map_iso e),
+  have := t0_space_of_injective_of_continuous e'.injective e'.continuous,
+  rw t0_space_iff_distinguishable at this,
+  exact this ⟨x, U.2⟩ ⟨y, hy⟩ (by simpa using h) h'
+end
+
 /-- A scheme `X` is integral if its carrier is nonempty,
 and `𝒪ₓ(U)` is an integral domain for each `U ≠ ∅`. -/
 class is_integral : Prop :=

src/logic/basic.lean

@@ -702,6 +702,13 @@ by rw [← decidable.not_and_distrib, decidable.not_not]
 
 theorem and_iff_not_or_not : a ∧ b ↔ ¬ (¬ a ∨ ¬ b) := decidable.and_iff_not_or_not
 
+@[simp] theorem not_xor (P Q : Prop) : ¬ xor P Q ↔ (P ↔ Q) :=
+by simp only [not_and, xor, not_or_distrib, not_not, ← iff_iff_implies_and_implies]
+
+theorem xor_iff_not_iff (P Q : Prop) : xor P Q ↔ ¬ (P ↔ Q) :=
+by rw [iff_not_comm, not_xor]
+
+
 end propositional
 
 /-! ### Declarations about equality -/

src/topology/separation.lean

@@ -137,6 +137,41 @@ end separated
 class t0_space (α : Type u) [topological_space α] : Prop :=
 (t0 : ∀ x y, x ≠ y → ∃ U:set α, is_open U ∧ (xor (x ∈ U) (y ∈ U)))
 
+lemma t0_space_def (α : Type u) [topological_space α] :
+  t0_space α ↔ ∀ x y, x ≠ y → ∃ U:set α, is_open U ∧ (xor (x ∈ U) (y ∈ U)) :=
+by { split, apply @t0_space.t0, apply t0_space.mk }
+
+/-- Two points are topologically indistinguishable if no open set separates them. -/
+def indistinguishable {α : Type u} [topological_space α] (x y : α) : Prop :=
+∀ (U : set α) (hU : is_open U), x ∈ U ↔ y ∈ U
+
+lemma t0_space_iff_distinguishable (α : Type u) [topological_space α] :
+  t0_space α ↔ ∀ (x y : α), x ≠ y → ¬ indistinguishable x y :=
+begin
+  delta indistinguishable,
+  rw t0_space_def,
+  push_neg,
+  simp_rw xor_iff_not_iff,
+end
+
+lemma indistinguishable_iff_closed {α : Type u} [topological_space α] (x y : α) :
+  indistinguishable x y ↔ ∀ (U : set α) (hU : is_closed U), x ∈ U ↔ y ∈ U :=
+⟨λ h U hU, not_iff_not.mp (h _ hU.1), λ h U hU, not_iff_not.mp (h _ (is_closed_compl_iff.mpr hU))⟩
+
+lemma indistinguishable_iff_closure {α : Type u} [topological_space α] (x y : α) :
+  indistinguishable x y ↔ x ∈ closure ({y} : set α) ∧ y ∈ closure ({x} : set α) :=
+begin
+  rw indistinguishable_iff_closed,
+  exact ⟨λ h, ⟨(h _ is_closed_closure).mpr (subset_closure $ set.mem_singleton y),
+      (h _ is_closed_closure).mp (subset_closure $ set.mem_singleton x)⟩,
+    λ h U hU, ⟨λ hx, (is_closed.closure_subset_iff hU).mpr (set.singleton_subset_iff.mpr hx) h.2,
+      λ hy, (is_closed.closure_subset_iff hU).mpr (set.singleton_subset_iff.mpr hy) h.1⟩⟩
+end
+
+lemma subtype_indistinguishable_iff {α : Type u} [topological_space α] {U : set α} (x y : U) :
+  indistinguishable x y ↔ indistinguishable (x : α) y :=
+by { simp_rw [indistinguishable_iff_closure, closure_subtype, image_singleton] }
+
 /-- Given a closed set `S` in a compact T₀ space,
 there is some `x ∈ S` such that `{x}` is closed. -/
 theorem is_closed.exists_closed_singleton {α : Type*} [topological_space α]
@@ -207,6 +242,34 @@ instance subtype.t0_space [t0_space α] {p : α → Prop} : t0_space (subtype p)
 ⟨λ x y hxy, let ⟨U, hU, hxyU⟩ := t0_space.t0 (x:α) y ((not_congr subtype.ext_iff_val).1 hxy) in
   ⟨(coe : subtype p → α) ⁻¹' U, is_open_induced hU, hxyU⟩⟩
 
+theorem t0_space_iff_or_not_mem_closure (α : Type u) [topological_space α] :
+  t0_space α ↔ (∀ a b : α, (a ≠ b) → (a ∉ closure ({b} : set α) ∨ b ∉ closure ({a} : set α))) :=
+begin
+  simp only [← not_and_distrib, t0_space_def, not_and],
+  apply forall_congr, intro a,
+  apply forall_congr, intro b,
+  apply forall_congr, intro _,
+  split,
+  { rintro ⟨s, h₁, (⟨h₂, h₃ : b ∈ sᶜ⟩|⟨h₂, h₃ : a ∈ sᶜ⟩)⟩ ha hb; rw ← is_closed_compl_iff at h₁,
+    { exact (is_closed.closure_subset_iff h₁).mpr (set.singleton_subset_iff.mpr h₃) ha h₂ },
+    { exact (is_closed.closure_subset_iff h₁).mpr (set.singleton_subset_iff.mpr h₃) hb h₂ } },
+  { intro h,
+    by_cases h' : a ∈ closure ({b} : set α),
+    { exact ⟨(closure {a})ᶜ, is_closed_closure.1,
+        or.inr ⟨h h', not_not.mpr (subset_closure (set.mem_singleton a))⟩⟩ },
+    { exact ⟨(closure {b})ᶜ, is_closed_closure.1,
+        or.inl ⟨h', not_not.mpr (subset_closure (set.mem_singleton b))⟩⟩ } }
+end
+
+lemma t0_space_of_injective_of_continuous {α β : Type u} [topological_space α] [topological_space β]
+  {f : α → β} (hf : function.injective f) (hf' : continuous f) [t0_space β] : t0_space α :=
+begin
+  constructor,
+  intros x y h,
+  obtain ⟨U, hU, e⟩ := t0_space.t0 _ _ (hf.ne h),
+  exact ⟨f ⁻¹' U, hf'.1 U hU, e⟩
+end
+
 /-- A T₁ space, also known as a Fréchet space, is a topological space
   where every singleton set is closed. Equivalently, for every pair
   `x ≠ y`, there is an open set containing `x` and not `y`. -/