l3lab/ntp-mathlib huggingface dataset:
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      

Proof state:
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
⊢ False

Next tactic:
obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      

Proof state:
case intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
⊢ False

Next tactic:
have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        exact Countable.union h Vct
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        

Proof state:
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
⊢ Set.Countable (U ∩ C)

Next tactic:
apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · 

Proof state:
case h
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
⊢ U ∩ C ⊆ E ∩ D ∪ V ∩ C

Next tactic:
rintro y ⟨yU, yC⟩
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          

Proof state:
case h.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
y : α
yU : y ∈ U
yC : y ∈ C
⊢ y ∈ E ∩ D ∪ V ∩ C

Next tactic:
by_cases h : y ∈ V
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · 

Proof state:
case pos
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h✝ : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
y : α
yU : y ∈ U
yC : y ∈ C
h : y ∈ V
⊢ y ∈ E ∩ D ∪ V ∩ C

Next tactic:
exact mem_union_right _ (mem_inter h yC)
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · 

Proof state:
case neg
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h✝ : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
y : α
yU : y ∈ U
yC : y ∈ C
h : y ∉ V
⊢ y ∈ E ∩ D ∪ V ∩ C

Next tactic:
exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        

Proof state:
case a
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
⊢ Set.Countable (E ∩ D ∪ V ∩ C)

Next tactic:
exact Countable.union h Vct
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        exact Countable.union h Vct
      

Proof state:
case intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
hU_cnt : Set.Countable (U ∩ C)
⊢ False

Next tactic:
have : U ∈ v := ⟨hUb, hU_cnt⟩
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        exact Countable.union h Vct
      have : U ∈ v := ⟨hUb, hU_cnt⟩
      

Proof state:
case intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
hU_cnt : Set.Countable (U ∩ C)
this : U ∈ v
⊢ False

Next tactic:
apply xD.2
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        exact Countable.union h Vct
      have : U ∈ v := ⟨hUb, hU_cnt⟩
      apply xD.2
      

Proof state:
case intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h : Set.Countable (E ∩ D)
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
hU_cnt : Set.Countable (U ∩ C)
this : U ∈ v
⊢ x ∈ V

Next tactic:
exact mem_biUnion this xU
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        exact Countable.union h Vct
      have : U ∈ v := ⟨hUb, hU_cnt⟩
      apply xD.2
      exact mem_biUnion this xU
    

Proof state:
case intro.intro.intro.refine'_2
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
this : ¬Set.Countable (E ∩ D)
⊢ ∃ y ∈ E ∩ D, y ≠ x

Next tactic:
by_contra! h
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        exact Countable.union h Vct
      have : U ∈ v := ⟨hUb, hU_cnt⟩
      apply xD.2
      exact mem_biUnion this xU
    by_contra! h
    

Proof state:
case intro.intro.intro.refine'_2
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
this : ¬Set.Countable (E ∩ D)
h : ∀ y ∈ E ∩ D, y = x
⊢ False

Next tactic:
exact absurd (Countable.mono h (Set.countable_singleton _)) this