Theorem: preimage_connectedComponent_connected
File: Mathlib/Topology/Connected/Clopen.lean

Proof steps:
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
⊢ IsConnected (f ⁻¹' connectedComponent t)
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
⊢ IsConnected (f ⁻¹' connectedComponent t)
tactic: have hf : Surjective f := Surjective.of_comp fun t : β => (connected_fibers t).1
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
⊢ IsConnected (f ⁻¹' connectedComponent t)
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
⊢ IsPreconnected (f ⁻¹' connectedComponent t)
tactic: refine ⟨Nonempty.preimage connectedComponent_nonempty hf, ?_⟩
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
⊢ IsPreconnected (f ⁻¹' connectedComponent t)
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
⊢ IsPreconnected (f ⁻¹' connectedComponent t)
tactic: have hT : IsClosed (f ⁻¹' connectedComponent t) :=
    (hcl (connectedComponent t)).1 isClosed_connectedComponent
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
⊢ IsPreconnected (f ⁻¹' connectedComponent t)
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
⊢ ∀ (u v : Set α),
    IsClosed u →
      IsClosed v →
        f ⁻¹' connectedComponent t ⊆ u ∪ v →
          Disjoint u v → f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: rw [isPreconnected_iff_subset_of_fully_disjoint_closed hT]
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u v : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
⊢ ∀ (u v : Set α),
    IsClosed u →
      IsClosed v →
        f ⁻¹' connectedComponent t ⊆ u ∪ v →
          Disjoint u v → f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: intro u v hu hv huv uv_disj
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: let T₁ := { t' ∈ connectedComponent t | f ⁻¹' {t'} ⊆ u }
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: let T₂ := { t' ∈ connectedComponent t | f ⁻¹' {t'} ⊆ v }
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: have hT₁ : IsClosed T₁ := (hcl T₁).2 (T₁_u.symm ▸ IsClosed.inter hT hu)
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
hT₂ : IsClosed T₂
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: have hT₂ : IsClosed T₂ := (hcl T₂).2 (T₂_v.symm ▸ IsClosed.inter hT hv)
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
hT₂ : IsClosed T₂
T_decomp : connectedComponent t ⊆ T₁ ∪ T₂
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
hT₂ : IsClosed T₂
T_decomp : connectedComponent t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: have T_disjoint : Disjoint T₁ T₂ := by
    refine Disjoint.of_preimage hf ?_
    rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq,
      inter_empty]
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
hT₂ : IsClosed T₂
T_decomp : connectedComponent t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
state_after: case inl
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
hT₂ : IsClosed T₂
T_decomp : connectedComponent t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : connectedComponent t ⊆ T₁
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v

case inr
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v
hT₁ : IsClosed T₁
hT₂ : IsClosed T₂
T_decomp : connectedComponent t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : connectedComponent t ⊆ T₂
⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v
tactic: cases' (isPreconnected_iff_subset_of_fully_disjoint_closed isClosed_connectedComponent).1
    isPreconnected_connectedComponent T₁ T₂ hT₁ hT₂ T_decomp T_disjoint with h h