case refine'_1
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ ∀ (t₁ : Set α),
    t₁ ∈ s \ {∅} → ∀ (t₂ : Set α), t₂ ∈ s \ {∅} → ∀ (x : α), x ∈ t₁ ∩ t₂ → ∃ t₃, t₃ ∈ s \ {∅} ∧ x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂

case refine'_2
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ t = generateFrom (s \ {∅})
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ IsTopologicalBasis (s \ {∅})
refine' ⟨_, by rw [sUnion_diff_singleton_empty, h.sUnion_eq], _⟩
no goals
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ ⋃₀ (s \ {∅}) = univ
rw [sUnion_diff_singleton_empty, h.sUnion_eq]
case refine'_1.intro.intro
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
⊢ ∃ t₃, t₃ ∈ s \ {∅} ∧ x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
case refine'_1
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ ∀ (t₁ : Set α),
    t₁ ∈ s \ {∅} → ∀ (t₂ : Set α), t₂ ∈ s \ {∅} → ∀ (x : α), x ∈ t₁ ∩ t₂ → ∃ t₃, t₃ ∈ s \ {∅} ∧ x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
rintro t₁ ⟨h₁, -⟩ t₂ ⟨h₂, -⟩ x hx
case refine'_1.intro.intro.intro.intro
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
t₃ : Set α
h₃ : t₃ ∈ s
hs : x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
⊢ ∃ t₃, t₃ ∈ s \ {∅} ∧ x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
case refine'_1.intro.intro
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
⊢ ∃ t₃, t₃ ∈ s \ {∅} ∧ x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x hx
no goals
case refine'_1.intro.intro.intro.intro
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t₁ : Set α
h₁ : t₁ ∈ s
t₂ : Set α
h₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
t₃ : Set α
h₃ : t₃ ∈ s
hs : x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
⊢ ∃ t₃, t₃ ∈ s \ {∅} ∧ x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
exact ⟨t₃, ⟨h₃, Nonempty.ne_empty ⟨x, hs.1⟩⟩, hs⟩
case refine'_2
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ generateFrom s = generateFrom (s \ {∅})
case refine'_2
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ t = generateFrom (s \ {∅})
rw [h.eq_generateFrom]
case refine'_2
α : Type u
t✝ : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
⊢ IsOpen t
case refine'_2
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ generateFrom s = generateFrom (s \ {∅})
refine' le_antisymm (generateFrom_anti <| diff_subset s _) (le_generateFrom fun t ht => _)
case refine'_2.inl
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
ht : ∅ ∈ s
⊢ IsOpen ∅

case refine'_2.inr
α : Type u
t✝ : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
he : t ≠ ∅
⊢ IsOpen t
case refine'_2
α : Type u
t✝ : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
⊢ IsOpen t
obtain rfl | he := eq_or_ne t ∅
no goals
case refine'_2.inl
α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
ht : ∅ ∈ s
⊢ IsOpen ∅
exact @isOpen_empty _ (generateFrom _)
no goals
case refine'_2.inr
α : Type u
t✝ : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
t : Set α
ht : t ∈ s
he : t ≠ ∅
⊢ IsOpen t
exact .basic t ⟨ht, he⟩