refine
  @TopologicalSpace.IsTopologicalBasis.mk ℝ tₛ open_intervals ?_ ?_ rfl
⊢ TopologicalSpace.IsTopologicalBasis open_intervals
case refine_1

⊢ ∀ t₁ ∈ open_intervals, ∀ t₂ ∈ open_intervals, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ open_intervals, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂

case refine_2

⊢ ⋃₀ open_intervals = Set.univ