refine
 @TopologicalSpace.IsTopologicalBasis.mk ℝ tᵤ upper_intervals ?_ ?_ rfl
⊢ TopologicalSpace.IsTopologicalBasis upper_intervals
case refine_1

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

case refine_2

⊢ ⋃₀ upper_intervals = Set.univ