openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /DenseWeakStrongContinuityPoints.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.SymmetricCorrection | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace ScalarConductivity | |
| open Filter Topology Set TopologicalSpace | |
| theorem dense_weakToStrong_continuityPoints | |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| [SeparableSpace V] (Z : Set (WeakSpace ℝ V)) (hZ : IsCompact Z) : | |
| Dense {z : Z | ContinuousAt | |
| (fun w : Z => (toWeakSpace ℝ V).symm w.val) z} := by | |
| have : CompactSpace Z := isCompact_iff_compactSpace.mp hZ | |
| apply dense_continuityPoints_of_closed_ball_preimages | |
| intro y r | |
| have hclosed : IsClosed ((toWeakSpace ℝ V) '' Metric.closedBall y r) := by | |
| apply closure_eq_iff_isClosed.mp | |
| rw [← (convex_closedBall y r).toWeakSpace_closure ℝ, | |
| Metric.isClosed_closedBall.closure_eq] | |
| convert hclosed.preimage continuous_subtype_val using 1 | |
| ext z | |
| simp only [Set.mem_preimage, Set.mem_image] | |
| constructor | |
| · intro hz | |
| exact ⟨(toWeakSpace ℝ V).symm z.val, hz, (toWeakSpace ℝ V).apply_symm_apply _⟩ | |
| · rintro ⟨x, hx, heq⟩ | |
| simpa only [← heq, LinearEquiv.symm_apply_apply] using hx | |
| end ScalarConductivity | |
| end | |
| end | |
| end OAI | |