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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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
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