File size: 2,550 Bytes
110f4f3 | 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology
open MeasureTheory
open scoped ContDiff ENNReal
open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory
open scoped Topology ContDiff Manifold Bundle ENNReal
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology Manifold
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Bundle
open scoped Topology Bundle
open scoped Topology
open Function Manifold Set
open Manifold Bundle
open scoped Manifold Bundle
namespace WeakMTWTransport
variable {B F : Type*} [TopologicalSpace B] [T2Space B]
[TopologicalSpace F] [T2Space F] {E : B → Type*} [∀ x, TopologicalSpace (E x)]
[TopologicalSpace (TotalSpace F E)] [FiberBundle F E]
lemma bundle_totalSpace_t2 : T2Space (TotalSpace F E) := by
apply t2Space_iff_disjoint_nhds.mpr
intro x y hxy
by_cases hb : x.proj = y.proj
· let e := trivializationAt F E x.proj
have hx : x ∈ e.source := e.mem_source.mpr (mem_baseSet_trivializationAt F E x.proj)
have hy : y ∈ e.source := e.mem_source.mpr (hb ▸ mem_baseSet_trivializationAt F E x.proj)
have he : e x ≠ e y := fun h => hxy (e.toOpenPartialHomeomorph.injOn hx hy h)
exact (e.toOpenPartialHomeomorph.continuousAt hx).disjoint
(disjoint_nhds_nhds.mpr he) (e.toOpenPartialHomeomorph.continuousAt hy)
· exact (FiberBundle.continuous_proj F E).continuousAt.disjoint
(disjoint_nhds_nhds.mpr hb) (FiberBundle.continuous_proj F E).continuousAt
end WeakMTWTransport
end
end OAI
|