| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
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| 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] |
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| 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 |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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