Theorems · Theorem · algebraic topology

VectorPrebundle.toVectorBundle

∀ {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [inst : NontriviallyNormedField R]
  [inst_1 : (x : B) → AddCommMonoid (E x)] [inst_2 : (x : B) → Module R (E x)] [inst_3 : NormedAddCommGroup F]
  [inst_4 : NormedSpace R F] [inst_5 : TopologicalSpace B] [inst_6 : (x : B) → TopologicalSpace (E x)]
  (a : VectorPrebundle R F E), VectorBundle R F E

Make a VectorBundle from a VectorPrebundle. Concretely this means that, given a VectorPrebundle structure for a sigma-type E -- which consists of a number of "pretrivializations" identifying parts of E with product spaces U × F -- one establishes that for the topology constructed on the sigma-type using VectorPrebundle.totalSpaceTopology, these "pretrivializations" are actually "trivializations" (i.e., homeomorphisms with respect to the constructed topology).

Defined in

Mathlib.Topology.VectorBundle.Basic

Cited by

1 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommMonoidModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpace