Theorems · Theorem · general topology
PartitionOfUnity.exists_isSubordinate
∀ {ι : Type u} {X : Type v} [inst : TopologicalSpace X] {s : Set X} [NormalSpace X] [ParacompactSpace X],
IsClosed s → ∀ (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ f, f.IsSubordinate UIf X is a paracompact normal topological space and U is an open covering of a closed set
s, then there exists a PartitionOfUnity ι X s that is subordinate to U.
- Defined in
- Mathlib.Topology.PartitionOfUnity
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.iUnionstatement and proof · cited by 2,483
- IsOpenstatement and proof · cited by 2,400
- IsClosedstatement and proof · cited by 1,639
- NormalSpacestatement and proof · cited by 84
- PartitionOfUnitystatement and proof · cited by 47
- BumpCoveringproof · cited by 46
- BumpCovering.IsSubordinateproof · cited by 15
- BumpCovering.toPartitionOfUnityproof · cited by 14
- ParacompactSpacestatement and proof · cited by 11
- PartitionOfUnity.IsSubordinatestatement and proof · cited by 10
Cited by1
Results whose statement or proof uses this declaration.
- exists_continuous_forall_mem_convex_of_localproof · cited by 1