Theorems · Theorem · manifolds

exists_contMDiffMap_zero_one_of_isClosed

∀ {E : Type uE} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type uH} [inst_2 : TopologicalSpace H]
  (I : ModelWithCorners ℝ E H) {M : Type uM} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
  [FiniteDimensional ℝ E] [IsManifold I (↑⊤) M] {n : ℕ∞} [T2Space M] [SigmaCompactSpace M] {s t : Set M},
  IsClosed s → IsClosed t → Disjoint s t → ∃ f, Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : M), f x ∈ Set.Icc 0 1

Given two disjoint closed sets s, t in a Hausdorff σ-compact finite-dimensional manifold, there exists an infinitely smooth function that is equal to 0 on s and to 1 on t. See also exists_contMDiff_zero_iff_one_iff_of_isClosed, which ensures additionally that f is equal to 0 exactly on s and to 1 exactly on t.

Defined in

Mathlib.Geometry.Manifold.PartitionOfUnity

Cited by

2 results in Mathlib

Foundations

Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceFiniteDimensionalIsManifoldT2SpaceSigmaCompactSpace