Theorems · Theorem · manifolds

exists_contMDiffMap_zero_one_nhds_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] [NormalSpace M] [SigmaCompactSpace M]
  {s t : Set M},
  IsClosed s →
    IsClosed t →
      Disjoint s t →
        ∃ f, (∀ᶠ (x : M) in nhdsSet s, f x = 0) ∧ (∀ᶠ (x : M) in nhdsSet t, f x = 1) ∧ ∀ (x : M), f x ∈ Set.Icc 0 1

Given two disjoint closed sets s, t in a Hausdorff normal σ-compact finite-dimensional manifold M, there exists a smooth function f : M → [0,1] that vanishes in a neighbourhood of s and is equal to 1 in a neighbourhood of t.

Defined in

Mathlib.Geometry.Manifold.PartitionOfUnity

Cited by

1 results in Mathlib

Foundations

Depth 297 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceFiniteDimensionalIsManifoldT2SpaceNormalSpaceSigmaCompactSpace