Theorems · Theorem · global analysis
IsOpen.exists_contDiff_support_eq
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {n : ℕ∞} {s : Set E},
IsOpen s → ∃ f, Function.support f = s ∧ ContDiff ℝ (↑n) f ∧ Set.range f ⊆ Set.Icc 0 1Given an open set s in a finite-dimensional real normed vector space, there exists a smooth
function with values in [0, 1] whose support is exactly s.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites102
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
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Top.topproof · cited by 9,680
- Norm.normproof · cited by 5,413
- ENatstatement and proof · cited by 4,985
- Set.rangestatement and proof · cited by 4,705
- NNRealproof · cited by 4,310
- mul_oneproof · cited by 3,885
- LE.le.transproof · cited by 3,151
- Filter.Eventuallyproof · cited by 3,134
Cited by2
Results whose statement or proof uses this declaration.
- ExistsContDiffBumpBase.u_existsproof · cited by 5
- IsOpen.exists_contMDiff_support_eq_auxproof · cited by 1