Theorems · Theorem · differential geometry

setOf_riemannianEDist_lt_subset_nhds

Deprecated since 2026-07-09Use setOfPred_riemannianEDist_lt_subset_nhds instead.

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_2} [inst_2 : TopologicalSpace H]
  (I : ModelWithCorners ℝ E H) {M : Type u_3} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
  [inst_5 : Bundle.RiemannianBundle fun x => TangentSpace I x] [inst_6 : IsManifold I 1 M]
  [IsContinuousRiemannianBundle E fun x => TangentSpace I x] [RegularSpace M] {x : M} {s : Set M},
  s ∈ nhds x → ∃ c > 0, {y | Manifold.riemannianEDist I x y < ↑c} ⊆ s

Alias of setOfPred_riemannianEDist_lt_subset_nhds. Any neighborhood of x contains all the points which are close enough to x for the Riemannian distance, ℝ≥0 version.

Defined in

Mathlib.Geometry.Manifold.Riemannian.Basic

Cited by

0 results in Mathlib

Foundations

Depth 282 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceBundle.RiemannianBundleIsManifoldIsContinuousRiemannianBundleRegularSpace