Theorems · Theorem · general topology

IsPathConnected.exists_path_through_family

∀ {X : Type u_1} [inst : TopologicalSpace X] {n : ℕ} {s : Set X},
  IsPathConnected s →
    ∀ (p : Fin (n + 1) → X),
      (∀ (i : Fin (n + 1)), p i ∈ s) → ∃ γ, Set.range ⇑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ Set.range ⇑γ

Defined in: Mathlib.Topology.Connected.PathConnected
Cited by: 2 results in Mathlib
Foundations: Depth 129 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace

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Cites22

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
Realstatement · cited by 25,697
TopologicalSpacestatement and proof · cited by 24,529
Set.Elemstatement and proof · cited by 7,166
Set.rangestatement and proof · cited by 4,705
unitIntervalstatement and proof · cited by 607
Pathstatement and proof · cited by 318
Fin.snocproof · cited by 113
Set.range_subset_iffproof · cited by 99
IsPathConnectedstatement and proof · cited by 65
Path.transproof · cited by 54

Cited by2

Results whose statement or proof uses this declaration.

IsPathConnected.exists_path_through_family'proof · cited by 1
PathConnectedSpace.exists_path_through_familyproof · cited by 0