eball_contractible — Mathlib

English

An emetric ball with positive radius is path-connected.

Русский

EMetric-окружность положительного радиуса является path-connected.

LaTeX

$$$\\text{IsPathConnected} (\\text{EMetric.ball } x r)$ with $r>0$$$

Lean4

theorem eball_contractible {x : E} {r : ENNReal} (hr : 0 < r) : ContractibleSpace (EMetric.ball x r) := by
  cases r with
  | top =>
    rw [eball_top_eq_univ, (Homeomorph.Set.univ E).contractibleSpace_iff]
    exact RealTopologicalVectorSpace.contractibleSpace
  | coe r =>
    rw [emetric_ball_nnreal]
    apply ball_contractible
    simpa using hr

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