isOpen_closedBall — Mathlib

English

For r ≠ 0, the closed ball is open.

Русский

Для r ≠ 0 замкнутый шар открыт.

LaTeX

$$r \neq 0 \Rightarrow IsOpen(\overline{\mathrm{ball}}(x,r))$$

Lean4

theorem isOpen_closedBall {r : ℝ} (hr : r ≠ 0) : IsOpen (closedBall x r) := by
  cases lt_or_gt_of_ne hr with
  | inl h => simp [closedBall_eq_empty.mpr h]
  | inr h =>
    rw [isOpen_iff]
    simp only [gt_iff_lt]
    intro y hy
    cases closedBall_eq_or_disjoint x y r with
    | inl hd =>
      use r
      simp [h, hd, ball_subset_closedBall]
    | inr hd => simp [closedBall_eq_of_mem hy, h.not_gt] at hd

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