isClopen_sphere — Mathlib

English

For r ≠ 0, the sphere { x : R : v(x) = r } is clopen.

Русский

При r ≠ 0 сфера { x : R : v(x) = r } является клозетной.

LaTeX

$$$\{x : R : v(x) = r\}$ isClopen$$

Lean4

/-- A sphere centred at the origin in a valued ring is clopen. -/
theorem isClopen_sphere {r : Γ₀} (hr : r ≠ 0) : IsClopen (X := R) {x | v x = r} :=
  by
  have h : {x : R | v x = r} = {x | v x ≤ r} \ {x | v x < r} :=
    by
    ext x
    simp [← le_antisymm_iff]
  rw [h]
  exact IsClopen.diff (isClopen_closedBall _ hr) (isClopen_ball _ _)

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