isNowhereDense_iff — Mathlib

English

If a set s is nowhere dense, then its closure is nowhere dense.

Русский

Если множество s нигде не плотное, то его замыкание also нигде не плотное.

LaTeX

$$IsNowhereDense(s) \rightarrow IsNowhereDense(\overline{s})$$

Lean4

/-- A closed set is nowhere dense iff its interior is empty. -/
theorem isNowhereDense_iff {s : Set X} (hs : IsClosed s) : IsNowhereDense s ↔ interior s = ∅ := by
  rw [IsNowhereDense, IsClosed.closure_eq hs]

Похожие утверждения