upperClosure_interior_subset' — Mathlib

English

If s is an upper set, and x lies in its closure, then for ε > 0 there exists y such that a small ball around y sits inside a ball around x and inside the interior of s.

Русский

Если s верхнее множество, и x лежит в замыкании s, то при ε > 0 существует y с шарамиBall(y, ε) ⊆ Ball(x, δ) и Ball(y, ε) ⊆ interior(s).

LaTeX

$$$\exists y\; (\overline{Ball}(y, δ/4) \subseteq \overline{Ball}(x, δ)) \land (\overline{Ball}(y, δ/4) \subseteq interior(s))$$$

Lean4

@[to_additive upperClosure_interior_subset]
theorem upperClosure_interior_subset' (s : Set α) : (upperClosure (interior s) : Set α) ⊆ interior (upperClosure s) :=
  upperClosure_min (interior_mono subset_upperClosure) (upperClosure s).upper.interior

Похожие утверждения