upperSemicontinuous_iff_isOpen_preimage

English

If s is open and y ≤ 0, then χ_s(y) is upper semicontinuous on t.

Русский

Если s открыто и y ≤ 0, χ_s(y) верхнеполупрерывна на t.

LaTeX

$$$\\mathrm{IsOpen}(s) \\land y \\le 0 \\Rightarrow \\mathrm{UpperSemicontinuousOn}(\\mathrm{indicator}(s, \\lambda x. y), t)$$$

Lean4

theorem upperSemicontinuous_iff_isOpen_preimage : UpperSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Iio y) :=
  ⟨fun H y => isOpen_iff_mem_nhds.2 fun x hx => H x y hx, fun H _x y y_lt => IsOpen.mem_nhds (H y) y_lt⟩

Похожие утверждения