upperSemicontinuous_iff_isClosed_preimage

English

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

Русский

Для открытого s 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_isClosed_preimage {f : α → γ} : UpperSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Ici y) :=
  by
  rw [upperSemicontinuous_iff_isOpen_preimage]
  simp only [← isOpen_compl_iff, ← preimage_compl, compl_Ici]

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