instClosedIicTopology — Mathlib

English

There is a version of tendsto_nhds_iff_lt stating equivalence with checking lt (less-than) relation for all elements using the lower lattice structure.

Русский

Существуют версии tendsto_nhds_iff_lt, эквивалентные проверке отношения меньше чем (lt) во всей структуре нижней решетки.

LaTeX

$$$\\forall {f},\\forall l,\\forall x,\\text{Iff }(\\mathrm{Tendsto}(f,l,nhds(x))) \\left(\\forall y, \\text{lt}(x,y) \\to \\text{Eventually }(f(z) < y)\\right)$$$

Lean4

/-- Left-infinite right-closed intervals (-∞,a] are closed in the upper topology. -/
instance : ClosedIicTopology α :=
  ⟨fun a ↦ isOpen_compl_iff.1 <| isOpen_iff_generate_Iic_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩⟩

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