ScottContinuous — Mathlib

English

The join operation is Scott-continuous: (b1,b2) ↦ b1 ⊔ b2 is Scott-continuous.

Русский

Операция объединения (join) в полной полевой решётке Scott–непрерывна: (b1,b2) ↦ b1 ⊔ b2.

LaTeX

$$$\\mathrm{ScottContinuous}\\left((b_1,b_2) \\mapsto b_1 \\sqcup b_2\\right).$$$

Lean4

/-- A function between preorders is said to be Scott continuous if it preserves `IsLUB` on directed
sets. It can be shown that a function is Scott continuous if and only if it is continuous w.r.t. the
Scott topology.
-/
def ScottContinuous (f : α → β) : Prop :=
  ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → IsLUB (f '' d) (f a)

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