isOpen_iff_ωScottContinuous_mem — Mathlib

English

For functions into Prop under IsScott, ωScott-continuity is equivalent to ordinary continuity.

Русский

Для функций в Prop при IsScott ωScott-переносимость равна обычной непрерывности.

LaTeX

$$$\omegaScottContinuous f \iff \text{Continuous } f$ for f: α \to Prop under IsScott.$$

Lean4

@[deprecated isOpen_iff_continuous_mem (since := "2025-07-02")]
theorem isOpen_iff_ωScottContinuous_mem {α} [OmegaCompletePartialOrder α] {s : Set (Scott α)} :
    IsOpen s ↔ ωScottContinuous fun x ↦ x ∈ s := by rfl

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