Scott Topological Spaces

A type of topological spaces whose notion of continuity is equivalent to continuity in ωCPOs.

Reference

• https://ncatlab.org/nlab/show/Scott+topology

@[simp]

theorem Topology.IsScott.ωScottContinuous_iff_continuous {α : Type u_1} [OmegaCompletePartialOrder α] [TopologicalSpace α] [IsScott α (Set.range fun (c : OmegaCompletePartialOrder.Chain α) => Set.range ⇑c)] {f : α → Prop} : OmegaCompletePartialOrder.ωScottContinuous f ↔ Continuous f

def Scott.IsωSup {α : Type u} [Preorder α] (c : OmegaCompletePartialOrder.Chain α) (x : α) : Prop

x is an ω-Sup of a chain c if it is the least upper bound of the range of c.

Equations

• Scott.IsωSup c x = ((∀ (i : ℕ), c i ≤ x) ∧ ∀ (y : α), (∀ (i : ℕ), c i ≤ y) → x ≤ y)

theorem Scott.isωSup_iff_isLUB {α : Type u} [Preorder α] {c : OmegaCompletePartialOrder.Chain α} {x : α} : IsωSup c x ↔ IsLUB (Set.range ⇑c) x

def Scott.IsOpen (α : Type u) [OmegaCompletePartialOrder α] (s : Set α) : Prop

The characteristic function of open sets is monotone and preserves the limits of chains.

Equations

• Scott.IsOpen α s = OmegaCompletePartialOrder.ωScottContinuous fun (x : α) => x ∈ s

theorem Scott.isOpen_univ (α : Type u) [OmegaCompletePartialOrder α] : IsOpen α Set.univ

theorem Scott.IsOpen.inter (α : Type u) [OmegaCompletePartialOrder α] (s t : Set α) : IsOpen α s → IsOpen α t → IsOpen α (s ∩ t)

theorem Scott.isOpen_sUnion (α : Type u) [OmegaCompletePartialOrder α] (s : Set (Set α)) (hs : ∀ t ∈ s, IsOpen α t) : IsOpen α (⋃₀ s)

theorem Scott.IsOpen.isUpperSet (α : Type u) [OmegaCompletePartialOrder α] {s : Set α} (hs : IsOpen α s) : IsUpperSet s

theorem isωSup_ωSup {α : Type u_1} [OmegaCompletePartialOrder α] (c : OmegaCompletePartialOrder.Chain α) : Scott.IsωSup c (OmegaCompletePartialOrder.ωSup c)