Documentation

Mathlib.Topology.OmegaCompletePartialOrder

Scott Topological Spaces #

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

Reference #

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@[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
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@[deprecated Topology.IsScott.ωScottContinuous_iff_continuous (since := "2025-07-02")]
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

Alias of Topology.IsScott.ωScottContinuous_iff_continuous.

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

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      theorem Scott.isOpen_univ (α : Type u) [OmegaCompletePartialOrder α] :
      IsOpen α Set.univ
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      theorem Scott.IsOpen.inter (α : Type u) [OmegaCompletePartialOrder α] (s t : Set α) :
      IsOpen α s → IsOpen α t → IsOpen α (s ∩ t)
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      theorem Scott.isOpen_sUnion (α : Type u) [OmegaCompletePartialOrder α] (s : Set (Set α)) (hs : ∀ t ∈ s, IsOpen α t) :
      IsOpen α (⋃₀ s)
      source
      theorem Scott.IsOpen.isUpperSet (α : Type u) [OmegaCompletePartialOrder α] {s : Set α} (hs : IsOpen α s) :
      IsUpperSet s
      source
      @[reducible, inline, deprecated Topology.WithScott (since := "2025-07-02")]
      abbrev Scott (α : Type u) :
      Type u

      A Scott topological space is defined on preorders such that their open sets, seen as a function α → Prop, preserves the joins of ω-chains.

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        @[reducible, inline, deprecated Topology.WithScott.instTopologicalSpace (since := "2025-07-02")]
        abbrev Scott.topologicalSpace (α : Type u) [OmegaCompletePartialOrder α] :
        TopologicalSpace (Scott α)

        Deprecated, use WithScott.

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          @[deprecated isOpen_iff_continuous_mem (since := "2025-07-02")]
          theorem isOpen_iff_ωScottContinuous_mem {α : Type u_1} [OmegaCompletePartialOrder α] {s : Set (Scott α)} :
          IsOpen s ↔ OmegaCompletePartialOrder.ωScottContinuous fun (x : Scott α) => x ∈ s
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          @[deprecated "Use `WithScott` API" (since := "2025-07-02")]
          theorem scott_eq_Scott {α : Type u_1} [OmegaCompletePartialOrder α] :
          Topology.scott α (Set.range fun (c : OmegaCompletePartialOrder.Chain α) => Set.range ⇑c) = Scott.topologicalSpace α
          source
          def notBelow {α : Type u_1} [OmegaCompletePartialOrder α] (y : Scott α) :
          Set (Scott α)

          notBelow is an open set in Scott α used to prove the monotonicity of continuous functions

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            @[deprecated isClosed_Iic (since := "2025-07-02")]
            theorem notBelow_isOpen {α : Type u_1} [OmegaCompletePartialOrder α] (y : Scott α) :
            IsOpen (notBelow y)
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            theorem isωSup_ωSup {α : Type u_1} [OmegaCompletePartialOrder α] (c : OmegaCompletePartialOrder.Chain α) :
            Scott.IsωSup c (OmegaCompletePartialOrder.ωSup c)
            source
            @[deprecated Topology.IsScott.ωscottContinuous_iff_continuous (since := "2025-07-02")]
            theorem scottContinuous_of_continuous {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : Scott α → Scott β) (hf : Continuous f) :
            OmegaCompletePartialOrder.ωScottContinuous f
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            @[deprecated Topology.IsScott.ωscottContinuous_iff_continuous (since := "2025-07-02")]
            theorem continuous_of_scottContinuous {α : Type u_1} {β : Type u_2} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β] (f : Scott α → Scott β) (hf : OmegaCompletePartialOrder.ωScottContinuous f) :
            Continuous f