mathlib documentation

topology.omega_complete_partial_order

Scott Topological Spaces

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

Reference

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def Scott.is_ωSup {α : Type u} [preorder α] (c : omega_complete_partial_order.chain α) (x : α) :
Prop

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def Scott.is_open (α : Type u) [omega_complete_partial_order 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.is_open_univ (α : Type u) [omega_complete_partial_order α] :
Scott.is_open α set.univ

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theorem Scott.is_open_inter (α : Type u) [omega_complete_partial_order α] (s t : set α) (a : Scott.is_open α s) (a_1 : Scott.is_open α t) :
Scott.is_open α (s ∩ t)

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theorem Scott.is_open_sUnion (α : Type u) [omega_complete_partial_order α] (s : set (set α)) (a : ∀ (t : set α), t ∈ s → Scott.is_open α t) :
Scott.is_open α (⋃₀s)

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def 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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@[instance]
def Scott.topological_space (α : Type u) [omega_complete_partial_order α] :
topological_space (Scott α)

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def not_below {α : Type u_1} [omega_complete_partial_order α] (y : Scott α) :
set (Scott α)

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

Equations
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theorem not_below_is_open {α : Type u_1} [omega_complete_partial_order α] (y : Scott α) :
is_open (not_below y)

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theorem is_ωSup_ωSup {α : Type u_1} [omega_complete_partial_order α] (c : omega_complete_partial_order.chain α) :
Scott.is_ωSup c (omega_complete_partial_order.ωSup c)

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theorem Scott_continuous_of_continuous {α : Type u_1} {β : Type u_2} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : Scott α → Scott β) (hf : continuous f) :
omega_complete_partial_order.continuous' f

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theorem continuous_of_Scott_continuous {α : Type u_1} {β : Type u_2} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : Scott α → Scott β) (hf : omega_complete_partial_order.continuous' f) :
continuous f