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Mathlib/ModelTheory/Topology/Types.lean

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@@ -10,6 +10,8 @@ public import Mathlib.ModelTheory.Types
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public import Mathlib.Topology.Bases
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public import Mathlib.Topology.Connected.TotallyDisconnected
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public import Mathlib.Topology.Compactness.Compact
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public import Mathlib.Topology.Connected.Separation
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public import Mathlib.Topology.Baire.LocallyCompactRegular
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/-!
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# Topology on the space of complete types

Mathlib/ModelTheory/Types.lean

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/-
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Copyright (c) 2022 Aaron Anderson. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Aaron Anderson, Anish Rajeev
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Authors: Aaron Anderson
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-/
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module
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public import Mathlib.ModelTheory.Satisfiability
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public import Mathlib.Topology.Separation.Profinite
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public import Mathlib.Topology.Connected.Separation
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public import Mathlib.Topology.Baire.LocallyCompactRegular
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/-!
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# Type Spaces
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- `FirstOrder.Language.Theory.typeOf` is the type of a given tuple.
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- `FirstOrder.Language.Theory.realizedTypes`: `T.realizedTypes M α` is the set of
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types in `T.CompleteType α` that are realized in `M` - that is, the type of some tuple in `M`.
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- `FirstOrder.Language.Theory.TypeSpace.basis` is the basis of the of the stone space of types
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over `(n : ℕ)` variables
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## Main Results
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- `FirstOrder.Language.Theory.CompleteType.nonempty_iff`:
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The space `T.CompleteType α` is nonempty exactly when `T` is satisfiable.
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- `FirstOrder.Language.Theory.CompleteType.exists_modelType_is_realized_in`: Every type is realized
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in some model.
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- `FirstOrder.Language.Theory.TypeSpace.basis` describes a topology that is totally separated,
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compact, and a Baire space
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## Implementation Notes
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universe u v w w'
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open Cardinal Set FirstOrder TopologicalSpace
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open Cardinal Set FirstOrder
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namespace FirstOrder
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