11/-
22Copyright (c) 2022 Aaron Anderson. All rights reserved.
33Released under Apache 2.0 license as described in the file LICENSE.
4- Authors: Aaron Anderson, Anish Rajeev
4+ Authors: Aaron Anderson
55-/
66module
77
88public import Mathlib.ModelTheory.Satisfiability
9- public import Mathlib.Topology.Separation.Profinite
10- public import Mathlib.Topology.Connected.Separation
11- public import Mathlib.Topology.Baire.LocallyCompactRegular
129
1310/-!
1411# Type Spaces
@@ -23,17 +20,13 @@ This file defines the space of complete types over a first-order theory.
2320- `FirstOrder.Language.Theory.typeOf` is the type of a given tuple.
2421- `FirstOrder.Language.Theory.realizedTypes`: `T.realizedTypes M α` is the set of
2522 types in `T.CompleteType α` that are realized in `M` - that is, the type of some tuple in `M`.
26- - `FirstOrder.Language.Theory.TypeSpace.basis` is the basis of the of the stone space of types
27- over `(n : ℕ)` variables
2823
2924 ## Main Results
3025
3126- `FirstOrder.Language.Theory.CompleteType.nonempty_iff`:
3227 The space `T.CompleteType α` is nonempty exactly when `T` is satisfiable.
3328- `FirstOrder.Language.Theory.CompleteType.exists_modelType_is_realized_in`: Every type is realized
3429 in some model.
35- - `FirstOrder.Language.Theory.TypeSpace.basis` describes a topology that is totally separated,
36- compact, and a Baire space
3730
3831 ## Implementation Notes
3932
@@ -51,7 +44,7 @@ This file defines the space of complete types over a first-order theory.
5144
5245universe u v w w'
5346
54- open Cardinal Set FirstOrder TopologicalSpace
47+ open Cardinal Set FirstOrder
5548
5649namespace FirstOrder
5750
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