We show that dp-minimal valued fields are henselian and that a dp-minimal field admitting a definable type V topology is either real closed, algebraically closed or admits a non-trivial definable henselian valuation. We give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure.Partially supported by ValCoMo (ANR-13-BS01-0006).
In this paper we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous "multi-valued functions". This generalizes known statements about weakly o-minimal, C-minimal and P-minimal theories.
We define the interpolative fusion [Formula: see text] of a family [Formula: see text] of first-order theories over a common reduct [Formula: see text], a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each [Formula: see text] is model-complete, [Formula: see text] coincides with the model companion of [Formula: see text]. By obtaining sufficient conditions for the existence of [Formula: see text], we develop new tools to show that theories of interest have model companions.
A first-order expansion of the R-vector space structure on R does not define every compact subset of every R n if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A ⊆ R k is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every R n can be constructed from A using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing dimension.Date: April 13, 2018. 2010 Mathematics Subject Classification. Primary 03C64 Secondary 03C45, 03E15, 28A05, 28A75, 28A80, 54F45.
For an arbitrary field K and K-variety V , we introduce the étale-open topology on the set V (K) of K-points of V . This topology agrees with the Zariski topology, Euclidean topology, and valuation topology when K is separably closed, real closed, p-adically closed, respectively. The étale open topology on A 1 (K) is not discrete if and only if K is large. Topological properties of the étale open topology corresponds to algebraic properties of K.As an application, we show that a large stable field is separably closed.
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