Tame topology over dp-minimal structures

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. This paper is a contribution to the study of generalizations and variations of o-minimality. O-minimality is a model-theoretic notion of tame geometry. Over an o-minimal structure definable functions are piecewise continuous and there is a well-behaved notion of dimension for definable sets. Conditions similar to o-minimality have been investigated, such as weak o-minimality and C-minimality, which imply analogous—though weaker—tameness properties. More recently, it was observed in the ordered case that a purely combinatorial condition, dp-minimality, is enough to imply such properties. The theory of dp-minimal ordered structures can be seen as a generalization of the theory of weakly o-minimal structures, see [Goo10] and [Sim11]. The present paper continues this line of work as our results hold over dp-minimal expansions of divisible ordered abelian groups. We work in a framework which includes both dp-minimal expansions of divisible ordered abelian groups and dp-mnimal expansions of valued fields. We work with a dpminimal structure M equipped with a definable uniform structure. We assume that the M does not have any isolated points and that every infinite definable subset of M has nonempty interior. It follows from work of Simon [Sim11] that these assumptions hold for dp-minimal expansions of divisible ordered abelian groups. It follows from the work of Johnson [Joh15] that our assumptions hold for any dp-minimal field which is not strongly minimal, in particular for any dp-minimal expansion of a valued field. Our main results are as follows: (1) Naive topological dimension, acl-dimension and dp-rank all agree on definable sets and are definable in families. (2) The dimension of the frontier of a definable set is strictly less then the dimension of the set. (3) A definable function is continuous outside of a set of smaller dimension. (4) Definable sets are finite unions of graphs of continuous definable correspondences U ⇒ Ml, U ⊆ Mk an open set. A correspondence is a continuous “multi-valued function”, this is made precise below. The last bullet is as close as we can get to cell decomposition. Note that we do not say anything about definable open sets. Cubides-Kovacsics, Darnière and Leenknegt [CKDL15] recently showed that (2)-(4) above hold for P-minimal expansions of fields. Dolich, Goodrick Date: September 28, 2015. Partially supported by ValCoMo (ANR-13-BS01-0006).