Tame Topology over dp-Minimal Structures

Abstract

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.

Defining integer-valued functions in rings of continuous definable functions over a topological field

“…Remark 2.4 These properties hold true in every dp-minimal expansion of a field which is not strongly minimal ( [SW18]). This implies and generalises known results on o-minimal, C-minimal and P -minimal fields (see [vdD98], [HM94], [HM97] and [CKDL17]).…”

“…However it is this slightly stronger statement with W d (X) which we need in Section 6. It appears in Proposition 4.6 of[SW18].…”

Definable groups in topological fields with a generic derivation

“…Finally, one has the following description of definable subsets of topological fields models of an L-open theory [2]; it is the analogue of the cell decomposition proven for dp-minimal fields (see [15,Proposition 4.1]). Before stating the result, we need to recall the notion of correspondences.…”

Strong theories of ordered Abelian groups

“…Other related recent work includes the study of dp-and inp-minimal ordered structures in [13] and [25] and the existence of Abelian, solvable and nilpotent definable envelopes of groups definable in NTP 2 theories by Hempel and Onshuus [16]. The new work by Simon and Walsberg on dp-minimal structures with tame topology in [26] also explores similar ideas, and it would be interesting to see how some of their results might be generalizable to the strong or finite dp-rank context.…”