Tame Topology over Dp-minimal Structures

TL;DR

This paper develops a framework for tame topology over dp-minimal structures, ensuring definable sets behave well with notions of dimension, continuity, and decomposition, generalizing several minimal theories.

Contribution

It introduces a set of assumptions that enable tame topology in dp-minimal structures, extending results known for o-minimal, C-minimal, and P-minimal theories.

Findings

Definable sets have a well-defined notion of dimension.

Definable functions are almost everywhere continuous.

Definable sets can be expressed as finite unions of graphs of continuous multi-valued functions.

Abstract

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.