NTP2 topological structures

TL;DR

This paper investigates the properties of NTP$_2$ expansions of ordered structures, showing they preserve constructibility and have well-behaved topological and definability features, extending to various structures including $(R,<,+)$ and $(Q_p,+, imes)$.

Contribution

It proves that NTP$_2$ expansions by constructible sets maintain constructibility and exhibit generic piecewise continuity, also classifying strong expansions and analyzing NTP$_2$ d-minimal structures.

Findings

NTP$_2$ expansions of $(R,<,+)$ by constructible sets only define constructible sets.

Definable functions in these structures are generically piecewise continuous.

NTP$_2$ expansions of $(Q_p,+, imes)$ are similarly constrained.

Abstract

A subset of a topological space is constructible if it is a finite Boolean combination of closed sets. We prove that every NTP$_2$ expansion of $(\mathbb{R},<,+)$ by constructible sets defines only constructible sets, and that definable functions are generically piecewise continuous. The result also holds for all NTP$_2$ expansions of $(\mathbb{Q}_p,+,\cdot)$, and all NTP$_2$ definably complete expansions of ordered groups. In the latter case, the structure is generically locally o-minimal, has definable choice, and carries a well-behaved notion of naive topological dimension. For NIP uniform topological structures, constructibility of definable sets is preserved in the Shelah expansion. We classify strong expansions of $(\mathbb{R},<,+)$ by constructible sets, and obtain results on NTP$_2$ d-minimal structures.