Dimension and topology in transserial tame pairs

TL;DR

This paper introduces a dimension concept in transserial tame pairs, showing it aligns with the order topology dimension, and explores their model-theoretic and topological properties, including local o-minimality and a definable Baire category theorem.

Contribution

It defines a unique, definable dimension in transserial tame pairs and demonstrates their local o-minimality and d-minimality, advancing understanding of their topological and model-theoretic structure.

Findings

Dimension equals the order topology dimension.

Transserial tame pairs are locally o-minimal and d-minimal.

A definable Baire category theorem is established.

Abstract

Every maximal Hardy field has a proper elementary differential subfield that is Dedekind complete in the maximal Hardy field. This pair of Hardy fields is a transserial tame pair, shown to have a complete and model complete elementary theory in arXiv:2408.07033. This paper introduces a dimension in transserial tame pairs and shows that it equals the (naive) dimension coming from the order topology. In particular, the dimension is definable in a certain sense, and is the unique definable dimension in a transserial tame pair. Further, transserial tame pairs are locally o-minimal and d-minimal. These properties and the dimension are used to establish topological properties of definable sets in transserial tame pairs, including a definable Baire category theorem.