Expansions which introduce no new open sets

TL;DR

This paper investigates conditions under which expanding a topological structure does not introduce new open sets, relating to o-minimal open core, and provides practical criteria and characterizations, especially for expansions by a generic predicate.

Contribution

It offers a general framework and easy-to-check conditions for when expansions preserve open sets, extending previous work on o-minimal open core and generic predicates.

Findings

Characterization of expansions with no new open sets

Conditions for expansions to preserve open sets in a general setting

Special case analysis for expansions by a generic predicate

Abstract

We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.