Open core and small groups in dense pairs of topological structures

Annals of Pure and Applied Logic 172 (1):102858 (2021)
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Abstract

Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.

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References found in this work

Expansions which introduce no new open sets.Gareth Boxall & Philipp Hieromyni - 2012 - Journal of Symbolic Logic 77 (1):111-121.
Weakly o-minimal nonvaluational structures.Roman Wencel - 2008 - Annals of Pure and Applied Logic 154 (3):139-162.
Stable theories with a new predicate.Enrique Casanovas & Martin Ziegler - 2001 - Journal of Symbolic Logic 66 (3):1127-1140.
Paires de structures o-minimales.Yerzhan Baisalov & Bruno Poizat - 1998 - Journal of Symbolic Logic 63 (2):570-578.

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