Groups, group actions and fields definable in first‐order topological structures

Abstract

AbstractGiven a group (G, ·), G⊆Mm, definable in a first‐order structure \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}=(M,\ldots )$\end{document} equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V⊆G and define a new topology τ on G with which (G, ·) becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from Mm. Likewise we topologize transitive group actions and fields definable in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}$\end{document}. These results require a series of preparatory facts concerning dimension functions, some of which might be of independent interest.