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Groups, group actions and fields definable in first‐order topological structures
Author(s) -
Wencel Roman
Publication year - 2012
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201200033
Subject(s) - mathematics , dimension (graph theory) , order (exchange) , group (periodic table) , transitive relation , topology (electrical circuits) , group action , topological group , combinatorics , function (biology) , pure mathematics , physics , quantum mechanics , finance , evolutionary biology , economics , biology
Abstract Given a group ( G , ·), G ⊆ M m , 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 M m . 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.