Submission date: 28 July 2007. Revised version: 9 February 2008.

Abstract:

We introduce a notion of Polish structure and, in doing so, provide a setting which allows the application of ideas and techniques from model theory, descriptive set theory, topology and the theory of profinite groups. We define a topological notion of independence in Polish structures and prove that it has some nice properties. Using this notion, we prove counterparts of some basic results from geometric stability theory in the context of small Polish structures. Then we prove some structural theorems about compact groups regarded as Polish structures: each small, $nm$-stable compact $G$-group is solvable-by-finite; each small compact $G$-group of finite ${\mathcal NM}$-rank is nilpotent-by-finite. Examples of small Polish structures and groups are also given.

Mathematics Subject Classification: 03C45, 03E15, 54H11, 54F15

Keywords and phrases: Polish group, independence relation

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