Galois groups as quotients of Polish groups

Krzysztof Krupiński; Tomasz Rzepecki

doi:10.1142/s021906132050018x

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arXiv · 1804.09247

Galois groups as quotients of Polish groups

Krzysztof Krupiński·Tomasz Rzepecki

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Abstract

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_\sigma$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $\emptyset$. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Andand Pillay, which says that for any strong type defined on a single complete type over $\emptyset$, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the aforementioned paper about bounded quotients of type-definable subgroups of definable groups.

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BibTeX RIS

Krzysztof Krupiński, Tomasz Rzepecki. 2018-04-24. Galois groups as quotients of Polish groups. https://doi.org/10.1142/s021906132050018x

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