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A López-Escobar theorem for metric structures, and the topological Vaught conjecture

Samuel CoskeyMartino Lupini — 2016

Fundamenta Mathematicae

We show that a version of López-Escobar’s theorem holds in the setting of model theory for metric structures. More precisely, let denote the Urysohn sphere and let Mod(,) be the space of metric -structures supported on . Then for any Iso()-invariant Borel function f: Mod(,) → [0,1], there exists a sentence ϕ of ω ω such that for all M ∈ Mod(,) we have f ( M ) = ϕ M . This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group...

Generalized Choquet spaces

Samuel CoskeyPhilipp Schlicht — 2016

Fundamenta Mathematicae

We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that κ < κ = κ . Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly 2 κ many such spaces up to homeomorphism. We also establish a Kuratowski-like...

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

Samuel CoskeyTamás MátraiJuris Steprāns — 2013

Fundamenta Mathematicae

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality 𝔭 ≤ 𝔟 does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting...

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