TL;DR
This paper explores the topological and definability properties of phi-types in NIP theories, demonstrating that M-invariant phi-types are Borel definable and form a Rosenthal compactum, indicating tameness.
Contribution
It establishes new topological and definability results for M-invariant phi-types in NIP theories, linking model theory with classical topology.
Findings
M-invariant phi-types are Borel definable in countable models
The space of M-invariant phi-types is a Rosenthal compactum
Topological tameness properties follow from these results
Abstract
We apply the work of Bourgain, Fremlin and Talagrand on compact subsets of the first Baire class to show new results about phi-types for phi NIP. In particular, we show that if M is a countable model, then an M-invariant phi-type is Borel definable. Also the space of M-invariant phi-types is a Rosenthal compactum, which implies a number of topological tameness properties.
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