Invariant sets in topology and logic

@article{Vaught1974InvariantSI,
  title={Invariant sets in topology and logic},
  author={Robert L. Vaught},
  journal={Fundamenta Mathematicae},
  year={1974},
  volume={82},
  pages={269-294}
}
  • R. Vaught
  • Published 1974
  • Mathematics
  • Fundamenta Mathematicae
Careful choices - a last word on Borel selectors
  • J. Burgess
  • Mathematics
    Notre Dame J. Formal Log.
  • 1981
Selector theory as surveyed in [13] and [14] deals with the following problem (instances of which arise in control theory, probability, mathematical economics, operator theory, etc.): We are given a
The effective Borel hierarchy
Let K be a subclass of Mod(L) which is closed under isomorphism. Vaught showed that K is Σα (respectively, Πα) in the Borel hierarchy iff K is axiomatized by an infinitary Σα (respectively, Πα)
Selectors for Borel sets with large sections
We prove a result asserting the existence of a Borel selector for a Borel set in the product of two Polish spaces. This subsumes a number of results about Borel selectors for Borel sets having large
An Application of Invariant Sets to Global Definability
TLDR
Vaught's "*-transform method" is applied to derive a global definability theorem of M. Makkai from a classical theorem of Lusin on countable-to-one continuous functions.
A Selector for Equivalence Relations with G δ Orbits
Assume A" is a Polish space and E is an open equivalence on X such that every equivalence class is a Gs set. We show that there is a Gs transversal for E. It follows that for any separable C*-algebra
A selector for equivalence relations with _{} orbits
Assume X is a Polish space and E is an open equivalence on X such that every equivalence class is a G6 set. We show that there is a G6 transversal for E. It follows that for any separable C*-algebra
On the measurability of orbits in Borel actions
We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure
The conjugacy problem for automorphism groups of countable homogeneous structures
TLDR
In each case, the precise complexity of the conjugacy relation in the sense of Borel reducibility of automorphism groups of a number of countable homogeneous structures is found.
A L\'opez-Escobar theorem for metric structures, and the topological Vaught conjecture
We show that a version of L\'opez-Escobar's theorem holds in the setting of logic for metric structures. More precisely, let $\mathbb{U}$ denote the Urysohn sphere and let
Notions of Relative Ubiquity for Invariant Sets of Relational Structures
TLDR
This work considers the collection of all L-structures on the set of natural numbers ω as a space as a compact metric space, and gives a notion of relative ubiquity, or largeness, for invariant sets of structures on ω.
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