We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the ℵ0-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.
The open dihypergraph dichotomy and the second level of the borel hierarchy
Carroy R.;
2020-01-01
Abstract
We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the ℵ0-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.
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