On effective sigma-boundedness and sigma-compactness

TL;DR

This paper extends known theorems on sigma-boundedness and sigma-compactness of certain definable sets in descriptive set theory, including generalizations to higher levels and models, with implications for understanding their structure.

Contribution

It provides new generalizations of classical theorems on sigma-boundedness and sigma-compactness for lightface \\Sigma^1_1 and \\Sigma^1_2 sets, including in the Solovay model.

Findings

Generalized Kechris's theorem to \\Sigma^1_2 sets.

Extended Louveau's theorem to \\Sigma^1_1 sets.

Proved results in the Solovay model.

Abstract

We prove several theorems on sigma-bounded and sigma-compact pointsets. We start with a known theorem by Kechris, saying that any lightface \Sigma^1_1 set of the Baire space either is effectively sigma-bounded (that is, covered by a countable union of compact lightface \Delta^1_1 sets), or contains a superperfect subset (and then the set is not sigma-bounded, of course). We add different generalizations of this result, in particular, 1) such that the boundedness property involved includes covering by compact sets and equivalence classes of a given finite collection of lightface \Delta^1_1 equivalence relations, 2) generalizations to lightface \Sigma^1_2 sets, 3) generalizations true in the Solovay model. As for effective sigma-compactness, we start with a theorem by Louveau, saying that any lightface \Delta^1_1 set of the Baire space either is effectively sigma-compact (that is, is equal to a countable union of compact lightface \Delta^1_1 sets), or it contains a relatively closed superperfect subset. Then we prove a generalization of this result to lightface \Sigma^1_1 sets.