On effective sigma-boundedness and sigma-compactness

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.Comment: arXiv admin note: substantial text overlap with arXiv:1103.106

Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy

We present a model of set theory, in which, for a given $n\ge2$, there exists a non-ROD-uniformizable planar lightface $\varPi^1_n$ set in $\mathbb R\times\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $\bf\Sigma^1_n$ sets with countable cross-sections are $\bf\Delta^1_{n+1}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level.Comment: A revised version of the originally submitted preprin