Point-cofinite covers in the Laver model

TL;DR

This paper investigates the properties of point-cofinite covers in the Laver model, establishing conditions under which certain sets of reals satisfy specific selection principles and their relation to cardinal invariants.

Contribution

It proves the consistency of various cardinality conditions for sets satisfying S1(Gamma,Gamma) in the Laver model, linking them to unbounded towers and Borel images.

Findings

Existence of sets of reals of size b satisfying S1(Gamma,Gamma) if an unbounded tower exists.

In Laver's model, all sets satisfying S1(Gamma,Gamma) can have size smaller than b.

In Laver's model, sets of size b have unbounded Borel images in w^w.

Abstract

Let S1(Gamma,Gamma) be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. b is the minimal cardinality of a set of reals not satisfying S1(Gamma,Gamma). We prove the following assertions: (1) If there is an unbounded tower, then there are sets of reals of cardinality b, satisfying S1(Gamma,Gamma). (2) It is consistent that all sets of reals satisfying S1(Gamma,Gamma) have cardinality smaller than b. These results can also be formulated as dealing with Arhangel'skii's property alpha_2 for spaces of continuous real-valued functions. The main technical result is that in Laver's model, each set of reals of cardinality b has an unbounded Borel image in the Baire space w^w.