Point-cofinite covers in the Laver model

Miller, Arnold; Tsaban, Boaz

doi:10.1090/S0002-9939-10-10407-9

Point-cofinite covers in the Laver model
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by Arnold W. Miller and Boaz Tsaban PDF

Proc. Amer. Math. Soc. 138 (2010), 3313-3321 Request permission

Abstract:

Let $\mathsf {S}_1(\Gamma ,\Gamma )$ 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. $\mathfrak {b}$ is the minimal cardinality of a set of reals not satisfying $\mathsf {S}_1(\Gamma ,\Gamma )$. We prove the following assertions:

If there is an unbounded tower, then there are sets of reals of cardinality $\mathfrak {b}$ satisfying $\mathsf {S}_1(\Gamma ,\Gamma )$.
It is consistent that all sets of reals satisfying $\mathsf {S}_1(\Gamma ,\Gamma )$ have cardinality smaller than $\mathfrak {b}$.

These results can also be formulated as dealing with Arhangel’skiĭ’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 $\mathfrak {b}$ has an unbounded Borel image in the Baire space $\omega ^\omega$.

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