## Point-cofinite covers in the Laver model.(English)Zbl 1200.03038

Summary: Let $${S}_1(\Gamma ,\Gamma )$$ 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 $${S}_1(\Gamma ,\Gamma )$$. We prove the following assertions: 6mm
(1)
If there is an unbounded tower, then there are sets of reals of cardinality $$\mathfrak{b}$$ satisfying $${S}_1(\Gamma ,\Gamma )$$.
(2)
It is consistent that all sets of reals satisfying $${S}_1(\Gamma ,\Gamma )$$ have cardinality smaller than $$\mathfrak{b}$$.
These results can also be formulated as dealing with Arkhangel’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 $$\omega ^{\omega }$$.

### MSC:

 03E35 Consistency and independence results 03E17 Cardinal characteristics of the continuum 26A03 Foundations: limits and generalizations, elementary topology of the line
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### References:

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