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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Point-cofinite covers in the Laver model


Authors: Arnold W. Miller and Boaz Tsaban
Journal: Proc. Amer. Math. Soc. 138 (2010), 3313-3321
MSC (2010): Primary 03E35, 26A03; Secondary 03E17
Published electronically: April 30, 2010
MathSciNet review: 2653961
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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:

  1. If there is an unbounded tower, then there are sets of reals of cardinality $ \mathfrak{b}$ satisfying $ \mathsf{S}_1(\Gamma,\Gamma)$.
  2. 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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Additional Information

Arnold W. Miller
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: miller@math.wisc.edu

Boaz Tsaban
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: tsaban@math.biu.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10407-9
PII: S 0002-9939(10)10407-9
Received by editor(s): October 21, 2009
Published electronically: April 30, 2010
Communicated by: Julia Knight
Article copyright: © Copyright 2010 American Mathematical Society