  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.

General Topology;Logic