  In this paper we prove that if there is a Borel-cover gamma-set of
cardinality the continuum, then there is one which is not hereditary. A set of
reals X is a Borel-cover gamma-set iff for every countable family of Borel sets
which is an omega-cover contains a gamma-cover. This is also denoted
S_1(Borel_omega, Borel_gamma). This result partially answers a question of
Bartoszynski and Tsaban. Tsaban points out that it also gives an example of set
which is both a gamma-set and sigma-set but is not hereditarily gamma, which
answers a question of Bukovsky, Reclaw, and Repicky.
