A Nonhereditary Borel-cover γ-set

In this paper we prove that if there is a Borel-cover γ-set of car-dinality the continuum, then there is one which is not hereditary. In this paper we answer some of the questions raised by Bartoszyński and Tsaban [1] concerning hereditary properties of sets defined by certain Borel covering properties. Define. An ω-cover of a set X is a family of sets such that every finite subset of X is included in an element of the cover but X itself is not in the family. Define. A γ-cover of a set X is an infinite family of sets such that every element of X is in all but finitely many elements of the family. Define. A set X is called a Borel-cover γ-set iff every countable ω-cover of X by Borel sets contains a γ-cover. These concepts were introduced by Gerlits and Nagy [5] for open covers. Being a Borel-cover γ-set is equivalent to saying that for any ω-sequence of countable Borel ω-covers of X we can choose one element from each and get a γ-cover of X – this is denoted S1(BΩ,BΓ). The equivalence was proved by Gerlitz and Nagy [5] for open covers but the proof works also for Borel covers as was noted in Scheepers and Tsaban [8]: Let Bn be Borel ω-covers of X. Since {U ∩ V: U ∈ U, V ∈ V} is an ω-cover if U and V are, we may assume that Bn+1 refines Bn. Let xn for n < ω be distinct elements of X and let B = {A \ {xn} : n < ω,A ∈ Bn} It is easy to check that B is an ω-cover of X. Now let C be a γ-subcover of B. Note that for any fixed n at most finitely many of the elements of C ca