The cardinal characteristic for relative γ-sets

For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γset in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p(2) and p(ω) in terms of definable free filters on ω which is related to the pseudo-intersection number p. We show that for every uncountable standard analytic space X that either p(X) = p(2) or p(X) = p(ω). We show that the following statements are each relatively consistent with ZFC: (a) p = p(ω) < p(2) and (b) p < p(ω) = p(2) First we remind the reader of the definition of a γ-set. An open cover U of a topological space X is an ω-cover iff for every finite F ⊆ X there exists U ∈ U with F ⊆ U . The space X is a γ-set iff for every ω-cover U of X there exists a sequence (Un ∈ U : n < ω) such that for every x ∈ X for all but finitely many n we have x ∈ Un, equivalently