The cardinal characteristic for relative gamma-sets

Abstract

For $X$ a separable metric space define $\pp(X)$ to be the smallest cardinality of a subset $Z$ of $X$ which is not a relative $\ga$-set in $X$, i.e., there exists an $\om$-cover of $X$ with no $\ga$-subcover of $Z$. We give a characterization of $\pp(2^\om)$ and $\pp(\om^\om)$ in terms of definable free filters on $\om$ which is related to the psuedointersection number $\pp$. We show that for every uncountable standard analytic space $X$ that either $\pp(X)=\pp(2^\om)$ or $\pp(X)=\pp(\om^\om)$. We show that both of following statements are each relatively consistent with ZFC: (a) $\pp=\pp(\om^\om) < \pp(2^\om)$ and (b) $\pp < \pp(\om^\om) =\pp(2^\om)$