Strongly unbounded and strongly dominating sets of reals generalized

Abstract

We generalize the notions of strongly dominating and strongly unbounded subset of the Baire space. We compare the corresponding ideals and tree ideals, in particular we present a condition which implies that some of those ideals are distinct. We also introduce ${\mathrm{DU}_\mathcal{I}}$-property, where ${\mathcal{I}}$ is an ideal on cardinal ${\kappa}$, to capture these two generalized notions at once. We use two player game defined in a Kechris’s paper (Trans Am Math Soc 229:191–207, 1977) to show that every ${\lambda}$-Suslin set with ${\mathrm{DU}_\mathcal{I}}$-property contains a perfect subset with ${\mathrm{DU}_\mathcal{I}}$-property, provided that ${\lambda}$ is sufficiently small.