Combinatorics of open covers (VIII)

For each space, Ufin(Γ,Ω) is equivalent to Sfin(Ω,Owgp) and this selection property has game-theoretic and Ramsey-theoretic characterizations (Theorem 2). For Lindelöf space X we characterize when a subspace Y is relatively Hurewicz in X in terms of selection principles (Theorem 9), and for metrizable X in terms of basis properties, and measurelike properties (Theorems 14 and 16). Using the Continuum Hypothesis we show that there is a subset Y of the Cantor set C which has the relative γ-property in C, but Y does not have the Menger property.