Strong γ-sets and other singular spaces

Whereas the Gerlits–Nagy γ property is strictly weaker than the Galvin–Miller strong γ property, the corresponding strong notions for the Menger, Hurewicz, Rothberger, Gerlits–Nagy (∗), Arkhangelʹskiıˇ and Sakai properties are equivalent to the original ones. The main result is that almost each of these properties admits the game theoretic characterization suggested by the stronger notion. We also solve a related problem of Kočinac and Scheepers, and answer a question of Iliadis.