  Whereas the Gerlits-Nagy gamma-property is strictly weaker than the
Galvin-Miller strong gamma-property, the corresponding strong notions for the
Menger, Hurewicz, Rothberger, Gerlits-Nagy (*), Arkhangel'skii 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 Kocinac and
Scheepers, and answer a question of Iliadis.
