Size direction games over the real line. II

Variants of two basic infinite games of perfect information are studied. A notion of continuous strategy for the player S (Size) is shown to be related to a notion of convergence norm for sequences of reals. With each such norm, a variant of each of the basic games is associated in which the size player has to see that each play obeys the norm. Restriction to choose only rational numbers is also imposed on S. Some games are completely solved, and in this case S has a winning strategy iff his set includes a perfect subset, and D has a winning strategy iff S's set is at most denumerable. Some other games, in which S has to choose only rationals and obey a norm, induce a hierarchy structure on the class of nowhere dense perfect sets, that is embedded cofinally in the lattice of infinite sequences of integers modulo finite differences.