MEAGER NOWHERE-DENSE GAMES .4. N-TACTICS (CONTINUED)

We consider the infinite game where player ONE chooses terms of a stri ctly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after omega innings TWO's nowhere dense sets cover ONE's first category sets. then TWO wins. We prove a theor em which implies for the real line: If TWO has a winning strategy whic h depends on the most recent n moves of ONE only. then TWO has a winni ng strategy depending on the most recent 3 moves of ONE (Corollary 3). Our results give some new information concerning Problem 1 of [S1] an d clarifies some of the results in [B-J-S] and in [S1].