  We introduce two resource-bounded Baire category notions on small complexity
classes such as P, SUBEXP, and PSPACE and on probabilistic classes such as BPP,
which differ on how the corresponding finite extension strategies are computed.
We give an alternative characterization of small sets via resource-bounded
Banach-Mazur games.
  As an application of the first notion, we show that for almost every language
A (i.e. all except a meager class) computable in subexponential time,
P(A)=BPP(A). We also show that almost all languages in PSPACE do not have small
nonuniform complexity.
  We then switch to the second Baire category notion (called
locally-computable), and show that the class SPARSE is meager in P. We show
that in contrast to the resource-bounded measure case, meager-comeager laws can
be obtained for many standard complexity classes, relative to
locally-computable Baire category on BPP and PSPACE.
  Another topic where locally-computable Baire categories differ from
resource-bounded measure is regarding weak-completeness: we show that there is
no weak-completeness notion in P based on locally-computable Baire categories,
i.e. every P-weakly-complete set is complete for P. We also prove that the
class of complete sets for P under Turing-logspace reductions is meager in P,
if P is not equal to DSPACE(log n), and that the same holds unconditionally for
quasi-poly time.
  Finally we observe that locally-computable Baire categories are incomparable
with all existing resource-bounded measure notions on small complexity classes,
which might explain why those two settings seem to differ so fundamentally.
