  We prove three results on the dimension structure of complexity classes.
  1. The Point-to-Set Principle, which has recently been used to prove several
new theorems in fractal geometry, has resource-bounded instances. These
instances characterize the resource-bounded dimension of a set $X$ of languages
in terms of the relativized resource-bounded dimensions of the individual
elements of $X$, provided that the former resource bound is large enough to
parameterize the latter. Thus for example, the dimension of a class $X$ of
languages in EXP is characterized in terms of the relativized p-dimensions of
the individual elements of $X$.
  2. Every language that is $\leq^P_m$-reducible to a p-selective set has
p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with
a resource-bounded instance of the Point-to-Set Principle, this implies that if
NP has positive dimension in EXP, then no quasipolynomial time selective
language is $\leq^P_m$-hard for NP.
  3. If the set of all disjoint pairs of NP languages has dimension 1 in the
set of all disjoint pairs of EXP languages, then NP has positive dimension in
EXP.
