  We introduce the concept of effective dimension for a wide class of metric
spaces that are not required to have a computable measure. Effective dimension
was defined by Lutz in (Lutz 2003) for Cantor space and has also been extended
to Euclidean space. Lutz effectivization uses the concept of gale and
supergale, our extension of Hausdorff dimension to other metric spaces is also
based on a supergale characterization of dimension, which in practice avoids an
extra quantifier present in the classical definition of dimension that is based
on Hausdorff measure and therefore allows effectivization for small
time-bounds.
  We present here the concept of constructive dimension and its
characterization in terms of Kolmogorov complexity, for which we extend the
concept of Kolmogorov complexity to any metric space defining the Kolmogorov
complexity of a point at a certain precision. Further research directions are
indicated.
