The large inductive
dimension (Ind) can be extended, by transfinite induction, to all the ordinals. The
transfinite inductive dimension so obtained has been investigated by many authors.
Unfortunately, it does not possess many of the nice properties which are possessed
by the inductive dimension in finite-dimensional spaces. For instance, the
transfinite inductive dimension fails to be monotone for separable metric
spaces; and it fails to satisfy the sum theorem, even for compact metric
spaces.
This paper introduces a new transfinite dimension called D-dimension, which is
defined for all metric spaces. For finite-dimensional spaces, D-dimension equals Ind.
It is shown that D-dimension is also a monotone and local property and that
it satisfies several sum and product theorems. These properties lead to a
characterization of D-dimension.
