This paper continues the
discussion of a new transfinite dimension which was introduced by the author in
“D-dimension, I. A new transfinite dimension,” Pacific J. Math. vol. 26. In the first
part of this paper we show that, for a metric space X, D(X) is an ordinal if and only
if each closed subset Y ⊂ X contains a dense open (in Y ) subset each of whose
points has a finite-dimensional neighborhood. It follows that if X is complete
and separable, then X is weakly countable-dimensional (i.e. the union of a
countable number of closed finite-dimensional subsets) if and only if D(X) is
an ordinal. It is also shown that, if Ind(X) exists, then Ind(X) < D(X);
furthermore, if X is compact and Ind(X) does not exists, then D(X) is not an
ordinal. In the second part, it is proved that each weakly infinite-dimensional
separable metric space has a compactification with the same D-dimension;
an example is given to show that this is not true for all separable metric
spaces.
