The procedure for this decomposition was worded to de-emphasize the
possible presence of a denumerable (component of the) space. Such a
component may be given outright; otherwise, it results if the space was
not simply connected. Any denumerable space is zero dimensional, as may
be verified easily from the full information theoretic definition of
dimensionality.
The obvious way of disposing of a denumerable space is to use the
conventional mapping that converts a Stieltjes to a Lebesque integral,
using fixed length segments. (It can be shown that H is invariant
under such a mapping.) Unfortunately, while this mapping followed by
a repetition of the preceding procedure will always solve a given
problem (no new[20] denumerable component _need_ be generated on the
second pass), little insight is provided into the structure of the
resulting space. On the other hand, because channels under cascading
constitute a group, any such denumerable space is a representation of a
denumerable group.
[20] Only non-cyclic irreducible (wrt direct product) denumerable group
components of the old denumerable space will remain.