  While there is only one natural dimension concept for separable, metric
spaces, the theory of dimension in noncommutative topology ramifies into
different important concepts. To accommodate this, we introduce the abstract
notion of a noncommutative dimension theory by proposing a natural set of
axioms. These axioms are inspired by properties of commutative dimension
theory, and they are for instance satisfied by the real and stable rank, the
decomposition rank and the nuclear dimension.
  We add another theory to this list by showing that the topological dimension,
as introduced by Brown and Pedersen, is a noncommutative dimension theory of
type I C*-algebras. We also give estimates of the real and stable rank of a
type I C*-algebra in terms of its topological dimension.
