  We explore various limit constructions for C*-algebras, such as composition
series and inverse limits, in relation to the notions of real rank, stable
rank, and extremal richness. We also consider extensions and pullbacks. We
identify some conditions under which the constructions preserve low rank for
the C*-algebras or their multiplier algebras. We also discuss the version of
topological dimension theory appropriate for primitive ideal spaces of
C*-algebras and provide an analogue for rank of the countable sum theorem of
dimension theory. As an illustration of how the main results can be applied, we
show that a CCR algebra has stable rank one if and only if it has topological
dimension zero or one, and we characterize those sigma-unital CCR algebras
whose multiplier algebras have stable rank one or extremal richness. (The real
rank zero case was already known.)
