Projective Hilbert modules and sequential approximations

We show that when A is a separable C*-algebra, every countably generated Hilbert A-module is projective (with bounded module maps as morphisms). We also study the approximate extensions of bounded module maps. In the case where A is a σ-unital simple C*-algebra with strict comparison, and every strictly positive lower semicontinuous affine function on quasitraces can be realized as the rank of an element in the Cuntz semigroup, we show that the Cuntz semigroup is equivalent to unitarily equivalent classes of countably generated Hilbert A-modules if and only if A has stable rank one.