  It is shown that the Cuntz semigroup of a space with dimension at most two,
and with second cohomology of its compact subsets equal to zero, is isomorphic
to the ordered semigroup of lower semicontinuous functions on the space with
values in the natural numbers with the infinity adjoined. This computation is
then used to obtain the Cuntz semigroup of all compact surfaces. A converse to
the first computation is also proven: if the Cuntz semigroup of a separable
C*-algebra is isomorphic to the lower semicontinuous functions on a topological
space with values in the extended natural numbers, then the C*-algebra is
commutative up to stability, and its spectrum satisfies the dimensional and
cohomological conditions mentioned above.
