The kth finite subset space of a topological space X is the space exp_k X of
non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The
construction is a homotopy functor and may be regarded as a union of configuration spaces
of distinct unordered points in X. We show that the finite subset spaces of a connected
2-complex admit "lexicographic cell structures" based on the lexicographic order on I^2 and
use these to study the finite subset spaces of closed surfaces. We completely calculate the
rational homology of the finite subset spaces of the two-sphere, and determine the top
integral homology groups of exp_k Sigma for each k and closed surface Sigma. In addition,
we use Mayer-Vietoris arguments and the ring structure of H^*(Sym^k Sigma) to calculate the
integer cohomology groups of the third finite subset space of Sigma closed and orientable.