  This paper expands on and refines some known and less well-known results
about the finite subset spaces of a simplicial complex $X$ including their
connectivity and their top homology groups. It also discusses the inclusion of
the singletons into the three fold subset space and shows that this subspace is
weakly contractible but generally non-contractible unless $X$ is a co-$H$
group. Some homological calculations are provided.
