  We study topological properties of random metric spaces which arise by
Lambda-coalescents. These are stochastic processes, which start with an
infinite number of lines and evolve through multiple mergers in an exchangeable
setting. We show that the resulting Lambda-coalescent measure tree is compact
iff the Lambda-coalescent comes down from infinity, i.e. only consists of
finitely many lines at any positive time. If the Lambda-coalescent stays
infinite, the resulting metric measure space is not even locally compact. Our
results are based on general notions of compact and locally compact (isometry
classes of) metric measure spaces. In particular, we give characterizations for
general (random) metric measure spaces to be (locally) compact using the
Gromov-weak topology.
