Limits of closed subgroups of a locally compact group are naturally closed subsets of the ``hyperspace'' of closed subsets of a compact metric space. The space of closed subsets of a compact metric space carries the Hausdorff topology, so this space carries a natural compact topology. But what is it? Can one understand it? Can one, for instance, compute its cohomology?

There is something similar about the space of limits of dynamical systems. In a few cases, there is a serious hope of understanding it. I will try to define all these terms, and give some examples of how we could hope to understand these amazingly complicated spaces.