The Gromov-Hausdorff metric makes the set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is there a natural topology on the set of (compact?) topological spaces?

The Gromov-Hausdorff metric makes the set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is there a natural topology on the set of (compact?) topological spaces?

Edit: I am not too concerned about set-theoretic issues, but perhaps part of the problem is to find a special collection of topological spaces that do form a set and have a natural topological structure. I am more interested in the topological structure.