Space of spaces as a metric space (Q1971564)

Within Riemannian geometry, the author introduces a space of spaces, and tries to define a distance in it. His main idea is to define this distance via the spectrum of the Laplacian operator on the corresponding spaces. This means: Riemannian spaces are called to be close to each other if their spectra are close to each other. As the spectrum is a closed subset of the set of all reals, one needs a distance in the set of closed subsets in it. For this purpose, the author uses the Hausdorff distance, which he calls Gromov-Hausdorff distance; for the Hausdorff distance the author cites a book in French only, here is another reference for it, which comments also on the original papers by Hausdorff himself [\textit{H.-J. Schmidt}, Hyperspaces of quotient and subspaces, II: Metrizable spaces, Math. Nachr. 104, 281-288 (1981; Zbl 0523.54007)]. (The hyperspace is the space of all nonempty closed subsets of a topological space.) In Theorem 1, the author shows that two versions of his definition produce the same topology, and that this topology is paracompact (Theorem 2), locally compact (Theorem 3), and second countable (Theorem 4). However, he is erring when he believes that his proposal is the first one to give a mathematical definition of the distance between Riemannian manifolds. Example: In [\textit{M. Rainer}, \textit{H.-J. Schmidt}, Gen. Relativ. Gravitation 27, No. 12, 1265-1293 (1995; Zbl 0861.53066)] the set of all homogeneous Riemannian 3-spaces is equipped with a topology, which is nontrivial, because the corresponding topologies and isometry groups may change discontinuously. If one allows also inhomogeneous spaces, one gets the infinite-dimensional superspace, see e.g. [\textit{H.-J. Schmidt}, The metric in the superspace of Riemannian metrics and its relation to gravity, in `Differential geometry and its applications', Internat. conf., Brno, Czechoslovakia, 27 Aug.--2 Sept. 1989, 405-411 (1990; Zbl 0792.58009)]. (The superspace is the space of all metrics on a given differentiable manifold, and it is used since 1967 for models of quantum gravity).