Space of Spaces as a Metric Space

Abstract:

In spacetime physics, we frequently need to consider a set of all spaces (“universes”) as a whole. In particular, the concept of “closeness” between spaces is essential. However there has been no established mathematical theory so far which deals with a space of spaces in a suitable manner for spacetime physics.

Based on the scheme of the spectral representation of geometry, we construct a space S _N , which is a space of all compact Riemannian manifolds equipped with the spectral measure of closeness. We show that S _N can be regarded as a metric space. We also show other desirable properties of S _N , such as the partition of unity, locally-compactness and second countability. These facts show that the space S _N can be a basic arena for spacetime physics.