  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 of all compact Riemannian manifolds equipped with the spectral measure
of closeness. We show that this space of all spaces can be regarded as a metric
space. We also show other desirable properties of this space, such as the
partition of unity, locally-compactness and the second countability. These
facts show that this space of all spaces can be a basic arena for spacetime
physics.
