Space of spaces as a metric space

TL;DR

This paper constructs a mathematically rigorous metric space of all compact Riemannian manifolds using spectral measures, enabling a formal notion of 'closeness' between spaces for spacetime physics.

Contribution

It introduces a spectral-based framework to define a metric space of all spaces, establishing properties like local compactness and second countability for applications in physics.

Findings

The space of all spaces is a metric space with spectral closeness.

The space has desirable topological properties such as local compactness.

It provides a foundational structure for spacetime physics studies.

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 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.