Space of spaces as a metric space
نویسنده
چکیده
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 SN , which is a space of all compact Riemannian manifolds equipped with the spectral measure of closeness. We show that SN can be regarded as a metric space. We also show other desirable properties of SN , such as the partition of unity, locally-compactness and the second countability. These facts show that the space SN can be a basic arena for spacetime physics. Running Title: ‘Space of spaces as a metric space’ or ‘Space of spaces’
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