Abstract
Given a set X we construct a metric ρ on the set \( (\cal S)(X) \) of semi-metrics on X. We prove that ρ is complete and that a variety of interesting subsets of \( (\cal S)(X) \) are closed, giving rise to complete metric spaces of semi-metrics. In the second part we generalize this to a result about finite separating families of semi-metrics. In the third part of the paper we apply the results from the first part by constructing canonical metrics on spaces of riemannian metrics on an open manifold, which metricize some of the uniform structures defined in [3]. Finally we give some directions for possible applications.
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Salomonsen, G. On the Completions of the Spaces of Metrics on an Open Manifold II. Results. Math. 32, 100–114 (1997). https://doi.org/10.1007/BF03322530
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DOI: https://doi.org/10.1007/BF03322530