A classification of the cofinal structures of precompacta
The purpose of this paper is to classify the families of precompact (totally bounded) sets in general metric spaces up to cofinal equivalence, where two partial orders are cofinally equivalent if each admits a monotone map onto a cofinal subset of the other. Note that this then also classifies the families of compact sets of general \textit{complete} metric spaces. Since `all' cardinal functions coincide in the class of metric spaces and are equal to the density the latter features prominently in the results. Indeed, if \(X\) has a locally compact completion then its family of precompact sets is equivalent to the family of finite subsets of the cardinal \(d(X)\) (provided \(X\) itself is not precompact). In the more general case one obtains the product of the latter partial order and a subset of the set functions from \(\mathbb{N}\) to \([ld(\tilde X)]^{<\aleph_0}\), where \(ld(\tilde X)\) is the local density of the completion of \(X\); depending on whether the latter is attained or not one has the full set of functions or just the set of bounded functions. These results are applied in the computation of the cofinalities of the families in question and of the weight and character of topological groups of the form \(C(X,G)\) where the latter carries the compact-open topology.
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