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Totally bounded ultrametric spaces and locally finite trees
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We investigate the interrelations between the metric properties, order properties and combinatorial properties of the set of balls in totally bounded ultrametric space. In particular, the Gurvich-Vyalyi representation of finite, ultrametric spaces by monotone rooted trees is generalized to the case of totally bounded ultrametric spaces. It is shown that such spaces have isometric completions if and only if their labeled representing trees are isomorphic. We characterize up to isomorphism the representing trees of these spaces and, up to order isomorphism, the posets of open balls in such spaces.
Forward citations
Cited by 3 Pith papers
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A tree has a countable vertex set exactly when some labeling makes its generated ultrametric separable or locally finite, and local finiteness is governed by rays and star subgraphs.
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Konig's Infinity Lemma for locally finite ultrametric spaces generated by labeled trees
For separable ultrametric spaces generated by labeled trees, local finiteness holds exactly when the induced ultrametric on every ray and every star subgraph is locally finite.
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