Polish metric spaces with fixed distance set

Riccardo Camerlo; Alberto Marcone; Luca Motto Ros

doi:10.1016/j.apal.2020.102832

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arXiv · 1809.06588

Polish metric spaces with fixed distance set

Riccardo Camerlo·Alberto Marcone·Luca Motto Ros

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Abstract

We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with distances in $A$, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that $A$ must have in order that all Polish spaces with distances in that set belong to a given class, such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of $A$, of the relations of isometry and isometric embeddability between these Polish spaces.

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BibTeX RIS

Riccardo Camerlo, Alberto Marcone, Luca Motto Ros. 2018-09-18. Polish metric spaces with fixed distance set. https://doi.org/10.1016/j.apal.2020.102832

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