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Can we classify complete metric spaces up to isometry?

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Abstract

We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last twenty years: here we tried to present them in a unitary and organic way, sometimes with new and/or simplified proofs. The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research.

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Notes

  1. Of course simpleness and concreteness are vague concepts, but we will provide an exact mathematical formulation of what we have in mind later on.

  2. Such a Borel map may be obtained by first identifying through a Borel isomorphism the space I with the space \( \mathbb {R}_{> 0 } \) of positive reals numbers, and then associating to each \( r \in \mathbb {R}_{> 0} \) a compact metric space consisting of exactly two points at distance r.

  3. We denote by \( \omega \) the first infinite ordinal. As customary in set theory, when useful we identify it with the set \( \mathbb {N}\) of natural numbers.

  4. For example, when \( L = \{ R \} \) is the graph language consisting of just one binary relational symbol, then \( {\text {Mod}}^{{\aleph _0}}_\mathcal {L}= 2^{\mathbb {N}\times \mathbb {N}} \), and each \( x \in {\text {Mod}}^{\aleph _0}_\mathcal {L}\) gives rise to the \( \mathcal {L}\)-structure \( M_x = (\mathbb {N}, R^{M_x}) \), where for every \( n,m \in \mathbb {N}\) we set \( n \mathrel {R^{M_x}} m \iff x(n,m) =1 \).

  5. The relation \( E^\infty _G \) can be construed as the coordinatewise right action of G on the product \( F(G)^{\mathbb {N}} \) of countably many copies of the Effros Borel space F(G) . Strictly speaking, this is just a Borel action of G on a standard Borel space: nevertheless, by [18, Theorem 4.4.6] it is possible to equip the latter with a Polish topology (generating the same Borel sets) which turns the given action of G into a continuous one.

  6. Results establishing that \( E \le _B F \) for E a “concrete” classification problem and F a well-understood equivalence relation are often called classification theorems. In contrast, results establishing that \( F \le _B E \), especially when F is quite complicated, are often referred to as anti-classification theorems: this is because, as discussed, they show that no classification strictly simpler than F can be obtained.

  7. Technically speaking, \( \mathcal {M} \) contains only codes for nonempty Polish metric spaces: this can be easily fixed by e.g. adding some element of \( \mathbb {R}^{\mathbb {N}\times \mathbb {N}} {\setminus } \mathcal {M} \) to \( \mathcal {M} \) and use it as a code for the empty space. This minor issue does not arise with Coding 1 because we already have \( \emptyset \in F(\mathbb {U}) \).

  8. However, for every zero-dimensional Polish metric space \( X = (X,d) \) there is a complete ultrametric \( d' \) on X such that \( d' \) is compatible with d, i.e. it generates the same topology.

  9. The argument we are going to present here does not explicitly appear elsewhere, but it is suggested by Clemens himself at the beginning of [8, Section 2]. The construction from [8] (see also [18, Section 14.3]) is actually a further elaboration of the same argument: such more complicated construction can easily be generalized in order to obtain more results which, unfortunately, we have to omit here for the sake of conciseness.

  10. The problem of finding an optimal classification for zero-dimensional Polish metric spaces is still widely open—see the Main Open Problem 1 and the comment after it.

  11. As acknowledged in [20], the proof of Theorem 2.14(2) is actually due to G. Hjorth.

  12. See [20] for the definitions of pseudo-connected Polish metric spaces and pseudo-connected components of locally compact Polish metric spaces.

  13. Recall that we already know that compact Polish (ultra)metric spaces are concretely classifiable by Theorem 1.1.

  14. One such example is given by \( \cong _{\mathsf {GRAPHS}} \), which is absolutely \( {\varvec{\Delta }}^1_2 \) reducible to isometry on the set \( \mathrm {WF} \) of well-founded trees on \(\mathbb {N}\), but is not Borel reducible to it by Proposition 5.2.

  15. The countability of \(\alpha \) is a necessary condition here because since \( \mathbb {R}\) is separable there are no uncountable strictly decreasing sequences of reals.

  16. With a more involved argument, one can actually show that \( {\cong ^i_{D'}} \sim _B {(\cong ^i_D)^\omega } \) (see [9, Theorem 5.13]).

  17. By definition, \( \kappa ^{< \kappa } = \sup \{ \kappa ^\lambda \mid \lambda \text { is a cardinal } < \kappa \} \). Equivalently, \( \kappa ^{< \kappa } \) is the cardinality of the set of sequences \( \kappa ^{< \alpha } \) introduced at the beginning of Sect. 3 when setting \( \alpha = \kappa \); this is why we can unambiguously confuse the two notations.

  18. By [28], an analogue \( \mathbb {U}_\kappa \) of the Urysohn space for complete metric spaces of density character \( \kappa \) exists if and only if \( \kappa ^{< \kappa } = \kappa \). And even when such a \( \mathbb {U}_\kappa \) exists, the space \( F(\mathbb {U}_\kappa ) \) of its closed subspaces endowed with the Effros \( \kappa ^+ \)-Borel structure (i.e. the smallest \( \kappa ^+ \)-algebra on \( F(\mathbb {U}_\kappa ) \) generated by the sets as in (1.3)) would be a \(\kappa ^+ \)-Borel space, but it is not clear whether it is standard or not. For these reasons, a generalization of Coding 1 to our new setup seems not the right move.

  19. This is why we do not need to form any quotient of \( (\kappa , d_c ) \) before completing it to the space \( M_c \) (compare this with the construction \( c \mapsto M_c \) in Coding 2): the fact that we now want to deal only with spaces of density character exactly \( \kappa \) enables us to always require that distinct elements of \( \kappa \) have positive distance. In the countable case, instead, this was not possible because there we wanted to include also codes for finite spaces.

  20. This situation is absent in the countable/separable context, as any Borel function between two standard Borel (\( \omega \)-)spaces is definable using reals and (countable) ordinals as parameters, and thus it belongs to any inner model containing the same reals of the universe, such as \( \mathrm {L}(\mathbb {R}) \).

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  1. Dipartimento di matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 10123, Turin, Italy

    Luca Motto Ros

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Correspondence to Luca Motto Ros.

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The author was supported by the Young Researchers Program “Rita Levi Montalcini” 2012 through the project “New advances in Descriptive Set Theory” for this research.

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Motto Ros, L. Can we classify complete metric spaces up to isometry?. Boll Unione Mat Ital 10, 369–410 (2017). https://doi.org/10.1007/s40574-017-0125-1

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