Borelness of Moduli Spaces of Metrics Implies Separability

Abstract

Let X be a metrizable space, and let Met(X) denote the space of metrics compatible with the topology of X, regarded as a subspace of the space of continuous pseudometrics with the topology induced by the supremum metric. We first prove that if Z is a discrete space whose cardinality is the first uncountable cardinal, then Met(Z) is not Borel. As a consequence, if Met(X) is Borel, then X is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space X is completely metrizable if and only if X is σ-compact. We also establish non-Archimedean analogues for spaces of ultrametrics. We further prove componentwise versions of these results, where two metrics belong to the same component if their distance is finite.

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BibTeX RIS

BibTeX

@misc{ishiki2026borelness,
  title = {{Borelness of Moduli Spaces of Metrics Implies Separability}},
  author = {Yoshito Ishiki and Tomoki Uda},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2608.19618},
  eprint = {2608.19618},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.19618},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Borelness of Moduli Spaces of Metrics Implies Separability
AU  - Yoshito Ishiki
AU  - Tomoki Uda
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.19618
UR  - https://arxiv.org/abs/2608.19618
AB  - Let X be a metrizable space, and let Met(X) denote the space of metrics compatible with the topology of X, regarded as a subspace of the space of continuous pseudometrics with the topology induced by the supremum metric. We first prove that if Z is a discrete space whose cardinality is the first uncountable cardinal, then Met(Z) is not Borel. As a consequence, if Met(X) is Borel, then X is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space X is completely metrizable if and only if X is σ-compact. We also establish non-Archimedean analogues for spaces of ultrametrics. We further prove componentwise versions of these results, where two metrics belong to the same component if their distance is finite.
ER  -

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