AI 中文总结
该研究证明离散空间Met(Z)非Borel集，Met(X)为Borel集时X可分，结合Koshino定理得到有界相容度量空间完全可度量化等价于Xσ-紧，还建立了超度量空间的非阿基米德类比结果。
AI 中文摘要
设X为可度量化空间，Met(X)表示与X拓扑相容的度量构成的空间，视为带上确界度量拓扑的连续伪度量空间的子空间。我们首先证明，若Z是基数为阿列夫1的离散空间，则Met(Z)不是Borel集。由此可得，若Met(X)是Borel集，则X是可分的。结合Koshino的定理，我们的结果表明：可度量化空间X上有界相容度量空间完全可度量化当且仅当X是σ-紧的。我们还为超度量空间建立了非阿基米德类比结果。
英文摘要
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 supremum-metric topology. We first prove that if Z is a discrete space of cardinality aleph-one, 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 sigma-compact. We also establish non-Archimedean analogues for spaces of ultrametrics.
Comments10 pages. This paper builds upon arXiv:2608.11023(https://arxiv.org/abs/2608.11023)
