代数无关距离与刚性度量
Algebraically independent distances and rigid metrics
- Tokyo Metropolitan University（东京都立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究代数无关距离与刚性度量，证明强零维空间上度量可被代数无关距离度量一致逼近，并得到刚性度量在紧空间上的稠密性。
AI中文摘要：
我们研究这样的度量：其在不同两点子集上的距离在有理数域上是代数无关的。我们证明，在基数至多为连续统的强零维可度量化空间上，每个相容度量都可以被具有此性质的相容度量一致逼近。若该空间是完全可度量的，则逼近度量也可选为完备的。对于每个σ-紧可度量化空间，具有代数无关距离的度量在一致拓扑中构成一个G_δ集。我们还研究刚性度量，即其唯一的双射自等距为恒等映射。在每个局部紧波兰空间上，刚性真度量在真相容度量中构成一个G_δ集。利用Niemiec的一个定理，我们得到在至少含三个点的紧可度量化空间上，刚性度量的一致稠密性。进而通过紧完备化，对任何至少含三个点的空间上的每个全有界相容度量，可获得刚性逼近。
英文摘要：
We study metrics whose distances on distinct two-point subsets are algebraically independent over the rationals. We prove that every compatible metric on a strongly zero-dimensional metrizable space of cardinality at most continuum can be uniformly approximated by compatible metrics with this property. If the space is completely metrizable, the approximating metrics can also be chosen complete. For every $σ$-compact metrizable space, the metrics with algebraically independent distances form a $G_δ$ set in the uniform topology. We also study rigid metrics, whose only bijective self-isometry is the identity. On every locally compact Polish space, rigid proper metrics form a $G_δ$ set among proper compatible metrics. Using a theorem of Niemiec, we obtain uniform density of rigid metrics on compact metrizable spaces with at least three points. Passing to compact completions then yields rigid approximations of every totally bounded compatible metric on any space with at least three points.
