Algebraically independent distances and rigid metrics

Abstract

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.