Strongly rigid metrics in spaces of metrics

TL;DR

This paper explores the properties of strongly rigid metrics in zero-dimensional spaces, showing their density and prevalence in certain classes of metrizable spaces, and characterizing metrics with trivial isometry groups.

Contribution

It extends Janos's 1972 result to the space of metrics, demonstrating density and comeagerness of strongly rigid metrics in strongly zero-dimensional spaces.

Findings

Strongly rigid metrics are dense in the space of metrics for strongly zero-dimensional spaces.

In certain spaces, strongly rigid metrics form a comeager set.

Metrics with no nontrivial self-isometries are also comeager in these spaces.

Abstract

A metric space is said to be strongly rigid if no positive distance is taken twice by the metric. In 1972, Janos proved that a separable metrizable space has a strongly rigid metric if and only if it is zero-dimensional. In this paper, we shall develop this result for the theory of space of metrics. For a strongly zero-dimensional metrizable space, we prove that the set of all strongly rigid metrics is dense in the space of metics. Moreover, if the space is the union of countable compact subspaces, then that set is comeager. As its consequence, we show that for a strongly zero-dimensional metrizable space, the set of all metrics possessing no nontrivial (bijective) self-isometry is comeager in the space of metrics.