On comeager sets of metrics whose ranges are disconnected

Abstract

For a metrizable space X, we denote by Met(X) the space of all metric that generate the same topology of X. The space Met(X) is equipped with the supremum distance. In this paper, for every strongly zero-dimensional metrizable space X, we prove that the set of all metrics whose ranges are closed totally disconnected subsets of the line is a dense Gδ subspace in Met(X). As its application, we show that some sets of universal metrics are meager in spaces of metrics.