On comeager sets of metrics whose ranges are disconnected

Abstract

For a metrizable space $X$, we denote by $\mathrm{Met}(X)$ the space of all metric that generate the same topology of $X$. The space $\mathrm{Met}(X)$ 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 $G_{\delta}$ subspace in $\mathrm{Met}(X)$. As its application, we show that some sets of universal metrics are meager in spaces of metrics.