On comeager sets of metrics whose ranges are disconnected (Q2687327)

Let \((X, \tau)\) be a metrizable topological space. Let us denote by \(\operatorname{Met}(X)\) the set of all metrics that generate the topology \(\tau\) and define a mapping \[ \mathcal{D}_X \colon \operatorname{Met}(X) \times \operatorname{Met}(X) \to [0, \infty) \] by \[ \mathcal{D}_X(d, \rho) = \sup_{x, y \in X} |d(x, y) - \rho(x, y)|. \] Then \(\mathcal{D}_X\) is a metric on \(\operatorname{Met}(X)\). The author prove the following. \medskip \textbf{Theorem 1.} Let \((X, \tau)\) be a strongly \(0\)-dimensional metrizable topological space and let \(DC(X) \subseteq \operatorname{Met}(X)\) be the set of all metrics with closed totally disconnected ranges. Then \(DC(X)\) is a dense \(G_{\delta}\) subspace of \((\operatorname{Met}(X), \mathcal{D}_X)\). The second theorem of the paper connected with gap-like metrics. Recall that a metric \(d\) on \(X\) is said to be gap-like if, for every point \(p \in X\), the set \(\{d(p, x) \colon x \in X\}\) is not dense in any neighborhood of \(0\) in \([0, \infty)\). \textbf{Theorem 2.} Let \((X, \tau)\) be a strongly \(0\)-dimensional metrizable topological space and let \(\operatorname{GL}(X) \subseteq \operatorname{Met}(X)\) be the set of all gap-like metrics. Then \(\operatorname{GL}(X)\) is a comeager subset of \((\operatorname{Met}(X), \mathcal{D}_X)\). Theorems 1 and 2 can be considered as a generalization of Theorem 7 of the paper [\textit{K. Broughan}, Bull. Aust. Math. Soc. 25, 133--142 (1982; Zbl 0465.54017)].