Does there exist a metric under which $\mathbb{R}$ is incomplete?
2026-04-12 15:16:57.1776007017
Does there exist a metric under which $\mathbb{R}$ is incomplete?
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I assume you want the metric to induce the usual topology on $\mathbb R$, otherwise the question becomes silly. So, you are essentially looking for a homeomorphic space to $\mathbb R$ which is metrizable and incomplete. Just notice that $\mathbb R$ is homeomorphic to $(0,1)$ with the standard topology.