Let (X,$\rho$) be a general metric space where $\rho$ is a bounded metric, that is, $\exists M\in\mathbb{R}$ s.t. $\forall x,y\in X$ $\rho(x,y)<M$. Now let $\sigma$ be a metric equivalent to $\rho$. Must $\sigma$ be a bounded metric?
2026-02-23 10:00:12.1771840812
Is boundedness conserved under equivalent metrics?
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No. For instance, $d(x,y) = |\tan^{-1}(x) - \tan^{-1}(y)|$ is a bounded metric on $\mathbb R$ that generates the standard topology.