let $d(x,y):=|x-y|$ is there a distance function $d'$ on $\Bbb R$ s.t. $d'(x,y) \geq d(x,y)$ so that $(\Bbb R, d')$ is compact ?
2026-04-02 03:34:43.1775100883
Is there a distance function so that $(\Bbb R,d') $ is compact?
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Hint. The condition $d'(x,y)\ge d(x,y)$ means that none of the finite-radius open balls around $0$ according to $d'$ is all of $\mathbb R$. But their union is, so together they form an open cover ...