Since $\mathbb{R}^n$ is an infinite space, I thought it is never compact no matter what metric it is endowed with. But, so what is the point in checking for non compactness of $\mathbb{R}^n$ with different metrics, does it not derive directly from the definition of the space?
2026-04-18 06:47:07.1776494827
Compactness of $\mathbb{R}^n$
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The concept of compactness depends on the topology of the space.
For a finite dimensional space all the p-norms are strongly equivalent, this includes the Euclidean metric, the taxicab distance, ... So the space is not compact with any of these metrics.
Maybe considering a weird metric (possibly one that does not come from a norm) would give you the desired result, but $\mathbb{R}^n$ won't be compact with any of the usual metrics.