In his book Poincaré's Legacies, Terence Tao writes on p. 215:
Since all metrics are essentially equivalent on compact spaces, we see that <...>
What exactly does he mean by that? Could someone give a reference? I had problems finding one.
2026-03-25 04:00:10.1774411210
Why are all metrics are essentially equivalent on compact spaces?
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Some properties of metric spaces $(X,d)$ depend on the choice of metric $d$ rather than the induced topology. Consider for example the question of whether $(X,d)$ is complete. Even if $d'$ and $d$ generate the same topology, it does not follow that $(X,d)$ being a complete metric space implies $(X,d')$ is.
For example $(0,1)$ and $\mathbb R$ are homeomorphic topological spaces, but only the second is complete. We could use this fact to put an equivalent metric on $(0,1)$ that makes it complete. However this would be changing the metric space.
All equivalent metrics being uniformly equivalent prevents this sort of problem happening.