Let $X$ be a compact metric space under a certain metric $d_1$.

Let $d_2$ be a metric equivalent to $d_1$. Is it true that $(X,d_2)$ is a compact metric space?

If so how do I go about proving it?

  • $\begingroup$ Compactness is a property of the topology. Equivalent metrics generate the same topology. $\endgroup$ – DanielWainfleet Apr 18 '18 at 5:27

Yes, it is true, since asserting that the metrics are equivalent means that the topologies are the same.


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