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Let $X = (0,1]$, $d_1$ be the usual metric on $X$, and $d_2$ be a discrete metric on $X$. Which of the following is true:

  1. $(X,d_1)$ is compact, but $(X,d_2)$ is not.
  2. $(X,d_2)$ is compact, but $(X,d_1)$ is not.
  3. Both $(X,d_1)$ and $(X,d_2)$ are compact.
  4. Neither $(X,d_1)$ nor $(X,d_2)$ is compact.
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  • $\begingroup$ What's your definition of compactness? $\endgroup$ – Ravi Sep 10 '17 at 17:17
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Hint: For the usual metric consider the open cover consisting of the sets $(1/n,1]$ for $n \geq 1$.

For the discrete metric consider the open cover consisting of all of the singletons.

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