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Given a metric space $(M, d)$ such that any subset E of M is compact. is M a finite set?

I think it is false but cannot find a counterexample.

nondefinite
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1 Answers1

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No, it is true. notice that every compact set in metric space is closed. So, every subset of $M$ is closed .So, $M$ is closed. Then every subset is open. Consider the following $$W=\{\{x\}\colon x\in M\}$$ is an open cover for $M$. So, if every element omitted from $W$ the reminder can not cover $M.$ I hope that helps

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