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On pg 45 of Baby Rudin we have:

22. A metric space is called separable if it contains a countable dense subset.

24. Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.

25. Prove that every compact metric space $K$ ... is separable.

Since "countable" is defined in Rudin to exclusively mean "infinite and countable", then clearly the results don't hold for finite metric spaces, since that would imply an infinite set is a subset of a finite one.

What am I misunderstanding?

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    $\begingroup$ He should have included finite as OK too. $\endgroup$ – Henno Brandsma Sep 28 '14 at 10:56
  • $\begingroup$ Clearly, "countable" in 22 means "finite or infinite countable"... $\endgroup$ – Did Sep 28 '14 at 10:56
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Indeed, Walter Rudin, in the Principles of Mathematical Analysis, p.25, defines as countable set one which is equinumerous to $\mathbb N$. In the same page, he defines what is an at most countable set.

So, in 22 the term countable should be replaced by at most countable to be consistent with itself.

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