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Do countable perfect sets exist? I know that they are uncountable in Rk, but what about metric spaces in general?

  • I don't really know a lot of Topology, just the basics which are covered in Rudin's Principles of Mathematical Analysis. –  Nov 14 '16 at 15:31

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A perfect set in a complete metric space has cardinality at least $2^\omega=\mathfrak{c}$; there’s a proof here. However, the set $[0,1]\cap\Bbb Q$ is a closed subset of the space $\Bbb Q$ of rational numbers that has no isolated points, and it is of course countable, so completeness is required to ensure the larger cardinality.

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Brian M. Scott
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  • @Noah: That was my first thought, but then I wanted to make it clear that the space itself wasn’t an aberrant special case. – Brian M. Scott Nov 14 '16 at 15:01
  • So, a perfect set is uncountable if and only if the space is complete? –  Nov 14 '16 at 15:28
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    @Eternity: It goes only one way: you can have an uncountable perfect set in a space that’s not complete. Take for your space $[0,1]\cup\Bbb Q$, for instance; this is not complete, but $[0,1]$ is an uncountable perfect subset. But completeness rules out anything smaller than $\mathfrak{c}$ as the size of a perfect set. – Brian M. Scott Nov 14 '16 at 15:39