Tom
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 Nov 19 comment Can someone check if my proof is sufficient enough? Removed the mention of latex, since you've now formatted it more legibly. Nov 19 revised Can someone check if my proof is sufficient enough? deleted 218 characters in body Nov 19 answered Can someone check if my proof is sufficient enough? Nov 19 awarded Analytical Nov 19 awarded Supporter Nov 19 comment Odd Corollary in Baby Rudin Chapter 2, Question 28 "In $\mathbb{R}^k$, countable sets are not perfect." - yes, that's why I had to use $\mathbb{Q}$ as a metric space in its own right to make the point. I was just trying to show that the decomposition in Q.28 was extraneous and that what really matters is the uncountability of perfect sets in $\mathbb{R}^k$. Nov 19 comment Odd Corollary in Baby Rudin Chapter 2, Question 28 (of $\mathbb{R}^k$, I mean) Nov 19 comment Odd Corollary in Baby Rudin Chapter 2, Question 28 There's nothing in the result proved in Q.28 to say the countable part can't itself be perfect. An example is $\mathbb{Q}$, considered as a metric space in its own right. You can decomposing the whole of $\mathbb{Q}$ (closed) into the empty set and $\mathbb{Q}$ itself, which is both countable and perfect. To prove the corollary you need the additional fact that perfect subsets of $\mathbb{R}$ are uncountable. However, that is all you need (it's trivial to prove the corollary from it alone) and I still can't see how the decomposition in Q.28 contributes in any way. Nov 19 revised Odd Corollary in Baby Rudin Chapter 2, Question 28 added 1 characters in body Nov 19 awarded Editor Nov 19 comment Odd Corollary in Baby Rudin Chapter 2, Question 28 To be clear, I'm not saying I have difficulty seeing that the corollary is true - it's obviously true. I have difficulty seeing in what way it is considered to be a "corollary" of Q.28. Nov 19 revised Odd Corollary in Baby Rudin Chapter 2, Question 28 added 10 characters in body Nov 19 comment Odd Corollary in Baby Rudin Chapter 2, Question 28 But that doesn't use the main body of the question, to which it is supposed to be a corollary: "every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable". Nov 19 awarded Student Nov 19 asked Odd Corollary in Baby Rudin Chapter 2, Question 28