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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?
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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
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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
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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