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Here's Prob. 18 in the Exercises after Chapter 2 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition.

Is there a non-empty perfect set in $\mathbb{R}^1$ which contains no rational number?

Here's Definition 2.18 (h) in Baby Rudin.

Let $X$ be a metric space. A subset $E$ of $X$ is said to be perfect if $E$ is closed and if every point of $E$ is a limit point of $E$.

And, here's Definition 2.18 (d) in Rudin.

Let $X$ be a metric space. A subset $E$ of $X$ is said to be closed if every limit point of $E$ is a point of $E$.

Moreover, I know that the Cantor set --- discussed in 2.44 in Rudin --- is an example of a perfect set in $\mathbb{R}^1$ which contains no segment (i.e. no open interval).

Of course, any non-empty perfect set in a metric space must necessarily be infinite.

How to go about finding an answer to Rudin's question?

Saaqib Mahmood
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  • Use a method similar to that of "removing middle thirds" when constructing the Cantor set. Insure the $n$'th removed set (now open intervals with irrational endpoints) contains the $n$'th term of some enumeration of the rationals. – David Mitra Apr 06 '16 at 11:04
  • ${n\sqrt2:n\in\mathbb{N}}$ – almagest Apr 06 '16 at 11:07
  • Erm, make sure that after removing the $n$'th set, the first $n$ terms of the enumeration have been removed. You also want to start in a closed interval with irrational endpoints. – David Mitra Apr 06 '16 at 11:15
  • @almagest your set is not perfect, I'm afraid. For example, if we take a neighborhood of radius less than $\frac{ \sqrt{2}}{3}$ of the point $4 \sqrt{2}$, then this neighborhood contains no point of the set other than the point $4 \sqrt{2}$ itself. – Saaqib Mahmood Apr 06 '16 at 11:18
  • @DavidMitra I would be really grateful if you could take time elaborating your comments into a detailed answer. All I've been able to read from your hints is to start with an interval such as $[ \sqrt{2}, \sqrt{3} ]$. The set of all the rational numbers in this interval is countable; so we can arrange these numbers in a sequence $(r_n)_{n \in \mathbb{N}}$. What next? – Saaqib Mahmood Apr 06 '16 at 11:22
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    Let $I=[\sqrt2,\sqrt3]$. Remove an open interval contained in $I$ with irrational endpoints, not $\sqrt 2$, or $\sqrt 3$, that contains $r_1$. This splits $I$ into two parts. In each of the two halves, remove appropriate open sets so that $r_2$ and $r_3$ are removed (they may have already been removed, which is ok). Continue... – David Mitra Apr 06 '16 at 11:33
  • This question and answers offer a very different approach. – Brian M. Scott Apr 07 '16 at 03:32

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