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Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals.

(Or prove that it does not exist).

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What is a perfect set? Also, this looks like a homework problem. –  Kevin H. Lin Jul 28 '10 at 21:36
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Why are you asking if you apparently know the answer? –  Mariano Suárez-Alvarez Jul 28 '10 at 21:37
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@Kevin: Line Bundle is not asking homework questions - no-one would have homework on so many different areas at once –  Casebash Jul 28 '10 at 21:37
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@Kevin, a perfect set is one which is equal to its derived set, as in en.wikipedia.org/wiki/Perfect_set –  Mariano Suárez-Alvarez Jul 28 '10 at 21:38
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Mariano has started a discussion at meta: meta.math.stackexchange.com/questions/313/… –  Kevin H. Lin Jul 28 '10 at 21:40

7 Answers 7

An easy example comes from the fact that a number with an infinite continued fraction expansion is irrational (and conversely). The set of all irrationals with continued fractions consisting only of 1's and 2's in any arrangement is a perfect set of irrational numbers.

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This is a really nice answer! –  Akhil Mathew Jul 29 '10 at 4:52
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Beautiful answer! –  BBischof Jul 30 '10 at 7:16

It can be proven that the Cantor set is perfect. Certainly, this contains infinitely many rationals. How about modifying the construction of the Cantor set by defining: $I_1 = [\sqrt{2},\sqrt{2}+1/3] \cup [\sqrt{2}+2/3,\sqrt{2}+1]$, $I_2 = [\sqrt{2},\sqrt{2}+1/9] \cup [\sqrt{2}+2/9,\sqrt{2}+1/3]\cup[\sqrt{2}+2/3,\sqrt{2}+7/9]\cup[\sqrt{2}+8/9,\sqrt{2}+1]$, etc and setting $P = \cap_{i=1}^\infty I_i$? Each of end points of any interval that appears in the construction is a member of $P$ and is irrational. However, is it true that all the members of $P$ must be an end point of a certain interval? I am tempted to think so because we can prove that $P$ does not contain any interval.

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There are only countably many endpoints, but a nontrivial perfect set is uncountable. –  JDH Jul 30 '10 at 1:45

Consider the set of reals x whose binary expansion, if you look only at the even digit places, is some fixed non-eventually-repeating pattern z. This is perfect, since we have branching at the odd digits, but they are all irrational, since z is not eventually repeating.

You can draw a picture of this set, and it looks something like the Cantor middle third set, except that you divide into four pieces, and take either first+third or second+fourth, depending on the digits of z.

Another solution: Begin with an interval having irrational endpoints, and perform the usual Cantor middle-third construction, except that at stage n, be sure to exclude the n-th rational number (with respect to some fixed enumeration), using a subinterval having irrational endpoints. By systematically excluding all rational numbers, you have the desired perfect set of irrationals.

(Hi François!)

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Hi Joel! Nice answer! –  François G. Dorais Jul 30 '10 at 18:30

It is well-known that $C$ is homeomorphic to $C \times C$, where $C$ is the Cantor set, as both are zero-dimensional compact metric spaces without isolated points. So $C$ contains uncountably many disjoint homeomorphic copies of $C$ and all but countably many of them can contain rationals...

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This is now my favorite example of a nonconstructive proof. –  Bryan Sep 9 '14 at 20:36
    
@Bryan Thx. It shows the power of the characterisation theorem nicely, I think. It's a folklore argument, don't know whose it is originally. –  Henno Brandsma Sep 9 '14 at 20:43

Let $C$ represent the Cantor set. Consider the set $C+\alpha$, where $\alpha=\sum_{n=1}^{\infty}10^{2^n}$.

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And this works because...? –  Andres Caicedo Feb 4 '11 at 6:22
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See this: math.stackexchange.com/q/381690/462 I do not know of an explicit $\alpha$ that works. –  Andres Caicedo May 5 '13 at 19:37

This is exercise 18 from chapter 2 of Rudin's Principles of Mathematical Analysis. Although this exercise comes with no hint, there is a previous problem(Ex.17) which can count as one:

  • Let $E$ be the set of all $x\in[0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Is $E$ perfect?

This can be seen by constructing $E$ in a similar fashion as the Cantor set; as when one considers the Cantor set using ternary expansions.

Now, define $a=\sum_{k=1}^{\infty}(10)^{-k^2}=\sum_{k=1}^{\infty}a_k/10^{k}$. We claim that $E+a$ contains no rational.

Suppose $r\in E+a$ is rational. $r=x+a$ for some $x\in E$. Let $\sum_{k=1}^\infty r_k/10^{k}$ and $\sum_{k=1}^\infty x_k/10^k$be the decimal expansion of $r$ and $x$, resp.; without allowing a trailing infinite sequence of $9'$s. Then $\sum_{k=1}^\infty (x_k+a_k)/10^k$ is the decimal expansion of $r$; since $x_k+1,x_k\neq 9$ for all $k$.

As $r$ is rational there are some $M,k>1$ such that for all $m\geq M$, $r_m=r_{m+k}$. Let $m>1$ be such that $2m+1>M,k$, then $m^2>2m+1>k$, and thus $r_{m^2}=r_{m^2+k}$. However, $r_{m^2}=x_{m^2}+1\in\{5,8\},$ and as $k<2m+1=(m+1)^2-m^2$, we get $a_{m^2+k}=0$, so $r_{m^2+k}=x_{m^2+k}\in \{4,7\}$.

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Let A be an open subset of R of finite measure and containing Q. This is possible because Q is countable. Let B=R\A. Now B is closed, and uncountable (because it has infinite measure). Let C be the family of open real intervals that, each, have countable intersection with B. Then UC (the union of the members of C) is equal to UD where D is a countable subset of C, so B has countable intersection with UC. The uncountable closed set E= B\UC is perfect. Indeed, if p belongs to E, and V is an open Real interval containing p, the intersection of E with V contains an uncountable closed subset of E.

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