# Some Questions about zero-dimensional subsets of the unit interval related to cantor set

Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these sets are zero dimentional.I have three questions about these sets.

Q1.Is it true that the sets $K$ and $H$ are topologically Homeomorphic?

Q2.Is the space $K$ order isomorphic to $\mathbb{P}$ ?(I mean the existence of a monotonically increasing function from $K$ onto $\mathbb{P}$)

Q3.Is $H$ the union of countably many disjoint intervals from $\mathbb{P}$ ?

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The standard "devil's staircase" Cantor function $f$ is a monotonically increasing function from $K$ onto $\mathbb P$. One way to see this is that a member $x$ of the Cantor set has a base-3 expansion $.d_1 d_2 \ldots$ where all $d_j \in \{0,2\}$, and $f(x)$ has the base-2 expansion $.b_1 b_2 \ldots$ where $b_j = d_j/2$. $x$ is irrational iff $d_1, d_2, \ldots$ is not eventually periodic iff $b_1, b_2, \ldots$ is not eventually periodic iff $f(x)$ is irrational.

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Ouch!! my answer came 20 mins late.. –  i707107 May 9 '12 at 18:10
Hello dear Robert. thank you very much for your description.This function is very beautiful and well-known. But How about The $Q1$ and $Q3$. what do you think about these Questions. Is it true that there is a homeomorphism between $H$ and $K$? –  Ali Reza May 9 '12 at 18:10
Thank you Dear !707107. How about you. Do you think that the answer of Robert Israel is complete? –  Ali Reza May 9 '12 at 18:14
I answered for Q3, not sure about Q1. –  i707107 May 9 '12 at 18:18

For Q2, consider a mapping $\phi:K\rightarrow \mathbb{P}$ defined by $$\phi( \sum_{n=0}^{\infty} \frac{2a_n}{3^n})=\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$ where $a_n$ is a sequence entirely consisted of 0 and 1, and not periodic.

For Q3, the answer is yes. H is just intersection of $\mathbb{P}$ and the complement of cantor set. The complement of cantor set is union of countably many disjoint open intervals.

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Thank you dear unKnown friend. You have also given the same beautiful and well-known answer. But the important part of this question is $Q1$.how do you think about it? –  Ali Reza May 9 '12 at 18:24
It will be a pain to write down, but Q1 can be solved by cutting P into countable pieces. Since both P and H are countable disjoint union of intervals having rational endpoints. They have to be homeomorphic. –  i707107 May 11 '12 at 1:22