## Are the rationals homeomorphic to any power of the rationals ?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (for the cantor set $C$). And then I got stuck, when I considered the rationals. So the question is:

Is $\mathbb{Q}^2$ homeomorphic to $\mathbb{Q}$ ?

-

## 3 Answers

Yes, Sierpinski proved that every countable metric space without isolated points is homeomorphic to the rationals: http://at.yorku.ca/p/a/c/a/25.htm .

An amusing consequence of Sierpinski's theorem is that $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}$. Of course here one $\mathbb{Q}$ has the order topology, and the other has the $p$-adic topology (for your favourite prime $p$) :-)

-

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

Yes, they are homeomorphic. To construct a homeomorphism from $\mathbb Q$ to $\mathbb Q^2$, one can proceed roughly as follows: express $q\in \mathbb Q$ as a continued fraction $[a_0, a_1,a_2,...]$ (of finite length) and associate with it the pair $([a_0,a_2,...], [a_1,a_3,...])$.

Mind that this is a homeomorphism, but not an isometry (cf comment on Tom's answer).

I vaguely remember that there is a ceneral Theorem in point set topology stating that all coutable topological spaces "of the same kind as $\mathbb Q$" are homeomorphic.

-
Why the construction does not work for $\mathbb R$ instead of $\mathbb Q$? – Wadim Zudilin May 26 2010 at 12:40

1
Each positive rational has two different continued fraction expansions. 5+1/2 = [5,2] maps to ([5],[2]) = (5,2). But 5+1/2 = [5,1,1] also, which maps to ([5,1],[1]) = (6,1). – Gerald Edgar May 26 2010 at 13:20

1
@Xandi: On the contrary, it works with infinite continued fractions to show that the space of irrationals is homeomorphic to its square. – Gerald Edgar May 26 2010 at 13:22

1
Sry but i was wondering why this is - surjective: only $a_0$ might be negative; hence you never get $a_1$ negative and hence the map is not surjective. - continuous : Note that $a_n:=[1;n]=1+1/n$ converges to $1$, but $([1],[n])=(1,n)$ diverges. I don't know how to repair this. Still it would be nice to have an explicit homeomorphism. – HenrikRüping May 26 2010 at 14:22

2
The irrationals have a nice characterisation as well (the rationals are the unique countable metric space without isolated points): the irrationals are the unique 0-dimensional [base of clopen sets] separable metric space that is nowhere locally compact [no non-empty open set has compact closure]. It is a way of showing that the irrationals are homeomorphic to N^N and hence to any finite or countable power of itself. – Henno Brandsma May 26 2010 at 17:55
show 6 more comments

I don't think so: the completion of $\mathbb{Q}^2$ is $\mathbb{R}^2$, so that a homeomorphism $\mathbb{Q}^2\to\mathbb{Q}$ would give a homeomorphism $\mathbb{R}^2\to\mathbb{R}$?

-

5
Your argument only shows that there is no isometry. – Xandi Tuni May 26 2010 at 12:35
You're quite right. I thought it sounded too good to be true... – Tom Smith May 26 2010 at 12:36

1
This will give a really good exercise for a topology course! – Xandi Tuni May 26 2010 at 12:48

3
It also shows there is no homeomorphism uniformly continuous in both directions. – Gerald Edgar May 26 2010 at 13:24
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 29, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9464564323425293, "perplexity_flag": "middle"}