# Non homeomorphic spaces

1. May 25, 2013

### hedipaldi

Why are the irrationals R-Q and the product space (R-Q)XQ not homeomorphic?
The first space i Baire space.may be the second space is not?

2. May 25, 2013

### Office_Shredder

Staff Emeritus
Their completions aren't homeomorphic, I think that does the trick but maybe there's some weird counterexample

3. May 25, 2013

### micromass

The completions of $\mathbb{Q}$ and $\{x\in \mathbb{Q}~\vert~0<x<1\}$ aren't homeomorphic either, even though the two spaces are homeomorphic. The problem is that completion is a metric concept and not a topological concept.

I think looking at Baire spaces is the way to go

4. May 26, 2013

### Bacle2

Well, how about from the perspective that R-Q can be embedded in R, but , at least that I can

tell, (R-Q)xQ cannot?

5. May 26, 2013

### hedipaldi

why not?
it seems that a homeomorphism actually can be defind by using continous fractions.Isn't it?

Last edited: May 26, 2013
6. May 26, 2013

### ForMyThunder

(R-Q)XQ isn't Baire. To prove that look at "slices".

7. May 26, 2013

### hedipaldi

what is "slices" ? can you give me a link for the proof?
Thank's a lot.

Last edited: May 26, 2013
8. May 26, 2013

### ForMyThunder

Take the sets (R-Q)x{q} for rational q. This is a countable set of closed sets with nonempty interior but their union is the entire space. Hence, it isn't Baire.

9. May 26, 2013

### hedipaldi

You mean with EMPTY interior right?