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Topic: N^N
Replies: 10   Last Post: May 12, 2013 9:34 AM

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,383 Registered: 1/8/12
N^N
Posted: May 2, 2013 10:59 PM

On Tue, 30 Apr 2013, Butch Malahide wrote:
> On Apr 29, 11:28 pm, Butch Malahide <fred.gal...@gmail.com> wrote:
> > On Apr 29, 9:17 pm, William Elliot <ma...@panix.com> wrote:
> >

> > > Why in N^N homeomorphic to R\Q?
> >
> > R\Q is homeomorphic to the space of *positive* irrational numbers. The
> > simple continued fraction expansion is a natural bijection between the
> > positive irrationals and the space N^N.

>
> Where by "positive irrationals" I mean irrationals between 0 and 1.

How so? The continued fraction for positive integers a1, a2,..
[a1, a2,.. ] = a1 + 1/(a2 + 1/(a3 + ..))

Would not those continued fractions not be
limited to (0,1) but to (0,oo)?

> > I'll bet that that bijection is a homeomorphism. It would be really
> > amazing if it weren't, wouldn't it?

>

Date Subject Author
4/29/13 William Elliot
4/29/13 Tanu R.
4/30/13 Butch Malahide
4/30/13 Butch Malahide
5/2/13 William Elliot
5/3/13 Butch Malahide
5/3/13 William Elliot
5/3/13 Butch Malahide
5/10/13 Leon Aigret
5/11/13 William Elliot
5/12/13 Leon Aigret