Date: May 3, 2013 4:51 AM
Author: William Elliot
Subject: Re: N^N
On Thu, 2 May 2013, Butch Malahide wrote:

> On May 2, 9:59 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)?

>

> By (0,oo) do you mean (1,oo)?

>

Whoops. Indeed yes, (1,oo).