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

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William Elliot

Posts: 1,551
Registered: 1/8/12
Re: N^N
Posted: May 3, 2013 4:51 AM
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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).



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