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

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William Elliot

Posts: 1,618
Registered: 1/8/12
N^N
Posted: May 2, 2013 10:59 PM
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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?

>



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