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? >