
Re: Stone Cech
Posted:
Mar 22, 2013 10:25 PM


On Mar 22, 4:34 am, quasi <qu...@null.set> wrote: > Butch Malahide wrote: > >Butch Malahide wrote: > >>Butch Malahide wrote: > >>>quasi wrote: > >>>>Butch Malahide wrote: > >>>>>quasi wrote: > >>>>>>quasi wrote: > >>>>>>>quasi wrote: > >>>>>>>>quasi wrote: > >>>>>>>>>Butch Malahide wrote: > >>>>>>>>>>David C. Ullrich wrote: > >>>>>>>>>>>Butch Malahide wrote > >>>>>>>>>>>>William Elliot wrote: > >>>>>>>>>>>>>David Hartley wrote: > >>>>>>>>>>>>>>William Elliot wrote: > > >>>>>>>>>>>>>>>Perhaps you could illustrate with the five > >>>>>>>>>>>>>>>different one to four point point > >>>>>>>>>>>>>>>compactifications of two open end line > >>>>>>>>>>>>>>>segements. > > >>>>>>>>>>>>>>(There are seven.) > > >>>>>>>>>>>>>Ok, seven nonhomeomophic finite Hausdorff > >>>>>>>>>>>>>compactications. > > >>>>>>>>>>>>How many will there be if you start with n segments > >>>>>>>>>>>>instead of 2? > > >>>>>>>>>>>Surely there's no simple formula for that? > > >>>>>>>>>>> ... > > >>>>>>>>>> ... > > >>>>>>>>>>I wasn't necessarily expecting a *complete* answer, > >>>>>>>>>>such as an explicit generating function. Maybe someone > >>>>>>>>>>could give a partial answer, such as an asymptotic > >>>>>>>>>>formula, or nontrivial upper and lower bounds, or a > >>>>>>>>>>reference to a table of small values, or the ID number > >>>>>>>>>>in the Encyclopedia of Integer Sequences, or just the > >>>>>>>>>>value for n = 3. (I got 21 from a hurried hand count.) > > >>>>>>>>>For n = 3, my hand count yields 19 distinct > >>>>>>>>>compactifications, up to homeomorphism. > > >>>>>>>>>Perhaps I missed some cases. > > >>>>>>>>I found 1 more case. > > >>>>>>>>My count is now 20. > > >>>>>>>I found still 1 more case. > > >>>>>>>So 21 it is! > > >>>>>>>But after that, there are no more  I'm certain. > > >>>>>>Oops  the last one I found was bogus. > > >>>>>>So my count is back to 20. > > >>>>>Hmm. I counted them again, and I still get 21. > > >>>>>4 3component spaces: OOO, OO, O, . > > >>>>>7 2component spaces: OO, O, O6, O8, , 6, 8. > > >>>>>10 connected spaces: O, , 6, 8, Y, theta, dumbbell, and > >>>>>the spaces obtained by taking a Y and gluing one, two, or > >>>>>all three of the endpoints to the central node. > > >>>>Thanks. > > >>>>It appears I missed the plain "Y", but other than that, > >>>>everything matches. > > >>>>So yes, 21 distinct types. > > >>> For n = 4 I get 56 types. If I counted right (very iffy), > > >> Found two more. Never mind! > > >And now I get 61. The hell with it. > > Ullrich predicted it (hopeless squared). > > For small n, say n < 10, it might be feasible to get the counts > via a computer program, but my sense is that the development of > such a program would be fairly challenging. If I get a chance, > I may give it a try.
Now I get 2, 7, 21, 61, 180 for the first five values. Doesn't match anything in the OEIS. I did the work slowly and carefully (took about an hour and a half), and so I'm reasonably sure my numbers are correct.

