
Re: Stone Cech
Posted:
Mar 21, 2013 2:34 PM


On Mar 21, 5:01 am, quasi <qu...@null.set> 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.

