quasi
Posts:
12,067
Registered:
7/15/05


Re: Stone Cech
Posted:
Mar 21, 2013 3:10 PM


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 btained 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.
quasi

