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Topic: Stone Cech
Replies: 49   Last Post: Mar 28, 2013 12:15 PM

 Messages: [ Previous | Next ]
 Butch Malahide Posts: 894 Registered: 6/29/05
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 non-homeomophic finite Hausdorff compactications.
>
> >>>>> >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 3-component spaces: OOO, OO|, O||, |||.

7 2-component 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.

Date Subject Author
3/14/13 William Elliot
3/14/13 fom
3/15/13 fom
3/16/13 William Elliot
3/15/13 David C. Ullrich
3/17/13 William Elliot
3/17/13 David C. Ullrich
3/17/13 fom
3/18/13 David C. Ullrich
3/18/13 fom
3/18/13 David Hartley
3/19/13 William Elliot
3/19/13 David Hartley
3/19/13 William Elliot
3/20/13 Butch Malahide
3/20/13 David C. Ullrich
3/20/13 Butch Malahide
3/20/13 Butch Malahide
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 Butch Malahide
3/21/13 quasi
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 quasi
3/22/13 David C. Ullrich
3/22/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 Butch Malahide
3/23/13 David C. Ullrich
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/22/13 Butch Malahide
3/23/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 quasi
3/23/13 Butch Malahide
3/23/13 Butch Malahide
3/24/13 quasi
3/24/13 Frederick Williams
3/24/13 quasi
3/25/13 Frederick Williams
3/28/13 Frederick Williams
3/25/13 quasi
3/19/13 David C. Ullrich